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Preface The area of Reliability has become a very important and an active area of research. This is clearly evident fro...

Author: N. Balakrishnan | C. R. Rao

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Preface

The area of Reliability has become a very important and an active area of research. This is clearly evident from the large body of literature that has been developed in the form of books, volumes and research papers since 1988 when the previous Handbook of Statistics on this area was prepared by P. R. Krishnaiah and C. R. Rao. This is the reason we felt that this is indeed a right time to dedicate another volume in the Handbook of Statistics series to highlight some recent advances in the area of Reliability. With this purpose in mind, we solicited articles from leading experts working in the area of Reliability from both academia and industry. This, in our opinion, has resulted in a volume with a nice blend of articles (33 in total) dealing with theoretical, methodological and applied issues in Reliability. For the convenience of readers, we have divided this volume into 13 parts as follows: I Reliability Models II Life Distributions III Reliability Properties IV Reliability Systems V Progressive Censoring VI Analysis for Repairable Systems VII Analysis for Masked Data VIII Analysis for Warranty Data IX Accelerated Testing X Destructive Testing XI Test Plans XII Software Reliability XIII Inferential Methods We hope that this broad coverage of the area of Reliability will not only provide the readers with a general overview of the area but also explain to them what the current state is in each of the topics listed above. We express our sincere thanks to all the authors for their fine contributions and for helping us in bringing out this volume in a timely manner. Our special thanks go to Ms. Nicolette van Dijk for taking a keen interest in this project and also for helping us with the final production of this volume. N. Balakrishnan C. R. Rao

Contributors

J. A. Achcar, ICMC, University of $8o Paulo, C.P. 668, 13560-970, Säo Carlos, SP, BraziI, e-mail: [email protected] (Ch. 29) R. Aggarwala, Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4, e-mail." [email protected] (Ch. 13) R. Agrawal, GE Corporate Audit Staff, Fairfield, CT 06432-1008, USA, e-mail: [email protected] (Ch. 27) P. A. Akersten, Center for Dependability and Maintenance, Luleä University of Technology, Luleä, Sweden, e-mail: [email protected] (Ch. 16) E. K. AL-Hussaini, Department of Mathematics, University of Assiut, Assiut 71516, Egypt, e-mail." [email protected] (Ch. 5) S. Aki, Division of Mathematical Science, OsaÆa University, Graduate School of Engineering Science, Toyonaka, Osaka 560-8531, Japan, e-mail: [email protected] (Ch. 11) M. Asadi, Department of Statistics, University of Isfahan, Isfahan 81744, Iran, e-mail: [email protected] (Ch. 7) N. Balakrishnan, Department of Mathematics and Statistics, McMaster University, Hamilon, Ontario, Canada L8S 4Kl, e-mail: [email protected]. mcmaster.ca (Chs. 1, 14, 23) U. Balasooriya, Department of Statistics and Applied Probability, National University of Singapore, Lower Kent Ridge Road, Singapore 119260, e-mail: [email protected] (Ch. 15) M. Banerjee, Center for Health Care Effectiveness Research, Wayne State University, Detroit, MI 48201, USA, e-mail." [email protected] (Ch. 19) A. P. Basu, Department of Statistics, University of Missouri at Columbia, Columbia, MO 65211-0001, USA, e-mail: [email protected] (Ch. 2) S. Basu, Division of Statistics, Northern Illinois University, DeKalb, IL 601152854, USA, e-mail: [email protected] or [email protected] (Ch. 19) B. Bergman, Division of Total Quality Management, Chalmers University of Technology, Gothenburg, Sweden, e-mail." [email protected] (Ch. 16) W. R. Blischke, Emeritus Professor, Department of Information and Operations Management, University of Southern California, Los Angeles, CA 90089-1421, USA, e-mail: [email protected] (Ch. 20)

xix

xx

Contributors

A. Chatterjee, Department of Statistics, Burdwan University, Burdwan 713104, W. Bengal, India, e-mail." [email protected] or [email protected] (Ch. 4) E. Cramer, Department of Mathematics, University of Oldenburg, D-26111 Oldenburg, Germany, e-mail: [email protected] (Ch. 12) N. Doganaksoy, GE Corporate Research and Development, K14C35, Niskayuna, N Y 12309, USA, e-mail: [email protected] (Chs. 26, 27) N. Ebrahimi, Division of Statistics, Northern Illinois University, DeKalb, IL 60115-2854, USA, e-mail: [email protected] (Ch. 31) B. J. Flehingert, IBM Research Division, Mathematieal Sciences Department, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, N Y 10598, USA (Ch. 18) S. K. Ghosh, Department of Statisties, University of North Carolina, Raleigh, NC 27695, USA, e-mail." [email protected] (Ch. 31) E. Gouno, Department of Applied Statistics (SABRES), University of South Brittany, Rue Yves Mainguy, Tohannic, F56 000 Vannes, France, e-mail: [email protected] (Ch. 23) G. Y. Hong, Institute of Information and Mathematical Sciences, Massey University, Auckland, New Zealand, e-mail: [email protected] (Ch. 28) K. Hussein, University of Wisconsin, Department of Mathematics, 400 University Drive, Fond du Lac, WI 54935 USA, e-mail: [email protected] (Ch. 33) R. A. Johnson, Department of Statistics, University of Wisconsin, 1210 W. Dayton Street, Madison, WI 53706-1685, USA, e-mail: [email protected] (Ch. 24) U. Kamps, Department of Mathematics, University of Oldenburg, D-26111 Oldenburg, Germany, e-mail: [email protected] (Ch. 12) N. Kannan, Department of Mathematics and Statistics, University of Texas at Sah Antonio, San Antonio, TX 78249, USA, e-mail." [email protected] (Ch. 14) Md. R. Karim, Department of Statistics, University of Rajshahi, Rajshahi- 6205, Bangladesh, e-mail: [email protected] (Ch. 21) L. B. Klebanov, Faculty of Mathematics and Mechanics, Division of Statistics and Probability, St. Petersburg State University, St. Petersburg 198904, Russia, e-mail: [email protected] (Ch. 9) B. Klefsjö, Center for Dependability and Maintenance, Lule{t University of Technology, Luleä, Sweden, e-mail." [email protected] (Ch. 16) K. B. Kulasekera, Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA, e-mail: [email protected] (Ch. 30) C. D. Lai, Institute of Information Sciences and Technology, Massey University, Palmerston North, New Zealand, e-mail: [email protected] (Ch. 3) N. Limnios, Département Génie Informatique, Division Mathématiques Appliquées, Université de Technologie de Compiègne, B.P. 20 529, 60205 Compiègne Cedex, France, e-mail: [email protected] (Ch. 1) W. Lu, Department of Statistics, University of Wisconsin, 1210 W. Dayton Street, Madison, WI 53706-1685, USA, e-mail: [email protected] (Ch. 24) M. Mazumdar, Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA, e-mail: [email protected] (Ch. 25)

Contributors

xxi

G. C. McDonald, Director, Enterprise Systems Lab, General Motors Research and Development Center, MC #480-106-359, 30500 Mound Road, Warten, MI 48090-9055, USA, e-mail." [email protected] (Ch. 17) J. Mi, Department of Statistics, Florida International University, University Park, Miami, FL 33199, USA, e-mail: [email protected] (Ch. 8) N. A. Mokhlis, Department of Mathematics, Faculty of Science, Ain-Shams University, Cairo, Egypt, e-mail: [email protected] (Ch. I0) S. P. Mukherjee, Department of Statisties, Calcutta University, Calcutta 700019, W. Bengal, India, e-mail: [email protected] or [email protected] (Ch. 4) D. N. P. Murthy, Department of Mechanical Engineering, University of Queensland, Brisbane, Queensland 4072, Australia, e-mail: [email protected] (Chs. 3, 20) P. R. Nelson, Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA, e-mail: [email protected] (Ch. 30) W. Nelson, Consultant, 739 Huntingdon Drive, Sehenectady, N Y 12309, USA, e-mail: [email protected] (Ch. 22) S. Panchapakesan, Department of Mathematics, Southern Illinois University at Carbondale, Carbondale, IL 62901-4408, USA, e-mail." [email protected] (Ch. 33) C. Papadopoulos, D@artement Génie Informatique, Division Mathématiques Appliquées, Université de Technologie de Compiègne, B.P. 20 529, 60205 Compiègne Cedex, France, e-mail." [email protected] (Ch. 1) J. Rajgopal, Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA, e-mail: [email protected] (Ch. 25) B. Reiser, Department of Statistics, University of Haifa, Haifa, Israel (Ch. 18) S. E. Rigdon, Department of Mathematics and Statistics, Southern Illinois University at Edwardsville, Edwardsville, IL 62026-1653, USA, e-mail: [email protected] (Ch. 2) A. Sen, Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA, e-mail: [email protected] (Ch. 19) M. Shaked, Department of Mathematics, University of Arizona, Tuscon, AZ 85721, USA, e-mail: [email protected] (Ch. 6) D. N. Shanbhag, Department of Probability and Statistics, University of Sheffield, Sheffield, $3 7RH, England, UK, e-mail." [email protected] (Ch. 7) F. Spizzichino, Dipartimento di Matematica, Universita degli Studi di Roma ùLa Sapienza", Piazzale Aldo Moro, 2, 00185 Roma, Italy, e-mail." [email protected] (Ch. 6) J. Stein, G.E. Corporate Research and Development, Building K1-4C27A, One Research Circle, Niskayuna, N Y 12309, USA, e-mail: [email protected] (Ch. 26) K. S. Sultan, Department of Mathematics, AI-Azhar University, Nasr City, Cairo 11884, Egypt (Ch. 5)

xxii

Conlributors

K. Suzuki, Department of Systems Engineering, The University of ElectroCommunications, 1-5-1 Chofugaoka, Chofu-city, Tokyo 182-8585, Japan, e-mail: [email protected] (Ch. 21) G. J. Szekely, Department of Mathematics and Statistics, Math Science Building, Bowling Green State University, Bowling Green, OH 43403-0221, USA, e-mail." [email protected] (Ch. 9) L. Wang, Graduate School of Information Systems, University of ElectroCommunications, Tokyo 182-8585, Japan, e-mail: [email protected] (Ch. 21) M. Xie, Department of IndustriaI and Systems Engineering, The National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, e-mail: [email protected] or [email protected] (Chs. 3, 28) E. Yashchin, IBM Research Division, Mathematical Sciences Department, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, N Y 10598, USA, e-mail: yashchi@us,ibm.com (Ch. 18) S. Zacks, Department of Mathematical Sciences, Binghamton University, Binghamton, N Y 13902-6000, USA, e-mail: [email protected] (Ch. 32)

N. Balakrishnan and C. R. Rao, eds., Handbookof Statistics, Vol. 20 © 200i Elsevier Science B.V. All rights reserved.

| 1

Basic Probabilistic M o d e l s in Reliability

N. Balakrishnan, N. L i m n i o s and C. Papadopoulos

Notation SMP EMC r.v,

RE

R(t)

A(t) A

M(t) MTTF MTTR MUT MDT MTBF

N(t) ~.(t) Qij(t) p8

H,(O ~ij(t)

semi-Markov process embedded M a r k o v chain r a n d o m variable relative error reliability function pointwise availability function limit availability maintainability function system failure rate function mean time to failure mean time to repair mean up time mean down time mean time between failures number o f j u m p s of the semi-Markov process in the time interval

(0,t] number of visits of the semi-Markov process into the state i in the time interval (0, t] semi-Markov kernel: discrete state space case; i E E, j E E, t E IP,+ transition function of the M a r k o v chain (Jn): discrete time case distribution function of sojourn time in the state i, i E E M a r k o v renewal function: discrete state space case;

i EE, j EE, t E IR+ #U #u mi

«(x,; t E ,r) Q1 -x Q2 Q(,)

initial law mean hitting time of the SMP into the state j, starting in state i mean hitting time of the E M C into the state j, starting in state i mean first jump time under IPi or mean sojourn time in state i the a-algebra generated by the family of r a n d o m variables (Xt; t E I) Stieltjes convolution of two semi-Markov kernels on E nth fold Stieltjes convolution of the semi-Markov kernel Q, n E N Stieltjes convolution product

2 «

a.s. N N*

IP, IR+ lA

N. Balakrishnan, N. Limnios and C. Papadopoulos

matrix Stieltjes convolution product almost surely the set of natural numbers: {0, 1 , 2 , . . } the set of positive natural numbers: { 1 , 2 , . . } the set of real numbers the set of nonnegative real numbers: [0, ec) indicator (of characteristic) function of a subset A; lA(x)=

l(x)

d ___+ a.s. _____+

N(0, 1)

ifxŒA ifx~A

the Heavyside function on IR;

l(x)= 1s~/"

{1 0

I1

0

ifx_>0

ifx<0

is an s-dimensional column vector ( 1 , . . , 1 , 0 , . . , 0) ', with r l's and s - r 0's; if r = s, we write 1« (or simply 1) in place of 1~« convergence in distribution convergence almost sure weak convergence of random elements standard normal r.v. (mean # = 0; variance a 2 = 1)

1. Introduction The aim of this chapter is to give the basic probabilistic models in reliability. We present thus the discrete time Markov chains (DTMC), the continuous time Markov chain (CTMC) and the semi-Markov model in continuous time. In the case of DTMC, where the reliability is modeled in discrete time, the formulation is simple and can be used to model reliability in a first approach. Moreover, in most cases the numerical accuracy of this formulation is good. In the case of CTMC we give explicit formulae for reliability-related indicators in continuous time. As far as semi-Markov processes are concerned, we give an explicit formulation of reliability-related indicators in continuous time and of finite state space. We also give statistical estimation for reliability and availability. We then continue with the basics of Monte Carlo methods that are used quite often in reliability theory. The simplicity of a Monte Carlo method, its efficiency on higher dimensions as weil as its ability to model any arbitrary system, are the main advantages of this method. We present the basic idea of Monte Carlo method that was originally used to estimate integrals, and we continue with the presentation of simple algorithms for the simulation of discrete and continuous random variable (r.v.), as well as D T M C and CTMC. We discuss the problem of rare event estimation and we briefly review the basic principles of the well-known variance reduction methods. Importance sampling is also quite useful in reliability systems where some basic system's parameters can be estimated by simulating the corresponding model over regenerative cycles.

Basic probabilistic models in reliability

3

We terminate with two algorithms for the simulation of semi-Markov processes and the results obtained for a simple three-state system. This chapter principally consists of two parts: the first one containing the basic probabilistic models in reliability, while the second one deals with mainly Monte Carlo simulation of reliability systems.

2. D i s e r e t e time M a r k o v ehains and reliability

2.1. Basics o f M a r k o v chains 2.1.1. Definitions

Let X = (Xn, n E N) be an E-valued stochastic process defined on (~2, ~ , IP). DEFINITION 2.1. The stochastic process X (DTMC)

is a discrete time M a r k o v il, f o r all n E N , and f o r all i,j, io, i l , . . . , in 1 in E, we have

IP(Xn+~=jIX0 = =

lP(Xn+l

chain

i0,X1 = i l , . . . ,Xn-1 = in-l,Xn = i) =

jIXn =

i) = Pn(i,j) .

(1)

The family of probabilities Pn(i,j) is called the transition probability function. In the case where this function is independent of the time n, we say that the Markov chain X is a (time) homogeneous Markov chain and we put P ( i , j ) := Pn(i,j) for all i , j E E. In the finite case we will refer to the transition probability family P mostly as the transition matrix P where the (i,j) entry is P(i,j).

The Chapman-Kolmogorov equation can be written as follows: Pn+m(i,j) = ~

Pn(i,k)pm(k,j)

,

(2)

kcE

where P n ( i , j ) is the transition probability from state i to state j in n steps of time, i.e., P n ( i , j ) = IP(Xn =jIÆ0 = i) .

Let us also denote by ~ the initial distribution of X, i.e., for any i E E, ~(i) = IP(XO = i) .

PROPOSITION 2.1. For all n >_ 1 and all i0, il, . . . , in E E, we have: 1. IP(Xo = io,X1 = i l , . . . ,Xn 1 = in-l,Xn = in) = c t ( i o ) P ( i o , i l ) . . . P ( i n - l , i n ) ;

2. 1P(Xù+I -----i l , . . . ,Xn+k-~ = ik-l,Xn+k = ikIXn = io) = P(io, il)...P(ik-1, ik); 3. ]P(Xn+m = jlXm

=

i) = IP(Xn = j[Xo = i) = p n ( i , j ) .

4

N. BaIakrishnan, N. Limnios and C. Papadopoulos

PROPOSITION 2.2. 1. The sojourn time o f the system into the state i E E is a geometric r.v. with parameter P ( i , j ) . p(ij) 2. The probability that the system enters stare j when it leaves stare i is' 1-p(i,i)"

The above two propositions allow us to simulate by M o n t e Carlo m e t h o d s a M a r k o v chain. EXAMPLE 2.1. (Binary c o m p o n e n t ) Consider a binary c o m p o n e n t (or system) having an up (functioning) state and a down (failure) state that we denote by 1 and 0, respectively. At time n = 0 the c o m p o n e n t starts functioning until its failure at a r a n d o m time, S1 say, where it is replaced instantaneously by an identical one which lasts a r a n d o m time, $2 say, and so on. The lifetime distribution of the c o m p o n e n t s is 0 = (On,n E N*), i.e. IP(S1 = n ) = 0 , The r.v.'s U1, U 2 , . . defined by: U~ = Sn - Sn 1, (n _> 1, So = 0), are i.i.d, and represent the lifetimes of the successive components. Assume now that the distribution 0 is the geometric one with p a r a m e t e r p. F o r a time n >_ 0, define the r.v. X~ with values in E = {0, 1}. The event {X~ = 1} means that the c o m p o n e n t is functioning at time n and the event {X~ = 0} means that the c o m p o n e n t is failed at time n. Then it is clear that the stochastic process X = (Xù, n >_ 0) is a M a r k o v chain with state space E, transition matrix P

~(~~ ~) q

1-q

and initial distribution (a, 1 - ~) (c~ E [0, 1]). After some calculus we obtain the following spectral representation for the powers of the transition matrix P: pn =

1- p q

p 1

q

1 --p+q

-~ p+q

-q

(3) The state probability vector is (4)

P(n) = (P1 (n),P2(n)) , Pl(n) -

q P+q

~-

(1 - p

-

q)n

(p~ - q(1 - ~)) .

(5)

P+q

2.1.2. Recurrent and transient states A state of a M a r k o v chain m a y be either transient or recurrent. This is a f u n d a m e n t a l notion in the study of M a r k o v chains. L e t j be a fixed stare and the r.v. S j, s J , . . , represent the times of the 1st, 2nd,...

return to stare j. Define also U~ - S~ - S~_I, J n = 1,2,..

(take S° = So), to be the

B a s i c p r o b a b i l i s t i c m o d e l s in reliability

5

times between two consecutive returns to state j, which are called recurrence times of state j and

Nr:=~ l{xk=j}

,

(6)

k=l

N/~.:= ~

l(X k ~=i,Xk=j} ,

(7)

k=l

Nj := ev~ ,

(8)

Nij:=N,f

(9)

,

p,y := ]Pi(S{ < oo) ,

(10)

B/j:= IEiS{ .

(11)

DEFINITION 2.2. A state i is called a recurrent s t a t e i f Pii = 1. On the other hand, if Pii < 1, then i is called a transient stare. I f i is a recurrent state lhen either ~ii < oo, and it is called recurrent positive state or #ii = OO, and it is called null recurrent state. PROPOSlTION 2.3.

1. A state i E E is transient iff ( i f and only if) lPi(Ni = + o o ) = 0 or / y ~ n P " ( i , i ) < oo. 2. A state i E E is recurrent ifflPi(Ni = +oo) = 1 or iff ~ n P n ( i , i ) = O0.

PROPOSITION 2.4. I f i and j are two recurrent states, then N n - -

n N n

--

a., l --~ - - » [lii

ij ~.s. _ / . . ~

a s 17 ---+

1

--+ v ~ , 1 , - - ,

n

O0

(12)

»

asn--+oc

.

(13)

#ii

2.1.3. Stationary probability and asymptotic behavior DEFINITION 2.3. A probability distribution on E, is called stationary or invariant with respeet to the M a r k o v chain X (or to the transition probability P ) , if f o r all j E E, we have Z

re(i)P(i,j) = ~(j') .

(14)

lEE

Eq. (14) can be written in matrix form as reP : re .

PROPOSlTION 2.5. For every recurrent state i, we have: re(i) = l /#ii. For each state i, deßne di = g.c.d. {n : n > 1,P"(i,i) > 0}.

6

N. Balakrishnan, N. Limnios and C. Papadopoulos

DEFINITION 2.4. A state i E E is said to be periodic if di > 1, and aperiodic if di = 1. DEFINITION 2.5. A recurrent positive and aperiodic state is called an ergodic state. An irreducible M a r k o v chain with all its states aperiodic is called an ergodic Markov chain. PROPOSITION 2.6 (Ergodic t h e o r e m for M a r k o v chains). For an ergodic M a r k o v chain we have P~(i,j) --+ re(j),

as n --+ oc .

PROPOSITION 2.7. I f E is finite and the chain is irreducible and aperiodic, then exists and pn converges toward FI = l~z with an exponential rate.

2.2. Reliability in countable time 2.2.1. Introduction The reliability of a system can be studied in countable time too; for example in N instead of IR+. This lies in its easy formulation and calculus. This countable formulation that can be used as a first approach, m a y also be, in m a n y cases, sufficient. Let us consider a c o m p o n e n t (of a system) that is observed in times n = 0, 1 , 2 , . . and suppose that at time n it occupies stare x a m o n g a n u m b e r of possible stares given by its stare space. In the time interval (n, n + 1), where the c o m p o n e n t is not observed, it can change or not its state with probabilities p and q = 1 - p , respectively. We can then define the "failure rate", the reliability, etc. Let us define the N - v a l u e d r.v. T which denotes the lifetime of the a b o v e c o m p o n e n t , then the failure rate is defined as follows: )o(n) := IP(T = n]T >_ n)

(15)

for any n E N. The sequence 2 = (2(n),n E N ) is a failure rate o f a c o m p o n e n t or a system if 0 _< 2(n) _< 1 and ~n>0 2(n) = +oo. Moreover, the probability of (ailure at time n _> 0 is

(]ln-1 f ( n ) = IP(T = n ) =

)

\ ~ . ö [ 1 - 2(i)]

)~(n)

(16)

with the convention that H ó (') = 1. The reliability at time n > 0 is given by

n i--0

R(n) = IP(r > n) = H [ x - 2(i)1 .

If R(m) > 0, we have

(17)

Basic probabilistic models in reliability f(n)

;(~)

-

R(~)

7

(18)

for all 0 < n < m.

2.2.2. Muhistate Markov systems Consider a c-order system (C, (p), where C - { 1 , . . ,c} stands for the set of its components (c E N*) and (p for its structure function. Ler S be the state space of the system and of its components. (There is no loss of generality by considering the same state space for all components and for the system.) For example S = {0, 1 , . . ,M}, where M is the perfect state of the components and of the system and 0 the complete failure. When M = 1 we say that the system is binary with 1 the working state and 0 the failure state. Let X i = (Xi, n E N) be an S-valued stochastic process describing the behavior of the component i, and ( X 2 , . . ,X~) be the SC-valued stochastic process. This last process describes jointly the states of the system components. Consider now a one-to-one mapping h : S c -+ E, where E = { 1 , . . , s}, s = # S « and the process Xn = h ( X 2 , . . ,X~). In general we take h to be the lexicographic order of the set S c. EXAMPLE 2.2. Consider a second-order binary system. We then have: c = 2, S = {0, 1}, S 2 = {(0,0),(0, 1), (1,0),(1, 1)} and E = {1,2,3,4}. If the components are independent and the above processes X i are Markov, then the process X is a Markov chain too. The following proposition gives a more precise description. PROPOSITION 2.8. Consider two independent Markov cha&s, X 1 and X 2 say, $1 and $2 valued with transition functions p1 and p2, respectively. Then the (X1,X 2) is an S 1 x S2-valued Markov chain with transition function P, with P((i,j), (k, g)) -p l ( i , k ) p 2 ( j , g ) , for all i,k E S 1 and j, g E S 2. The above proposition can be generalized for more than two processes. In the case of binary systems, we have to partition the whole stare space E into disjoint sets, U and D say, where U includes the working states (up stares) and D the failure states (down states), (i.e. U U D -- E, U N D = ~ and U ¢ (~, D ¢; (~.) The reliability-related indicators at time n > 0, become: • Reliability: R(n) = IP(Vr E [0, n] M N,Xv E U). • Availability (pointwise): A(n) = 1P(Xn E U). • Maintainability: M(n) -- 1 -1P(Vv E [0, n I N N,X~ c D). Consider now a binary multistate system (BMS) with state space E={1,..,s}, up states U = { 1 , . . , r } and down states D = { r ÷ l , . . . , s } , described by an E-valued Markov chain X, with transition probability matrix P and initial distribution vector et.

8

N. Balakrishnan, N. Limnios and C. Papadopoulos

Let us consider the following partition of P and c~corresponding to the U and D sets:

p = (Ph

\

P21

=(<

P12 "~ P22 J~

~~)

•

We have the following results. PROPOSITION 2.9. For the above B M S , we have: • Availability: A(n) = c~Pnls,r. • Reliability: R(n) = cqP~~lr. • Maintainability: M(n) = 1 - a2P~21s_r. 2.2.3. Hitting times Besides the main quantities of interest in reliability theory, other quantities such as the m e a n hitting time of failure stares, are also important. In what follows we will define these m e a n hitting times in the f r a m e w o r k of M a r k o v chains. • • • • •

Mean Mean Mean Mean Mean

time to failure ( M T T F ) . time to repair ( M T T R ) . up time ( M U T ) . down time (MDT). time between failure (MTBF).

Ler T and Y be the hitting times of sets D and U, respectively, i.e., T = inf{k >_ 0 : X~ E D} and Y=inf{k>0:X~EU} with inf0 = ec. We have the following explicit formulas. PROPOSITION 2.10. (19)

MTTF := IET = ~1(I - P11) l l r , MTTR := IEY = ~2(I

-- P22)-lls-r

.

(20)

I f the M a r k o v chain describing the above B M S is an ergodic chain, with stationary probability row vector 7z = (7zl, 7z2) (a partition following the U and D sets), we have the following result. PROPOSITION 2.1 1. 7"Cllr M U T := ~E3~[Tl -- rtiP211r

(21)

Basic probabilistic models in reliability

9

/r2 ls-r ~2P121 r ,

M D T := IEB2[Y]

(22)

where fil and fl2 are the input distributions to the sets U and D, respectively, under re. For the M T B F we have the following representation." MIBF = MUT + MDT .

(23)

PROPOSITION 2.12. The variance o f the hitting time T o f the set D is given by Vari(T) = V(i) - (L1(i))2 ,

(24)

where V = [IEI(T2), ..,IEm(T2)] ' = ( I - P l l ) - a [ l + 2 P l l ( I []EI(T),..,IEm(T)]'= (I-Pll)-11, vector a.

where a' means

Pll) 11] and LI =

the transpose

vector o f

EXAMPLE 2.3. In the case of the binary c o m p o n e n t presented in the example of Section 2.1.1, we have for the initial distribution vector (e, 1 - e ) : Reliability: R(n) = e(1 - q)", q (1 - p - q ) " Availabiaity: A(n) -- - + P+q P+q

(p~ - q(1 - c~)),

Maintainability: M ( n ) = 1 - (1 - «)(1 - q)", MTTF = MUT

-

MTTR = MDT =

1 1 -p'

l 1-q

2.3. Non-homogeneous M a r k o v chains and reliability Consider a M a r k o v chain X = (Xù, n E IN) whose transition function depends on time n, i.e. Pù(i,j) = IP(Xn+I = jlXn = i) for all i , j E E and n E il,I. Then X is called a n o n - h o m o g e n e o u s discrete time M a r k o v chain ( N H D T M C ) . Define also, for m > n > 0, the transition probabilities Pù,m(i,j) = IP(Xm = jlXù = i) and, for n _> 0, P~,ù(i,j)

=

l{i=j}

•

10

N. Balakrishnan, N. Limnios and C. Papadopoulos

We have Pn,n+l (i, j) = Pn (i, j). Moreover, the C h a p m a n - K o l m o g o r o v equation is

Pù,m(i,j) = Z Pm,r(i, k)Pr,m(k,j) kEE

forn
r-1

Pn,n+r(id) -- I1 Pn+~(i,j) k=0

Let Œ be an initial distribution on E. Then, using the conditional probability formula and the M a r k o v property, the following relations can be easily derived: PROPOSITION 2.13. For all n >_ 1 and all io, il,..., in C E, we have:

1. IP(Xo = io,X1 = i l , . . . ,Xn-1

=

in 1,Xn = in) = o~(io)Po(io,

2. IP(X~+I = i l , . . . ,X~+k-1 = i ~ - l , X n + k ----i~lX~ = io)

il)"

"Pn-l(in-1, in).

=Pn(io,il)'" "Pn+k-l(ik-l,ik).

EXAMPLE 2.4. Consider an N H D T M C X with transition probabilities Pn =

n+2 1 1+ n+--~ ~

•

Then for any initial distribution e, we have

(i Pn,m =

11@+ / n+21 21 n+2 2 q- ~ /

The following results can also be f o u n d in Platis et al. (1998). PROPOSITION 2 . 1 4 .

The availability of an N H D T M C system is given by n-1

n >_ 1,

2 j ~ ~ «(J),

n = 0

A(n) = { Œ[L=0Pkl~,r, PROPOSITION 2.15.

R(n) = { PROPOSITION 2.16.

The reliability of an N H D T M C system is given by o~ "Fln l

U

1 Ilk=0 Pk L ,

n >_ 1,

The maintainability of an N H D T M C system is given by

n 1 D Pk 1~_~, n _> 1, A(n) = { 1 -- ~X2 Hk=0

1 - ~ j c D c~Ü),

n = 0 .

Basic probabilistic models in reliability

11

The M T T F and M T T R of an N H D T M C system are given by

PROPOSITION 2.17.

M T T F = Cq ( I X /

MTTR

= ~2 ( I

\

oe k D)

+Z~P~~=0 1~~

}~XAMPLE 2.5. Consider an absorbing N H D T M C matrices, for n = 0, 1 , 2 , . .

with transition probability

( (n+l)a , (n+l)a ~ 1_~) Pn =

1

1

(n+l)b0 (n+l)b0

1

~

j

,

where a > 1 and b > 1. The stare space partition is: U = {1,2} and D = {3}. Then, we have: 2(a + b) ~-I

A(n)=R(n)-

n[anbù_l ,

n> 1

and A(0) = R(O) = 1. M o r e o v e r , MTTF=I+

2b .1+~ [e~ - 1 ] a+b

.

3. Continuous time Markov chains and reliability

3.1. Definition This section deals with continuous time M a r k o v chains, i.e. I = IP,+. As in the previous section we will consider here a probability space (f~, ~ , IP) where we define an E-valued stochastic process X = (X(t), t c IP,+). The state space E here is at m o s t a countabte space. DEFINITION 3.1. The stochastic process X is a continuous time Markov chain

( C T M C ) if, for all j E E, and all t, h >_ O, we have Ip(X(t+h) =j]X(u),u < t) = Ip(X(t+h) =jIX(t)),

(a.s.) .

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We consider here time h o m o g e n e o u s processes in which case the transition function Pt,t+h(i,j) := IP(X(t) = jIX(t) = i), i E E, j E E and t > h > 0 is independent of the time t. Thus we can put Pt,t+h(i,j)= Ph(i,j). The C h a p m a n K o l m o g o r o v equation in this case can be written as follows:

12

N. Balakrishnan, N. Limnios and C. Papadopoulos

Pt+h(i,j) = Z P t ( i , k ) P h ( k , j ) ,

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kEE

which in matrix form can be written as

Pt+h - PtPh •

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Thus the family (Pt, t _> 0) forms a semi-group. We will study here Markov processes w/th semi-group satisfying the following property:

limp,(i,j) = l{i_j} t+0

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for all i,j E E and the semi-group will be called standard. If limtloPt(i,i) = 1 is satisfied uniformly with respect to i E E, then the semi-group will be called uniform.

3.2. Reliability in continuous t/me Let us follow the same formulation of the reliability-related indicators as in the case of DTMC. The difference is that the t/me here is continuous. Consider a CTMC X, w/th state space E = ( 1 , . . , s}, generator A, transit/on function Pt and initial distribution 0{. As in the case of DTMC, consider the up states set U = { 1 , . . , r } , and the down states set D = {m + 1 , . . , s } and the part/tion of the generator and the initial distribution vector following U and D: A=

(An

A12)

\A21

A22

0{ = (0{1

0{2) .

Availability: PROPOSITION 3.1. The (pointwise) availability of a C T M system is g/ven by

A(t) = 0{etAls,r .

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The steady-state availability, denoted by Aoo, is given by Aoo : Z ~ ( k )

: 7c-ls,r •

kEU

Reliability: PROPOSITION 3.2. The system's reliability is R(t) = 0{~etanlr.

Maintainability: PROPOSITION 3.3. The maintainability is g/ven by M(t) = 1 - 0{2etA221s_r.

Hitting times: For the mean hitting times that were defined in the previous section we have the following:

Basic probabilistic models in reliability

13

PROPOSITION 3.4. MTTF = - CqAllllr, MTTR

= -

Œ2A~lls r •

PROPOSmON 3.5. I f the C T M C X is ergodic, then 7"Cll m

MUT - - -

7ZlA21 l r '

7"C21N_m

MDT -

7z2A121s-r

and M T B F = M U T

+ MDT.

PROPOSITION 3 . 6 . Var(T) = 2«lAi-~l - ( C q A l ? l ) 2 .

3.3. Distributions o f phase type Phase-type distributions (of Ph-distributions) play a very important role in reliability theory for two reasons. The first one is because of a property making this family to be a dense set in the set of all distributions on IR+, while the second one is due to the fact that we have an explicit formulation of the basic system operation in reliability as parallel system, series system, redundant cold standby system, etc. (see Neuts, 1981; Asmussen, 1987). Consider a M a r k o v process, X say, with generator A and stare space E = {1,..,N+ 1} with the following partition: U = { 1 , . . , N } and D = { N + 1}. Suppose that U is a transient class and state N + 1 is an absorbing state. Following U and D the generator and the initial distribution vector can be partitioned as follows:

A:(~ A0 0) and (c~,~ZN+I). DEFIMTION 3.2. A distribution function x H F(x) on [0, oc) is o f phase type ( P H distribution), if it is the distribution function o f the absorbing time o f a C T M C defined as above. The couple (Œ, T) is called a representation o f F. The distribution of absorbing time in the state F(x) = 1 - c~exrl . Properties (Neuts, 1981):

N + 1

is given, for x _> 0 , by

N. Balakrishnan, N. Limnios and C. Papadopoulos

14

1. It possesses an atom at x = 0, which is equal to C~N+I. The absolutely continuous part has density f , given by f(x) = F'(x) = aeXTTo, x > O. 2. The Laplace-Stieltjes transform/~ of F is given by /W(S) = ~ N + I -~- O~(SI --

T)-IT0,

for Re(s) _> 0

.

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3. The moments (no centered) are ]~!. z (-1)~n!(~T-'l), n > O. PROPOSITION 3.7 (Convolution of two distributions of phase type - N e u t s , 1981).

Consider two distributions of phase type, F and G say, with representations (c~,T) and (fi, S) of order M and N, then their convolution F * G is a phase-type distribution with representation (7, L), where

0:0) and 7 = (e, CgN+lfl)" Let us define now some elements of Kronecker's algebra useful for the next proposition. The operation ® is the Kronecker's sum of two matrices. Note d//m, the space of matrices of dimension m x n and let A c J g , , and B E Jmm; the Kronecker's sum is defined as follows:

A

= (A ®Ira) + (B®I,)

with ® the Kronecker's product. Let A E ,/¢lkt and B E ~/~mn, we have: A @ B E J~kxmflxn and

allB

...

a11B)

A®B= ak~B ...

aklB

PROPOSITION 3.8 (Formation of series and parallel systems - Neuts, 1981).

Consider two phase-type distributions F and G with representations (cq T) and (fl, S) of order M and N, respectively, then: 1. The distribution K given by K(x) = F(x)G(x) is a Ph-distribution of representation (7, L) of order MN + M + N with L = (T@So0 I®SOTo T O o 1 )

and 7 = [c~@ fi, fiN+lCq C~M+lg]

2. The distribution W defined by W(x) = 1 - ( 1 - F ( x ) ) ( 1 - G ( x ) ) is a Ph-distribution of representation (c~® ~, T @ S).

Basic probabilistic models in reliability

15

P~OPOSITION 3.9 (Asymptotic behavior - Neuts, 1981). Let F be a Ph-distribution of representation (cq T). I f T is irreducible, then F is asymptotically exponential, i.e.,

1 - F(x) = Ke -z~ + o(e -~=) , K > O, 2 > 0 with - 2 the eigenvalue of T having the greatest modulus of real part and K = c~v where v is the right eigenvector of T corresponding to the eigenvalue

4. Semi-Markov processes and reliability Some systems satisfy a M a r k o v property not for all points of time but only for a special family of increasing stopping times. These times are the state change times of the considered stochastic process.

4.1. Basic results and definition o f a Markov renewal process 4.1.1. Definition Consider an at most countable set, E say, a two-dimensional stochastic process (J,S) = (J~,S~, n E N), where the r.v. J~ take values in E. Consider also that the r.v. Sn takes values in IP,+ and satisfies 0 = So _< $1 _< $2 _< ... DEFINITION 4.1. The stochastic process (J, S) is called a Markov Renewal Process ( M R P ) if it satisfies the following relation: lP(J~+l = j, Sn+l <_ ttJo,J1,S1, . . Jn,Sn) = IP(J~+I = j, Sn+l <_ tl&,S~)

=: Q«o«ù+, (t - sù),

(~.~.)

for all n E N , j C E and t E IR+. The set E is called the state space of the MRP. F r o m these relations it is clear that (Jn) is a M a r k o v chain with state space E and transition probability P(i,j) = Qij(oc). It is called the embedded Markov chain. For every i E E, we have P(i, i) = O. Let us now define, the counting process (N(t), t >_ 0) associated to the point process (Sn, n _> 0), i.e., for each time t _> 0 the r.v. N(t) is

N(t) := sup{n : Sn <_ t} and define the continuous time process Z = (Z(t), t E IR+) by Z(t)

:=

JN(t) •

Then, the process Z is called semi-Markov process. Define also

N. Balakrishnan, N. Limnios and C. Papadopoulos

16

P~j(t) = IP(Z(t) = jlz(0)

= i),

Hi(t) = Z Qij(t)' j~E mi=

/o ~ [1 -

Hi(u)ldu

•

The M R P considered here will be strongly reguIar, i.e., for every t E IR+, N(t) < oc (a.s.), and for all j c E, mj < oc.

4.1.2. Particular cases 1. Discrete time M a r k o v chain

Qij(t) = P(i,j)l{t>_l}

for all i,j E E,

andt_>0

.

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2. Continuous time M a r k o v chain

Qij(t) = P(i,j)(1

-

e -2(i)t)

for all i,j C E, and t > 0 .

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3. Renewal processes: (a) Ordinary: it is an M P R with two states E = {0, 1}, P(0, 1) = P(1,0) = 1 and Q0~ (') = F(.), where F is the c o m m o n distribution of the inter-arrival times of the renewal process. (b) Modified or delayed: it is an M R P with E = {0,1,2}, P ( 0 , 1 ) = 1, P(1,2) = P(2, 1) = 1 and 0 elsewhere and Q01 (') = F0(.), Q12(') = Q21 (') = F(.), where F0 is the distribution function of the first arrival time and F is the c o m m o n distribution function of the other inter-arrival times of the renewal process. P(0,1)=P(1,0)= 1, and 0 elsewhere and (c) Alternating: E = { 0 , 1 } , Q01(') = F(.), Qa0(') = G(.), where F and G are the distribution functions corresponding to the odd and even inter-arrival times.

4.1.3. Markov renewal equation By a renewal argument, we obtain

Pij(t) = l{i:j}(1 - Hi(t)) +

~/0 ' Qik(ds)Pkj(t -- s)

,

which in matrix f o r m becomes e(t) = [I - H(t)] + Q . P(t) .

Solution: If supjHj(t) < 1 for some t > 0 and maxi,j SUpxI(I - H(x))(i,j)l _< 1, the solution of the above M R E exists, is unique and it is given by P(t) = (I - Q(t)) (-1) * (I - H(t)) ,

B a s i c p r o b a b i l i s t i c m o d e h ' in reliability

17

where ~9(t) = (I - Q(t)) (~) = Z Q ( " ) ( t ) n>O

is the M a r k o v R e n e w a l Function. Moreover, O!~)(t) = x-~U

Qik (t-u)Qkjdu, {0~k f ó~/~1/

ift>O ift
and (o)

Qij (t)

=

(1)

Qij (t)

l{i_j}l{t>o}, Qij(t)

.

4.1.4. L i m i t distributions and theorems

Here we give some limit theorems useful for the reliability analysis. More importantly, these theorems give an extension of the basic classical limit theorems in probability theory to the semi-Markov setting. THEOREM 4.1 (Steady-state distribution, Taga, 1963). 7rj = l i m Pij(t) = vjmj vm where

]Ei[S1]

v = (vi) :

is

aH

invariant

measur«

of

(Jn)

and m = (mi)

with

mi =

f~(1 - Hi(t))dt.

The following two theorems are straightforward applications of the Blackwell renewal theorem and of the key renewal theorem, respectively. THEOREM 4.2 (Blackwell type theorem, ~inlar, 1969). c

~,ü(t) - ~,,,(t-

«) ~

E,[sl] '

ast--+

(?~ .

THEOREM 4.3 (Key renewal type theorem, ~inlar, 1969). I f i persistent state and hi a direct R i e m a n n integrable function, then t

/o ~~~ldyl~~l~-~l ~ ~

1

oo

/o ~~~ld~

THEOREM 4.4 (Law of large numbers, Taga, 1963). 1

n

~~X~~~~[Xl],=

as t ~ o~ .

is non-periodic

N. Balakrishnan, N. Limnios and C. Papadopoulos

18

THEOREM 4.5 (Central limit theorem, Taga, 1963).

Xl + . . . + X, - hIE[X1] --+ N(O, 1) . nv/h~io-i

Under the hypotheses that E~[X1] < oc, and a2i = IEi([Y[ - SlIEr(X1)] 2) < ec where yi = 2j=s~ 1+1Xj, S~ = 0, n = 1 , 2 , . . . , then S i, is the recurrence time of state i for the Markov chain (J~). 4.1.5. A central limit theorem Let f be a real measurable function defined on E x E x IR. Define, for each t _> 0, the functional Wf(t) by N~j(t)

Wf(t)= Z

E f ( i , j , Xij~)

i,j n=l

when the series converges. Put:

/0 ~0

Aij=

f(i,j,x)dQij(x),

Ai=

Aij, j=l

0(3

Bij =

S

(f(i,j,x))ZdQij(x),

Bi = Z B i j j=l

and s

mi =

=

ii 5 ZAJ/ j=l

-

+

Br

i;/

)j

r--1 s

+ 2E Z

E AreAklti*i(#*ti + #i*k- I~e*k)/(#~r#*kk),

r=l g,~-i k~-i

mi B f - -_ - -

mf~--~ #ii

~2 z

.

~lii

THEOREM 4.6 (Pyke and Schaufele, 1964). Under the hypotheses that the above

moments are finite, we have that, as t -+ oc, t-1/Z[wf(t)- t.mf]

d N(O, Bf) .

Basic probabilistic models in reliability

19

4.1.6. A functional central limit theorem

Let W(t)

=

g(Z(u))du

be a functional of the semi-Markov process (Z(t), t >_ 0). Hypothesis (H): • MC (Jn) has a unique invariant measure ~;

• llp<~) (x,-)-H(x, a~) < oo. THEOREM 4.7 (Limnios and Oprisan, 1999a, b). Suppose that (H) isfulfilled and ~ n > l ~(n) 1/2 < oo and Var~X1 < oo. Put

s,, = ~

g(J~_,)x,~ - ~(g),

k=l

0n (t) = +

SE<

and (72 = Var~[g(Jo)X1] 2 q- 2 ~

Cov~(g(Jo)Xa,g(Jk j)Xk) •

k>_2

Then a 2 < oo, and for all probability measures # on #, IP~ o 0n~ ~ W provided that «2 > O. W is the Wiener measure and ~ denotes weak convergence. 4.1.7. Statistical inference

Consider an observation of a sample path of the semi-Markov process on [0, T], T is a fixed time. ~T

= { J 0 , J 1 , .. , J N ( T ) , X 1 , . . . , X N ( T ) }

•

Put k>l

N,j = N,«(r) := Z

l~j~ l=,,J~=j,~~_«~ •

k_>l

The empirical estimator of the semi-Markov kernel is: 1 ~

ôgj(t, T) = Ni ~

l{j~_,=+,J~:+X~<_t} .

N. Balakrishnan, N. Limnios and C. Papadopoulos

20

Estimators of other quantities are defined by replacing the semi-Markov kernel in the analytical expressions by the above estimator. THEOREM 4.8 (Moore and Pyke, 1961). The empirical estimator of the semiMarkov kernel." Qij(t, T) is uniformly strongly consistent, i.e. max sup IO«(t,T)-Qq(t)l

224 O,

asT--+oc

.

icl" O
THEOREM 4.9 (Ouhbi and Limnios, 1997). For all t E IP,+ max sup I Ôij(t, T) - ~pij(t) l .... » O, ij O
as T--+ oo .

THEOREM 4.10 (Ouhbi and Limnios, 1997).

B { O i j ( t , T) - ~Pij(t)} ~

N(O, az(t)),

a s T ----~ o(3~

where S

~72(t) = ~

S

~

t.t«q{ (tPiq * ~kj) 2 * Q«k(t) - [(~tiq * I/Ikj * Q«k)(t)] 2} .

q=l k=l

THEOREM 4.11 (Ouhbi and Limnios, 1997). A vailability: sup ]Äij(t, T) - Ai«(t)] O<_t
a.s.

---+ 0,

asT--+ec .

Reliability: sup [Rij(t, T) - Ru(t)[ O<_t<_L

a.s.

---+ 0,

asT--+e~

(L ~ ~,+).

4.2. Safety and reliability modeling In reliability problems, the state space E = { 1 , . . , s } is usually partitioned into the following two sets: • U = { 1 , . . , r), containing the up states, and • D = {r + 1 , . . , s), containing the down stares.

Thus, the Reliability-related indicators will be expressed as in Limnios (1996):

Basic probabilistic mode& in reliability

21

Reliability:

R(t) = ~ l ( I - Q l l ( t ) )

( 1) s r ( I - H l ( t ) ) l r

•

REMARK 4.1. T h e s a m e f o r m u l a as a b o v e gives the safety o f the s y s t e m w h e n the D set c o n t a i n s o n l y u n s a f e states.

Availability:

A(t) = ~(I - Q ( t ) ) (-1) -x (I - H(t))ls,,- .

Mean time to failure:

M T T F = cq(I - P l l ) - l m l

.

Mean time to repair:

M T T R = ~z2(I - P l l ) - l m 2

.

Mean up time:

MUT -

7qm1

7~2P211~

Mean Down Time: MDT -

g2m2 7rlP121D

5. Monte Carlo methods in reliability 5.1. Introduction A M o n t e C a r l o m e t h o d is u s u a l l y u s e d to p r o v i d e a n a p p r o x i m a t e s o l u t i o n to the f o l l o w i n g e s t i m a t i o n p r o b l e m : L e t h be a real function, h : IR --+ IR. We want to

B n d the value o f the integral

N. Balakrishnan, N. Limnios and C. Papadopoulos

22

=

ä•01

h(x)~.

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Even if the problem of estimation o f / h a s no probabilistic content at all, it can be easily transformed to a problem with an intrinsic probabilistic structure, i.e. I =

f0 ~h ( x ) d x = /0 ~h ( x ) f x ( x ) d x

.

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where f x ( x ) = l[0,1](x) is the density of a uniform U(0, 1) random variable X. Since I = lEIh(X)], the estimation of the integral can be carried out using probabilistic arguments. This fact constitutes one of the greatest advantages of the Monte Carlo simulation method, together with its efficiency in higher dimensions and of course, its obvious simplicity. A Monte Carlo simulation method uses pseudo-random numbers that have the property of being almost uniformly distributed in a given interval. These pseudorandom numbers are generated by some deterministic and not at all random sequence constituting the pseudo-random number generator. The most common pseudo-random number generators are the linear congruential ones, described by the following recursive formula: x0

initial value,

xn=(aXù-l+C) modulom,

n> 1 ,

where the initial value x0 is called the seed of the generator, and ~ and c are given positive integers. The elements of the generated sequence xn will then be approximately uniformly distributed in the interval {0, 1 , . . , m - 1}. Consequently, m is called the p e r i o d of the generator and surely, after a large number of trials (less than m) the sequence of number will be reproduced exactly the same. It is desirable, therefore, to choose ~, c and m in a way that the period of the generator is the largest possible, and in fact constructing " g o o d " pseudo-random number generators constitutes an active research area. As far as reliability theory is concerned, Monte Carlo simulation is a very efficient tool since it does not make any restrictive assumptions on the system model. We can thus simulate without any particular difficulty DTMC, CTMC, semi-Markov processes, as well as non-Markovian ones. The output of the simulation of a stochastic process is normally the trajectories describing the evolution of the system in time, i.e. the stare occupied by the system at any instant. No doubt, these simulated trajectories will be close to the real trajectories of the system depending of course on the system model and out ability to clearly describe it. Consequently, a number of simulated trajectories can be used in order to estimate system's parameters such as the reliability, the availability, the M T T F , the stationary distribution, etc. In the following, we give an algorithm describing how to simulate a discrete random variable. This algorithm will be useful to construct an algorithm for the simulation of DTMC.

Basic probabilistic models in reliability

23

5.1.1. Simulating discrete random variables In order to simulate a discrete r a n d o m variable X taking values in a set E = { 1 , 2 , . . ,s}, with distribution pi, i CE, i.e. Pi = IP(X = i), the following algorithm can be used: Algorithm 1 - Inverse transform method

Input data: The probability vector p = (Pi, i = 1 , . . , s). 1. Generate a uniform r a n d o m number in the interval (0,1), say u; 2. Find k, k >_ 1 such that k 1

k

ZPi<__u<~pi i=1

,

i=1

using the convention that ~~-1" = 0; 3. Set X = k. The previous algorithm can be ameliorated/modified depending on the distribution of the r a n d o m variable that we want to simulate (see Ross, 1990).

5.1.2. Simulating discrete time Markov chains Algorithm 2

Input data: The state space of the D T M C , its initial law #(.), as well as its transition matrix P. 1. Sample a r.v. X ~ # and put n = 0, Xn(co) = X(c~); 2. Generate a uniform r a n d o m number in the interval (0,1), say u; 3. Find k such that k-I

k

Z P(X~(o,),j) < ~ < ~ P(X~(oo),j) j=l

j-1

4. Set n = n + 1, Xn(o~) = k; 5. Repeat steps 2 4 for the number of jumps of the chain needed; 6. Output the sequence of states visited, described by Xn(c~). Different implementations of the previous algorithm are also possible. For example, given that we are in state i at time n, one could generate a geometric r a n d o m variable having parameter 1 - P(i, i) to find out when the chain will exit the current state i and then use the transformed transition matrix to find the state of the system in which the process will finally jump into. Consider again the binary system described in Section 2.1.1. One typical (simulated) trajectory of the corresponding M a r k o v chain is given in Figure 1. Moreover, independent simulations can be carried out in order to estimate the principal system's parameters.

N. Balakrishnan, N. Limnios and C. Papadopoulos

24

Sample trajectory of the binary system

E

õ o

-3 I

I

I

[

I

I

1

2

4

5 Time

6

9

10

Fig. 1. A typical trajectory of the binary system.

5.1.3. Simulating continuous random variables PROPOSITION 5.1 (see Ross, 1990). Let U be a uniform (0, 1) random variable. For any continuous function F the random variable X defined by X = F -1 (U) has distribution F (in this, F - l ( u ) is defined to be the minimum value of u such that F(x) = u). F o r example, if we want to generate a sample from an exponential rand o m variable with distribution function F(x) = 1 - e - a , x > O, 2 > O, all we have to do is to first generate a uniform r a n d o m n u m b e r u in (0,1) and then take

1 x = F - l ( u ) = - 7 l o g ( 1 - u)

B

A

This is very useful in the simulation o f C T M C , where the sojourn times in different states have indeed an exponential distribution.

5.1.4. Simulating continuous time Markov chains Algorithm 3 I n p u t data: The state space o f the C T M C , its initial law/~(.), the exit rates at each stare of the state space (q(x)), as well as the transition matrix P o f the embedded M a r k o v chain.

Basic probabilistic models in reliability

25

1. Sample a r.v. X ~ # and set t = 0, X(t, co) = X(co); 2. Generate -c an exponential r a n d o m variable with parameter q(X(t, co)); 3. Using the transition probability matrix of the embedded M a r k o v chain find the state y in which the chain will j u m p into; 4. S e t t = t + ~ , X ( t , co)=y; 5. Repeat steps 2-4 for the number o f j u m p s of the chain needed, or until the time t becomes greater than the observation period T.

6. Variance reduction methods Consider now the class of systems whose failure is an event of a small probability. Examples of this type of systems are computer systems and networks, nuclear stations, communication systems, etc. In order to analyze the behavior of such systems our principal tool will be the variance reduction methods. Throughout this chapter the term "rare" will imply a probability that will be "sufficiently" small. For example, in the communication systems field we are interested in the probability of error during the transmission of bits. This probability is often smaller than 10 .6 and doing the analysis of such a system by taking into account events of this type, becomes an extraordinary task. We have to underline however, that a definition of the rare event does not clearly exist, since it depends not only on the probability of the event but also on the system model. Moreover, the term "rare" reflects our difficulty and the effort in us obtaining estimates of the associated measure. Thus, the event of a failure of a two-component system having a failure probability of 10 -9 may be stated as "less rare" than the failure of a 300-component system whose corresponding failure probability is 10 s. Moreover, the notion of rare event changes as years pass by. It may also be the case that the system under question is quite complex containing a large number of components. To model a system having n components we need a state space of at least 2 n different states. The analysis becomes more complicated when the system at hand is, furthermore, non-Markovian in nature. In the case of a large state space, state lumping or state aggregation methods can be used (cf. Goyal et al., 1987; Kemeny and Snell, 1976) allowing us to reduce the dimension of the problem and treat a system $1 instead of the original S. However, these techniques fall into practical difficulties as a considerable amount of computer time and m e m o r y is still required. Furthermore, they are not easily applied to complicated system models and even if this is possible it may sometimes be quite difficult to assess the error incurred through the state aggregation process. Existing analytical methods are not efficient in such settings and we are quickly obliged to use simulation in order to estimate system's performance. In the case of rare events direct simulation is not efficient since, in order to get an idea about the probability of the event, we have to hit it several times and thus use a large number of samples. For instance, if we want a 20% error and 95% confidence interval for the estimation of a probability of the order of 10 6, we will certainly need at least 108 realizations.

26

N. Balakrishnan, N. Limnios and C. Papadopoulos

Given that direct simulation is too expensive in terms of the number of samples needed in order to obtain a given confidence level, other simulation methods have appeared to face this problem. These methods are known as variance reduction methods, and are quicker than the standard Monte Carlo simulation scheine and can be principally classified into four separate categories (see Bratley et al., 1987; Fishman, 1996; Ripley, 1987; Ross, 1990): 1. 2. 3. 4.

correlation methods; conditioning methods; methods of importance; others...

In the first category, we can find the method of antithetic variables together with the one of control variables. The underlying principal idea consists in using correlated (negatively or positively) random variables in order to reduce as rauch as possible the variance of the corresponding estimator. The second category comprises the method of conditioning as well as the method of stratified sampling, both of which are based on the formula of the conditional expectation/variance. The third one, deals with the method of importance sampling, a method that is actually used to estimate systems parameters especially in highly reliable markovian systems (see Glynn and Iglehart, 1989; Heidelberger, 1995). This method consists in modifying the original probabilistic dynamics of the system and carrying out the simulation using a new distribution. A compensatory factor, called the likeIihood ratio, will remove the bias introduced to the estimator by this change of measure. The aim of this method is to choose a new distribution that will result in a variance reduction. Normally, this distribution has to privilege the rare event under question and make it happen more frequently. In such cases, a significant amount of variance reduction may be obtained. However, a bad choice of this new distribution may give rise to an infinite variance and therefore we have to be very careful when doing such changes of measure. 6.1. Correlation methods 6.1.1. Antithetie variables Suppose that we are interested in estimating a parameter 0 associated to a given system model, and consider that 0 can be expressed as the expected value of a function of a random variable X. The natural way to estimate 0 would be to consider independent replications of the random variable X, say t " 1 , X 2 , . . ,X,, and take as estimator of 0 the quantity 1 ôù =~~x~.=,

which is an unbiased estimator for 0. Moreover its variance is given by

var[ô,,] - Var[xl] n

Basic probabilistic models in reliability

27

Consider now the case when n = 2k, with k E N and the estimator

ô'

1 ff--,x,1 + x~2 ~i=1

2

'

which is also an unbiased estimator for 0 and whose variance is given by Var[Ô'ù] = Var[X11 @212] 4k In such cases it will be advantageous to have the random variables Xtl,X~z negatively correlated, since then we will have that Var[Òùl < Var[Ôù]. This is exactly the case when Xll can be expressed as a function of m random numbers, Xll = h(U1, U 2 , . . , Um), while X12 = h(1 - UI, 1 - U2, . . , 1 - Um). Of course the two r a n d o m variables have the same distribution and given that the function h is monotone with respect to all of its arguments they are negatively correlated. In reliability systems, normally the function h stands for the structure function of the system which in most cases is monotone. Thus, using the method of antithetic variables we have the following two benefits: • the estimator has smaller variance than the direct estimator; • only half of the r a n d o m numbers is necessary to carry out the simulation.

6.1.2. Control variables Consider again the problem of estimation of 0 = lE[X] and let Y be a supplementary r a n d o m variable for which the expected value /~y = lE[Y] is already known. The method of control variables consists in using the information we have for the r.v. Y, in order to ameliorate out estimates of 0. Notice that the variance of the following unbiased estimator for 0: X + «(Y - ~y) ,

(35)

is equal to Var[X + «(Y - gy)] = Var[X] + «2 Var[Y] + 2« Cov[X, Y]

(36)

and thus, the value of c minimizing (36) is given by

e* -

Cov[X, Y] VarV]

Moreover, using such a value for the parameter c the variance of the estimator (35) beeomes

Var[X + c*(Y - U~)] = Var[X]

(Cov[x, y])2 VarV] '

which is evidently smaller than the variance of X itself.

28

N. Balakrishnan, N. Limnios and C. Papadopoulos

Even if using the method of control varies we have the benefit of effectively reducing the variance of the estimator, this method has the inconvenience to necessitate some additional information. We need to know for example Var(Y) and Cov(X, Y) which is not always the case. In practice, these quantities have to be estimated using the first samples of the simulation and the estimates obtained can be used to give an approximate value for c*. We can then carry out the rest of the simulation using this value. The name of control variables sterns from the fact that the random variable Y and the samples obtained by the simulation play the role of the correction/control factor. In other words, when the simulated Y value is greater than its already known expected value, then i f X and Y are positively correlated, X will have the tendency to be greater than its mean (0), also. However in this case, Cov(X, Y) will be positive making thus c* to be negative which will consequently adjust the value o f X + c * ( Y - tty) more closer to 0 than does X itself. Similar arguments can be given for the case when X and Y are negatively correlated or the observed value of Y is smaller than its known mean. No doubt, the efficiency of the method depends principally on the chosen control variable Y as well as on the precision we have in estimating the c* value. 6.2. Conditioning methods 6.2.1. Conditioning The conditioning method is based on the following expression for the variance: Var[X] = IE(Var[X]Y]) + Var(lE[X]Y]) .

(37)

Since the variance is always positive, the terms at the right of this last expression are all positive, implying thus that Var[X] _> Var(~[XrY]) .

(38)

However, since lE{lE[X]Y]} = lE[X] = 0, lE[X]Y] can also be used as an unbiased estimator of 0 and it is preferable to do so since its variance will be smaller than the one of the direct estimator as indicated by (38). In fact in this case, we prefer to simulate the random variable Y and by observing its values make conclusions about X. In this, we suppose that lE[X] Y] is known or can be easily determined from the simulation tun. This method may be quite efficient in the case where we are interested in estimating the failure probability of a k-out-of-n system. The variable Y will represent the n - 1 components and if we know the state of these components we will probably be in position to determine the state of the system. 6.2.2. Stratified sampIing Stratified sampling is a variance reduction method based on the same formula (37) as the conditioning method. The idea behind this method is to separate our original sample space into different strata and carry out independent simulations

Basic probabilistic models in reliability

29

in each of these• Then, by taking the expected value of the output of the simulations on all of the strata we can obtain the desired estimate of the system's parameter. However, the principal difference with the method of conditioning is the fact that the former uses (38) to prove the variance reduction while in the case of stratified sampling the basic argument is that Var[X] _> IE(Var[XIYI). Moreover, it is sometimes difficult to separate the sample space into strata and to define how many samples we have to take from each one in order to have the largest possible variance reduction. This depends clearly on the problem at hand and may be difficult to do in the general case, except in problems having an intrinsic layered structure. See Ross (1990), for the details and examples of this method.

6.3. Importance sampling 6.3.1. Relative error Consider now the problem of estimation of the following quantity: 7 = ~y[h(x)] =

h(x)f(x)d~,

(39)

O0

where X is a real random variable defined on a probability space (~2,~ , IP), having density f and h:lR--+ IR. In order to estimate ~ using Monte Carlo simulation we can generate an n-sample (~ol,..., con) issued from f and consider the following unbiased estimator:

n

•

In the case where h(x) = lA (x) with lA (x) = 1, if x ~ A and 0 otherwise, then 7 = IP(X ~ A). Moreover, the variance of the ~n estimator is equal to 7(1 - 7)/n, while the associated 100 x (1 - 6)% confidence interval will be

where z6/2 is defined by the equation 6/2 = P(Z > z6/2) and Z denotes a random variable having the standard normal distribution N(0, 1). If we are interested in constructing a confidence interval for 7 the natural way will be to continue the simulation until the interval's half width becomes less than ~c (~c E]0, 1D times the value of the parameter that we are trying to estimate. Thus, the stopping criterion for our simulation will be

z~/2

)~(1 - ~~) < ~~, n

which implies that z~/2

7

< ~c .

(40)

N. Balakrishnan, N. Limniosand C. Papadopoulos

30

Note however, that the relative error (RE) of the estimator ~ù, which is defined to be the ratio of its standard deviation to its expected value, will be given by (as n ~ +oc)

Bl~-~ù) REG)

= z6/~

-

-

~,

,'~ Z 6 / 2

1 ~,

^ n-,+o~ since ?.

"7 •

It is this last equation that clearly illustrates the inconvenience of using direct simulation: the relative error of the estimator remains without bounds, while the event becomes rarer and rarer (i.e. RE(~n) ~ +ec when 7 --+ 0). It also means that in order for Eq. (40) to be satisfied and thus obtain the desired relative precision of estimation, we have to considerably increase the size n of the sample. In other words, in order to estimate ? up to a certain level of precision, one has to increase the number of simulation runs as the probability of the event becomes smaller and smaller.

6.3.2. Some background theory Importance sampling is a method that may help us to overcome the previous difficulty. The basic idea of the method is to change the original probabilistic dynamics of the system, and modify at the same time the function to be integrated. This change of measure is illustrated as follows: =

f

+o~ h(x ) ~ f

j ~x)

' ( ~ ) & = ~s, Eh(x)L(x)]

,

(41)

where L(X) = f ( X ) / f ' ( X ) represents the corresponding likelihood ratio and the subscript f ' means that the expected value is now taken with respect to the new density fl. The name given to this method is due to the fact that the process is sampled in the areas that are more important for the estimation of 7, in the case where h(x) = lA(x) the areas where the event {X E A} is realized. Consequently, the new density f~ has to be chosen in a way to make the rare event under consideration more likely to occur. Since this change of measure introduces a bias to our estimation, the results obtained by the simulation have to be multiplied by the appropriate likelihood ratio. This term plays the role of the compensatory factor, since the system has been simulated using a probability measure that is not directly associated to the system's model. Eq. (41) is valid only in the case that f ( x ) > 0, for every x E IP, with f ( x ) > 0 and h(x) > 0, which implies that a possible value o f X under f , is also possible under S - It is possible however to have f ( x ) = 0 and f(x) > 0, for any x E IP, with h(x) = 0. By making this change of measure, the new unbiased estimator of 7 will be ~n(f') = ~ ~ L 1 h(oi)L(°»i), where the new n-sample (col,..., con) has now been generated using density S . Its corresponding variance is given by

Basic probabilistic models in reliability

Varf,[~n(f')] =

~ h(x)

= IEf[h(X)L(X)] - 72

31

S ( x ) d x _ ?2 (42)

The main aim of importance sampling is to find a suitable - and easily imp l e m e n t a b l e - new density f l in order to minimize the variance of 7n(ff) and by doing this, reduce the cost of the estimation procedure. Thus, using importance sampling the rare event has to be realized more often, meaning that its new probability taust be greater than the original one. The corresponding L term in Eq. (42) has to be kept as small as possible. In the case where the L term is uniformly less than one, then Varf,[~ù(f')l < 7 - 7 2 = Varf[Tn(f)] and we will certainly obtain a variance reduction. Another alternative would be to choose f~ in a way that lEf Ih(X)L(X)] is of the same order of magnitude as 72. In such cases the associated change of measure is sometimes called asymptotically efficient or asymptotically optimal (see the survey of Heidelberger, 1995 for a discussion on this matter). An optimal change of measure is defined to be a measure that results in a zero variance estimator for the unknown quantity (see Kuruganti and Strickland, 1995) and it always exists. For our example, this corresponds to choosing

f* (x) -- h(x)f(x) ?

(43)

Using the optimal change of measure for the simulation, the exact value of the parameter will be obtained in the first simulation tun. Unfortunately, it has the disadvantage of containing 7, the parameter that we are trying to estimate, making it thus not directly exploitable. Nevertheless, in some special cases, we can explicitly construct this optimal change of measure, which will enable us not only to estimate ? at a minimum cost, but also - and more importantly - to find its exact value, as a by-product of the intermediate calculations (see Kuruganti and Strickland, 1995, 1997). The conditions on the applicability as well as the theoretical framework behind importance sampling are given in Glynn and Iglehart (1989). In their work, importance sampling is extended to problems arising in the simulation of both discrete time and continuous time Markov processes, as well as in generalized semi-Markov processes. In the same paper, the authors discuss the problem of steady-state quantities estimation, that can be carried out by exploiting the regenerative structure of the Markov chain, as well as the estimation of transient quantities, where a different approach has to be used.

6.3.3. The optimal change of measure Let us consider again expression (43) and let h(x) = lA (x). In this case, the choice f* (x) = f ( x ) 1~ (x) 7

(44)

32

N. Balakr&hnan, N. Limnios and C. Papadopoulos

corresponds to the optimal change of measure associated to the estimation of 7 = IP(X E A). Even though Eq. (44) seems to be at first sight of no use, since it contains the unknown parameter, it has the benefit of providing us with a very useful insight concerning the choice of the new density fl(.). Indeed, as it is indicated in Strickland (1993) the following hold: • All the mass of the probability is concentrated on the rare event {X E A}, and consequently only those samples that correspond to the realization of this event will be produced when the optimal change of measure is used to carry out the simulation. • On A, the new density is exactly the conditional density of X, given that {X C A} has occurred: f ( X ) " IA(x) _ ~ f ( x [ X E A), f * (x) -- ~ ~- Ä) 1, o,

x E A, otherwise .

Therefore, the relative likelihood of the values of X on A, is exactly the same for the original as well as the new distribution dF*(xl) _ dF(xl) dF* (x2) dF(x2) '

where Xl,X 2 ŒA

and F(.) represents the cdf corresponding to the density f ( . ) .

6.4. M a r ovian systems and importance sampling Consider now a continuous time M a r k o v chain X = {Xt: t > 0}, with state space E = {0, 1 , . . , s } , s < +ec, infinitesimal (conservative) generator Q = {q(x,y): x , y c E}, and initial law #(.). The quantity q(x) = - q ( x , x ) represents the total rate out of state x. Suppose also that the stare space of the system is divided into two disjoint subsets, U and F, with U U F = E and U N F = (3. The set U = {0, 1 , . . , m } represents the set of operational states, while F = {m + l , . . . , s} stands for the set of failed states of the system. In state 0, all components are considered new. Define also 0 = To < T1 < ... < Tn < ..., the sequence of the successive j u m p times of the chain X. Then Y = {Yn, n > 0}, defined by B-Xtù,

n=0,1,..

,

will be the embedded discrete time M a r k o v chain associated to Xt. The elements ofits transition matrix P = { P ( x , y ) : x , y E E}, are given by P(x,y) = q ( x , y ) / q ( x ) , when x ¢ y and 0 otherwise. Importance sampling can be easily extended to the case of Markovian systems. In order to modify the probabilistic dynamics of the system, one has to basically modify the transition probability matrix or the generator of the process (see Glynn and Iglehart, 1989) and/or the initial distribution of the process.

Basic probabilistic models in reliability

33

The estimation problems may concern transient or steady-state quantities. In the first case, the regenerative structure of the chain is employed and the estimation problem is transformed to its analog over the regenerative cycles of the chain (see the following discussion), while in the second case the process is simulated for a given time horizon. Moreover, in case of simulation of a continuous time Markov chain, one part of the simulation is devoted to the simulation of the corresponding embedded Markov chain where the sequence of states visited by the process is generated. Then, given the sequence of states visited by the chain, the second part of the simulation concerns the generation of the associated (conditional) sojourn times in these states. This discrete time conversion always results in a variance reduction (see Goyal et al., 1992; Fox and Glynn, 1986).

6.4.1. Regenerative simulation Let -c be a stopping time for {Yn : n _> 0} which means that the realization or not of the event {z = n} may be determined by yn _ (Y0,. •, Yn). Note also En for the set of all possible paths of the chain Y until time n en - { / =

( y 0 , y , , . . ,yn): • c E} .

Then, the probability associated to any yn c En is given by P(Y") = #(Yo)P(Yo,Yl)...P(Yn-I,Yn) , where #(Y0) = P(Y0 = y0) is the initial law of the chain. Moreover, let Bn c f~n stand for the set of all paths for which {z = n}. We have the following proposition: PROPOSmON 6.1 (Goyal et al., 1992). Consider a discrete time Markov chain with transition probability matrix P. Ler P be the probability measure associated with the different trajectories of the chain and z a stopping time which is finite under P, with probability 1. Note also Z, a measurable function of Y~ .for which IEe [[Z(Y~)11 < ~ . Let P' be a new probability measure for whieh z is also finite with probability 1 and for any y~ ¢ Bù, p'(yn) • 0 whenever Z(yn)P(y n) ~ O. Then IEe[Z(Y~)] = IEp,[Z(Y ~) L(Y~)], with L(Y ~) = P ( S ) / P ' ( y n ) , for any yn C Bh. Remark however, that in this case it is not necessary for the new importance sampling measure to correspond to a time-homogeneous Markov chain. A different measure P' given by p,(yn) = P'(Yo)P'(Yl lYO)'" P'(Yn]Y0...Yn-1) , can also be used and it is called a "Dynamic Importance Sampling measure" (DIS, see Shahabuddin and Nakayama, 1993 and references therein). In this, P'(YnlY0..-Yn-1) represents the likelihood of the path Yn=yn given that En-1 = CVO,... ,Yn 1).

Consider now that we are interested in estimating the steady-state unavailability of the system c~that represents the fraction of time for which the system is

N. Balakrishnan, N. Limnios and C. Papadopoulos

34

considered failed. Let h(y) = 1/q(y) be the mean sojourn time in state y and let g(Y) = 1F(y)h(y). Then, we can write (see Crane and Iglehart, 1975) nz rv,~0-1

« = ~PL~k=0

g(:~k)]

(45)

] l P rV'VCo -1 h(rk)l L2Jk=O

Let us now define TB = inf{t > 0 : Xt E B}, the hitting time for B C E (with the convention that infO = +co) and r» = inf{n > 0 : I1, E B}, the number of jumps for Y to enter B C E. A somehow similar representation holds for the mean time to failure (MTTF) of the system, which may be written as (see Goyal et al., 1992) nz rv'~min(z0,ZF) - I h(Yk)] M T T F = ]Ep[TF] = ]te[min(T0, TF)] = uzP[2--*k=0 P(TF < To) ]lp[l{zF<~0} ] '

(46)

where P(rF < z0) = ]lp[l{zF
]lp[H]

'

where G and H for the case of steady-state unavailability and M T T F estimation are given above. In the case of M T T F estimation, our main difficulty is the estimation of the denominator that corresponds to a rare event, while for the numerator we can simply use direct simulation. In the case of unavailability estimation, importance sampling can be used to estimate the numerator while direct simulation may be suitable for the denominator. Thus, the two parts of the ratio can be simulated in an independent manner and a different number of simulation runs can be used to obtain estimates for ]lp[G] and ]lp[H 1. This is the principle behind the "Measure Specific Dynamic Importance Sampling" (MSDIS, see Goyal et al. 1987, 1992). Suppose that a total number of n regenerative cycles will be used for the simulation, where the first l(nJ cycles, with 0 < ~ < 1, will be generated using P1, while the rest F(1 - ()nj cycles will use P2 (P1 # P2). Let also L represent the likelihood ratio between the original and the new measure and note Gj,Lj, and Hj the samples of G, L, and H, respectively, from t h e / t h regenerative cycle. We can then construct the following estimator for q: (2}~~ GjLy)/L~nJ

0(,,~) = (~}'-r«,] HjL«)/F(1 -

()nl

"

This is a consistent estimator for ~, and we have (see Goyal et al., 1992):

Basic probabilistic models in reliability

35

lim 0~n r~ = with probability 1. Moreover: v~(0(n,~ ) - t/) ~ N(0, o-2(p1, P2)/]E22 [H]) ,

(47)

where " ~ " denotes the convergence in distribution and «2 (P1, P2)

-

Varp 1[GL] ~

+~2 Varp2 ~ - ~[HL]

(48)

Thus, in the case of steady-state unavailability estimation we can take P1 ¢ P and P2 = P, while in the case of M T T F estimation we can choose P1 = P and P2¢P. 6.4.2. Estimation of P("CF < "Co) Consider now Markovian systems where the failure rates of individual components are rauch smaller than the corresponding repair rates. For these systems the entrance into the failed subset of states may be considered as a rare event and a quantity of great interest is the probability P("CF < "CO)of failure during a cycle. The last one is associated to the M T T F of the system (see (46)), as well as to the unreliability of the system at time t (see Shahabuddin and Nakayama, 1993). In particular, the following proposition holds. PROPOSITION 6.2 (Shahabuddin, 1994). The probability 7 = P('CF < "Co) may be represented as aoer + o(er), where ao and r are positive constants. In this, e stands for the maximum failure rate of the components in the system and it is exactly this parameter that reflects the highly reliable nature of the system. For this reason, we call e the rarity parameter of the system. Consequently, Markovian systems can be classified in balanced systems, where all components have failure rates of the same order of magnitude and unbalanced systems otherwise. See Nakayama (1994, 1995, 1996) for a detailed description of the system model. Thus for the estimation of the probability P("CF < "Co), good importance sampling schemes are those making the first term in the expression of the variance of the importance sampling estimator to be close enough to e2r, the order of magnitude of 72. In such cases, the corresponding method will be said to possess the bounded relative error property, since the relative error of the importance sampling estimator will be bounded by a constant. The methods actually used in practice, where the total failure (repair) probability at each state is increased (decreased), are called failure biasing methods. They are distinguished from each other from the way the new failure probability is allocated to individual transitions. See the references for the different failure biasing methods. Nakayama (1996) gave necessary and sufficient conditions for a failure biasing method to have bounded relative error. These conditions are difficult to verify in

36

N. Balakrishnan, N. Limnios and C. Papadopoulos

practice, since one has to find the order of magnitude of a large number of paths of the process. Note also that the results obtained by any arbitrary simulation scheme can be ameliorated by eliminating all transitions to state 0. This is a direct consequence of the optimal change of measure (see Kuruganti and Strickland, 1997). EXAMPLE. Consider a 3-out-of-5:F Markovian system that is considered failed if at least 3 of its 5 independent components are failed. The state space of the system is E = {0, 1,2, 3}, where state 0 is the perfect state of the system (all components are operational) and state 3 is the failed state (F = {3}). The transition diagram of this system is illustrated in Figure 2 where the transition rates are equal to: 201 z 5 × 10 -4, -'~12 = 4 x 10 -4, /~23 = 3 x 10-4, #10 = 1 and [/21 = 2. Note that the repair rate in state 3 is of no use since the simulation ends the moment we enter state 3. We want to estimate the probability P(zF < z0) as well as the M T T F of the system using the ratio formula (46). The simulation results are given in Figure 3. At the left-hand side, we give the estimation results for P(zF < %) together with the ones obtained by an analytical method. In the same figure, we also represent the estimation results obtained by the same method, if before changing the transition probability matrix according to the Balanced Failure Biasing method, we eliminate all transitions to state 0. In both cases a significant amount of variance reduction is obtained. At the righthand side we represent the estimation of the M T T F of the system using the same methods. All the results presented correspond to a 95% confidence interval with 1% error level.

7. Simulation for semi-Markov systems In what follows, we present two different algorithms for the simulation of semiMarkov systems: the method of competing risk and the embedded Markov chain method. We give also simulation results for a simple 3-state semi-Markov system. 7.1. The method o f competing risk

The method of competing risk is based on the following proposition (see Opri~an, 1999).

~01

~2

initial state

z~3 3 :tate

~qo

t51

Fig. 2. A 3-out-of-5system.

Basic probabilistic models in reliability

37

Simulation results for t h e a-out-of-5 system ! ! ! !

x 10 l °

4.4

[

!

BFB(0.9) BFB(0.9) + no transitions Exact method

- -

to state

0

4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~

3.8

3.6

................................................................

8.4

: 3.2

i

I

1000

2000

x 10 -8

~

I

4000

5000 Iterations

6000

7000

8000

9000

!

!

Simulation results for t h e 3-out-of-5 system ! ! ! !

1

0ù9

3000

i

.......................

-

BFB(0.9) BFB(0.9)

-

.....

.. . . . . . . . .

, .........

to state

0.8

..

0.7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.~ 0.6

~

+ no transitions

.. . . . . . . . .

lOOOO

0

,

............................................................

E

~

0,5

......

". . . . . . . . . . . . . . . . . . .

. . . . .

.........

.

.

.

.

.

.

.

.

.

.

.

.

.

.

ò ku 0 . 4

.........................................................................

0.3

0.2

0.1

1000

2000

Fig.

3.

I

I

I

i

I

i

I

3000

4000

5000 Iterations

6000

7000

8000

9000

Simulation

results

for

the

3-out-of-5

system.

10000

38

N. Balakrishnan, N. Limnios and C. Papadopoulos

PROPOSITION 7.1 ( K o r o l y u k and Turbin, 1982). For all i c E, there exists a family o f independent random variables {~ik, k E E}, taking values in ~ + with the distribution functions Aik(t) =-

{ 1-

[ , Qikld~tl /f« > 0,

exp - fö I-H~(ù)J

0

(49)

otherwise .

The semi-Markov matrix Qij(t) is given by Qij(t) =

hi«(u)Aij(du),

i,j E E ,

where 1 - Hi(u)

hij(u) - 1 - Aij(u) -- IE[~j]z/j = u 1 and Iij is the indicator function o f the event {mink~Æ zik = zij}. The algorithm for the realization of one sample trajectory of the process is the following.

Algorithm 4. I n p u t data: The state space of the process, its initial law #(.), and the distribution functions Aij. 1. 2. 3. 4.

Sample a r.v. X ~ # and set t = 0, X(t, o~) = X(m); Set i = X(t, ~o); generate zifs (/" C E) using the distribution functions Aij; Set ~ = minj~s ~,ij, t = t + z, set X(t, o~) = arg minjcs Tij'~ Repeat steps 2-3 for the n u m b e r of j u m p s of the process needed, or until the time t becomes greater than the observation period T.

7.2. Embedded M a r k o v chain method This m e t h o d consists simply in using the transition probabilities of the e m b e d d e d M a r k o v chain in order to find the next state of the system and generating the Exp(0.001)

~x~~oo~~~®/we~~u~~~o~~~ Fig. 4. A 3 states semi-Markov system.

Basic probabilistic models in reliability

39

Availability for the semi-Markov system ! ! !

1

\

--......

0.99

. . . . . . . !. . . . . . .

!

!

Competin Risk Embedde~ Markov Chain Method

l --

Analytical solution

]

0.97

0.94[.......

i.........

i

!

.................................................

|

0.920"93I. ...... . . . . . .i. . . . . . . . . . . !.

0.91

.......

0

50

100

1 ~

!

!

i

i i

150

" >'~"~ . . . . . . .

i

200

250 Time

! .......

300

i. . . . . . . . .

........

350

400

450

Reliability for the semi-Markov system ~ ! ! ~

~

!

500

Competin q Risk Embedded Markov Chain Method

....

0.950.850.9

I

I

....................

0.6 0.75 07

0.65

0.6

50

100

150

200

250 Time

300

350

400

Fig. 5. Graphical comparison of the simulation methods.

450

500

40

N. Balakrishnan, N. Limnios and C. Papadopoulos

corresponding holding times to the states visited using the distribution functions F/j(t). This algorithm is similar to the algorithm used for the simulation of C T M C and is given below.

Algorithm 5. Input data: The state space of the process, its initial law /~(.), the transition probabilities Pij and the distribution functions ~j. 1. Sample a r.v. X ~ # and set t = 0, X(t, co) = X(co). 2. Set i = X(t, co); using the transition matrix of the embedded M a r k o v chain find the state j in which the process will jump into. 3. Generate z, the holding time in state i, using the distribution function F~j(t),

4. S e t t = t + ~ , X ( t ,

co)=j.

5. Repeat steps 2M for the number of jumps of the process needed, or until the time t becomes greater than the observation period T. EXAMPLE. Consider the semi-Markov system whose state diagram is given in Figure 4. The state space of this system is E = {0, 1,2}, with U = {0, 1} and F = {2}. The different laws governing the sojourn time of the process in different states are indicated in the figure. Thus the sojourn law of the system in state 0 is exponential with parameter 21 -- 0.001. When the system is in state 1 and jumps towards state 0, then the sojourn time has a Weibull distribution with parameters cq = 2.0, fil = 10.0, while when the system goes towards state 2, the sojourn time has a Weibull distribution with parameters «2 = 0.7,/~2 = 2.0. When the system is in state 2, only one transition is possible, the one going to stare 0. In this case, the sojourn time in state 2 has an exponential law with parameter 22 = 0.01. We have simulated the system described previously in order to estimate its availability A(t) = IP(X(t) E U), as well as its reliability R(t) = ]P(TF > t). The results of the simulation obtained for 100 iterations of the algorithm are given in Figure 5, where the results obtained by the previous two simulation methods are compared to the results obtained using an analytical method. For the embedded M a r k o v chain the transition probabilities were taken to be pl0 = 0.1048 and P12 = 0.8952. Note however that the time step of the algorithm was 0.2 time units.

Hitting times: Mean Mean Mean Mean

time to failure: M T T F = 1119.7, time to repair: M T T R = 100, up time: M U T = 1117.1, down time: M D T = 100.

References Asmussen, S. (1987). Applied Probability and Queues. Wiley, New York. Bratley, P., B. L. Fox and L. E. Schrage (1987). A Guide to Simulation. Springer, Berlin.

Basic probabilistic models in reliability

41

~inlar, E. (1969). Markov renewal theory. Adv. Appl. Probab. 1, 123 187. Crane, M. A. and D. L. Iglehart (1975). Simulating stable stochastic systems III, regenerative processes and discrete event simulation. Oper. Res. 23, 3345. Fishman, G. S. (1996). Monte Carlo. Concepts, Algorithms and Applications. Springer Series in Operations Research, Springer, New York. Fox, B. L. and P. W. Glynn (1986). Discrete time conversion for simulating semi-Markov processes. Oper. Res. Lett. 5, 191 196. Glynn, P. W. and D. L. Iglehart (1989). Importance sampling for stochastic simulations. Manage. Sci. 35, 1367 1392. Goyal, A., P. Heidelberger and P. Shahabuddin (1987). Measure specific dynamic importance sampling for availability simulations. In 1987 Winter Simulation Conference Proceedings, pp. 351-357. IEEE Press. Goyal, A., P. Shahabuddin, P. Heidelberger, V. F. Nicola and P. W. Glynn (1992). A unified framework for simulating Markovian models of highly reliable systems. IEEE Trans. Comput. 41(1), 36-51. Goyal, A., S. S. Lavenberg and K. S. Trivedi (1987). Probabilistic modeling of computer system availability. Ann. Oper. Res. 8, 285 306. Hammersley, J. M. and D. C. Handscomb (1964). Monte Carlo Methods. London, Methuen. Heidelberger, P. (1995). Fast simulation of rare events in queueing and reliability models. A C M Trans. Modeling Comput. Simul. 43-85. Ionescu, D. C. and N. Limnios, (Eds.) (1999). Statistical and Probabilistic Models in Reliability. Birkhäuser, Boston. Janssen, J. and N. Limnios, (Ed.) (1999). Semi-Markov Models and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands. Kemeny, J. G. and J. L. Snell (1976). Finite Markov Chains. Springer, Berlin. Korolyuk, V. S. and A. F. Turbin (1982). Markov Renewal Processes in Problems of Systems Reliability. Naukova Dumka, Kiev (in Russian). Kuruganti, I. and S. G. Strickland (1995). Optimal importance sampling for Markovian systems. In Proceedings of the 1995 IEEE Systems, Man and Cybernetics Conference. Kuruganti, I. and S. G. Strickland (1997). Optimal importance sampling for Markovian systems with applications to tandem queues. Math. Comput. Simul. 44(1), 61 80. Limnios, N. (1996). Dependability analysis of semi-Markov systems. Reliab. Eng. and Syst. Safety. Limnios, N. and G. Oprisan (1997a). A general framework for reliability and performability analysis of semi-Markov systems. In Eighth International Conference on ASMDA. Anacapri (Napoli). Italy, June 1997. Limnios, N. and G. Oprisan (1997b). A general framework for reliability and performability analysis of semi-Markov systems. Appl. Stochast. Models Data AnaL (to appear). Limnios, N. and G. Oprisan (1997c). Semi-Markov process to regard of their application. World Energy Syst. J. 1(1), 6zF75. Limnios, N. and G. Oprisan (1999a). Invariance principle for an additive functional of a semi-Markov process. Rer. Roumaine. Math. Pures Appl. 44(1), 75-83. Limnios, N. and G. Oprisan (1999b). Semi-Markov Processes and Reliability. Birkhäuser (to appear). Nakayama, M. K. (t994). A Characterization of the simple failure biasing method for simulations of highly reliable Markovian systems. A C M Trans. Modeling Comput. Simul. 4(1), 52-88. Nakayama, M. K. (t995). Asymptotics for likelihood ratio derivative estimators in simulations of highly reliable Markovian systems. Manag. Sci. 41, 52zP554. Nakayama, M. K. (1996). General conditions for bounded relative error in simulations of highly reliable Markovian systems. Adv. Appl. Prob. 28. Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. The John Hopkins University Press, Baltimore, MD. Opri~an, G. (1999). On the failure rate. In Statistical and Probabilistic Models in Reliability (Eds. Ionescu and Limnios).

42

N. Balakrishnan, N. Limnios and C. Papadopoulos

Ouhbi, B. and N. Limnios (1996). Non-parametric estimation for semi-Markov kernels with application to reliability analysis. Appl. Stoch. Models Data Anal. 12, 209-220. Ouhbi, B. and N. Limnios (1997). Estimation of kernels, Availability and Reliability functions of semi-Markov Systems, In Statistical and Probabilistic Models in Reliability (Eds. Ionescu and Limnios). Platis A., N. Limnios and M. Le Du (1998). Hitting time in a finite non-homogeneous Markov chain with applications. Applied Stoch. Models Data Anal. 14, 241-253. Pyke, R. (1961a). Markov renewal processes: definitions and preliminary properties. Ann. Math. Stat. 32, 1231-1242. Pyke, R. (1961b). Markov renewal processes with finitely many states. Ann. Math. Stat. 32, 1243-1259. Pyke, R. and R. Schaufele (1964). Limit theorems for Markov renewal processes. Ann. Math. Stat. 35, 1746-1764. Ripley, B. D. (1987). Stochastic Simulation. Wiley, New York. Ross, S. M. (1990). A Course in Simulation. Maxwelt MacMillan International Editions. Shahabuddin, P. (1994). Importance sampling for the simulation of highly reliable Markovian systems. Manag. Sci. 40, 333-352. Shahabuddin, P. and M. K. Nakayama (1993). Estimation of reliability and its derivatives for large time horizons in Markovian systems. In 1993 Winter Simulation Conference Proceedings, pp. 422429. IEEE Press. Strickland, S. G. (1993). Optimal importance sampling for quick simulation of highly reliable Markovian systems. In 1993 Winter Simulation Conference Proceedings, pp. 437~444. IEEE Press. Taga, Y. (1963). On the limiting distributions in Markov renewal processes with finitely many states. Ann. Inst. Star. Math. 15, 1-10.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

") /_...,

The Weibull Nonhomogeneous Poisson Process

Asit P. Basu and Steven E. Rigdon

The Weibull h a z a r d function is often parametrized as

h(x) = u

(x_k ~-1 \0/

,

x > 0

(1)

x > 0 .

(2)

or

h(x)

= ,~fix/~-x,

The n o n h o m o g e n e o u s Poisson process that has an intensity function o f f o r m (1) or (2) is often called the WeibuU proeess, or m o r e c o m m o n l y , the power law proeess. Such a process is often used to model the occurrence o f events in time, and in particular, to model the failure times of repairable systems. We therefore begin with a discussion of the h o m o g e n e o u s and n o n h o m o g e n e o u s Poisson processes.

1. The Poisson processes Let N(t) denote the n u m b e r of events that occur at or before time t. Such a r a n d o m variable is called a eounting proeess. W h e n the argument to N is an interval, such as (a,b], then N(a,b] is defined to be the n u m b e r o f events that occur in that interval. Thus, N(t) = N(0, tl. A counting process N(t) is said to be a Poisson process if: 1. N(0) = 0. 2. F o r any a < b _< c < d the r a n d o m variables N(a, b] and N(c, d] are independent. This property is called the independent inerements property. 3. There is a function 2, called the intensity fnnetion, such that

2(t)

= lim At-+O

P(N(t, t + At] =

1)

At

4. lim At~O

P(N(t,t+ At] > 2) At

= 0 .

The last property precludes the possibility o f simultaneous failures. 43

44

A. P. Basu and S. E. Rigdon

These four properties, as minimal as they seem, are enough to establish the property that the number of failures in the interval (a, b] has a Poisson distribution with mean equal to

Ja b E(N(a, b]) =

2(t)dt .

The proof involves recursively solving a system of differential equations; see Rigdon and Basu (2000) for the derivation. The function

A(t) =

~0t ,Z(x)dx,

which gives the expected number of events through time t, is called the mean funetion for the process. Clearly, A'(t) = 2(t). The nonhomogeneous Poisson process having an intensity function of the form

=0~,0/

,

t>0

(3)

or ,~(t) = ,t/~t ~-~,

t> o

is called the power law process or the Weibull nonhomogeneous Poisson process. This model has gone by many other names as well, including and most notably, the Weibull process. When fi < 1 the intensity is a decreasing function of t. In this case, failures will become less frequent as the system ages; this is reliability improvement. When fl > 1 the intensity is an increasing function of t, and in this case failures will become more frequent as the system ages. This is called deterioration. When fi = 1, then the intensity function is a constant. Thus, the homogeneous Poisson process is a special case of the power law process. The power law process can thereYore be used to model systems that improve, deteriorate, or remain steady over time, hut it cannot be used to model systems that improve for some intervals of t and deteriorate for other intervals. Some repairable systems have an intensity function that has the bathtub shape as shown in Figure 1. For small values of t, that is, when the system is young, the rate of occurrence of failures (ROCOF) is high and failures are frequent. After the bugs are removed, or after some of the weakest components fail, the R O C O F will be smaller, and it will remain at this level throughout its useful life. Then as the system ages, the R O C O F begins to increase. At this stage, the system is deteriorating. The two functions in Figure 1 look nearly identical, but there is an important difference in their interpretations. The bathtub intensity function indicates that the system will initially experience reliability growth. A few early failures will be followed by the useful life when failures occur at roughly a eonstant rate.

The Weibull nonhomogeneous Poisson process ~(t)

Bathtub intensity function

h(x)

45

Bathtub hazard function

<

#

I' t "~--Early failures'-'~ ~

Constant ~,----~ ~

Deterioration

"X ~-

Burn-in ~

~

Useful life ~

",*---- W e a r o u t

Fig. 1. Bathtub intensity and bathtub hazard functions. Eventually, as the system ages, the failures become more frequent. On the other hand, the bathtub hazard function indicates that there is a high chance that the system will fail (for the first and only time) early in its life. A few of the systems have serious defects that will cause early failures. Eventually, a working system will begin to wear out and the failure will ensue. The hazard function is the limit of a conditional probability. For a system that is wearing out, the probability of failure in (xo, xo + Ax] conditioned on survival past time x0 will be smaller than the probability of failure in (xl,xl + Ax] conditioned on survival past time xl, provided x0 < xl. There are therefore two bathtub curves: a bathtub intensity function for repairable systems and a bathtub hazard function for non-repairable systems. The bathtub hazard function expresses conditional probabilities of the one and only failure of the system. The bathtub intensity function indicates that the system will experience many failures early in its life, which will be followed by a time when the R O C O F is constant; finally, as the system ages, the failures will become more frequent. Although the power law process cannot model the bathtub-shaped intensity function, the bathtub curve concept helps to illustrate the difference between the interpretations of the intensity and hazard functions. In particular, it helps to illustrate the difference between the power law process (i.e., the Weibull nonhomogeneous Poisson process) and the Weibull distribution. The power law process is the nonhomogeneous Poisson process with intensity function 2(t) = (~/O)(t/O) p 1 and the Weibull distribution is that distribution having a hazard function of the form h(x) = (~/O)(x/O) ~ 1.

2. Models for the reliability of repairable systems The power law process can be used to model the occurrence of events in time. The most common application has been to model the failures of a repairable system. We assume that a failed unit is immediately repaired, or that the repair time is not counted in the operating time. If we also assume that a failed unit is brought back to exactly the same condition as it was just before the failure, then it is clear that the nonhomogeneous Poisson process is the appropriate model for the failure

46

A. P. Basu and S. E. Rigdon

times. This is called minimal repair. The assumption of minimal repair, therefore, leads to the nonhomogeneous Poisson process. This minimal repair model has also been called the bad-as-old model (Ascher, 1968). When a failed unit is repaired to a like new state, the distribution of the time to the next failure is the same as the distribution of the time to the first failure. This is called perfect repair, or renewal repair. A process for which the times between failure are independent and identically distributed is called a renewal process. Thus, the assumption of perfect repair leads to a renewal process. The renewal process model is also called the good-as-new model. Although the power law process is often used to model the failure times of repairable systems, it is much more versatile since it can model the occurrence of any kind of event in time.

3. Some data sets In this section, we give four examples of actual data. In three cases, the power law process is a reasonable model, and in the fourth case a nonhomogeneous Poisson process that is not of the power law form may be a reasonable model. EXAMPLE 1. Jarrett (1979) gave corrections and extensions to the famous data set - first presented by Maguire et al. (1952) - on the time intervals (in days) between coal mining disasters in the UK. A coal mining disaster was defined as an explosion leading to the deaths of at least 10 workers. There were 191 such disasters from 15 March 1851 to 22 March 1962, a time span covering 40,549 days. The first event was taken to be time zero, so the data set consists of 190 times between events. Figure 2 shows the event times on the horizontal axis and the cumulative number of events on the vertical axis. The smooth curve is the fitted power law process; we will discuss this fit in Section 5. The concave down relationship in Figure 2 indicates that the events were becoming less frequent. Thus, we would expect the growth parameter fl to be less than one. EXAMPLE 2. Table 1 shows the times of unscheduled maintenance for the photocopier in the Department of Mathematics and Statistics at Southern Illinois University Edwardsville. For a system like a copier, the number of copies made is a more appropriate measure of age than the actual calendar time. The first failure time on 17 November 1992 is taken as time zero. Figure 3 shows a nearly linear relationship, indicating that the copier neither deteriorated nor improved. Thus, we would expect the growth parameter to be nearly equal to one. EXAMPLE 3. On 17 January 1977, the state of Utah carried out the first execution in the United States in over nine years. Since that date, executions have become more and more frequent, totaling 618 through 7 March 2000. Table 2 lists the

47

The Weibull nonhomogeneous Poisson process

N(0

200 180

J

160

~Õ

..,i~.

° °

°,¢~

14o

3_2O

I¢) >

100

~

80

~0 40 20 5000

I0000

15000

20000

25000

30000

35000

40000

t

Time in days since March 15, 1851 Fig. 2. Times between coal mining disasters.

Table 1 Failure times for photocopier i

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Failure date

ci = copy

11-17-92 12-3-92 2-17-93 4-22-93 9-20-93 12-3-93 3-2-94 5-3-94 7-26-94 12-15-94 5-25-95 8-25-95 11-9-95 12-20-95 1-10-96 1-11-96 5-1-96 8-22-96 9-25-96

240497 243324 253625 266229 286734 298511 311662 320246 331053 352816 372329 383052 393774 398892 400200 400217 417702 432158 437374

ti = c i - 240497

xi = t i - ti_l

0 2827 13128 25732 46237 58014 71165 79749 90556 112319 131832 142555 153277 158395 159703 159720 177205 191661 196877

2827 10301 12604 20505 11777 13151 8584 10807 21763 19513 10723 10722 5118 1308 17 17485 14456 5216

count

d a t e s o f t h e first f o u r e x e c u t i o n s a n d t h e m o s t r e c e n t f o u r (as o f 7 M a r c h 2000). T h e c o n c a v e u p r e l a t i o n s h i p i n F i g u r e 4, w h i c h s h o w s a p l o t o f ti a g a i n s t i, indicates that executions have become more frequent in the US.

A. P. Basu and S. E. Rigdon

48

Bt) 2o _= õ i5 O

5 29 )

50000

i00000

150000

200000

g

N u m b e r o f copies Fig. 3. Failures of copy machine. Table 2 Executions in the US: 1977 present Date

Days since 1/17/79 = ti

Number i

17 January 1977 2 5 M a y 1979 22 October 1979 9 M a r c h 1981

24 February 2000 24 February 2000 1 M a r c h 2000 3 M a r c h 2000

1 2 3

858 1008 1512

614 615 616 617

8438 8438 8444 8446

N(t) 500

õ

500

õ 400

.~

300

200

co io0 2T

2o0o

400o

6oo0

Time in days since January 17, 1977 Fig. 4. Executions in the US since 1977.

sooo

t

49

The Weibull nonhomogeneous Poisson process

EXAMPLE 4. C r o w d e r et al. (1991) gave the times o f unscheduled maintenance on the USS Halfbeak main propulsion engine n u m b e r 3. The data are shown in Table 3. Figure 5 indicates that there was reliability deterioration, although there seems to be a change point rather than a s m o o t h change. 4. Point and interval estimation for the power law proeess

If events are observed until the nth failure, where n is fixed, then the resulting data set is said to be failure truneated. If events are observed until a fixed time, say t, then the data set is said to be time truneated. Since inference procedures for the parameters fl and 0 o f the power law process depend on the type o f truncation, we must consider these two cases separately. 4.1. Failure truncation

I f we observe a system until n events occur, then the joint p d f o f the event times 0 < t~ < t2 < . . . < tn is, assuming a n o n h o m o g e n e o u s Poisson process with intensity function 2(t), f ( t l , r a , . . . , tn) =

)~(ti

exp -

2(x)dx , O < t~ < t2 < . . . < tn .

W h e n the intensity function has the f o r m 2(t) = (Ô/O)(t/O) ô-1, we have

f(q,t2,..,tù)

= (I~-fl0 (~) p-~) e x p [ tù_f0-fl0 ( 0 ) ~ - l d x 1

--ti - 0n/~ \i=1 /

exp -

,

0 < tl < t2 < ... < tn .

The log-likelihood function is therefore Table 3 Failure times tl from USS Halfbeak. Read down the columns 1382 2990 4124 6827 7472 7567 8845 9450 9794 10848 11993

12300 15413 16497 17352 17632 18122 19067 19172 19299 19360 19686

19940 19944 20121 20132 20431 20525 21057 21061 21309 21310 21378

21391 21456 21461 21603 21658 21688 21750 21815 21820 21822 21888

21930 21943 21946 22181 22311 22634 22635 22669 22691 22846 22947

23149 23305 23491 23526 23774 23791 23822 24006 24286 25000 25010

25048 25268 25400 25500 25518

50

A. P. Basu and S. E. Rigdon

BO 8

8O

õ 8

6O

4O ù~ 2O |

•

5000

I

I

I

f

10000

15000

20000

25000

}1

Cumulative operating time Fig. 5. Failure data for USS Halfbeak number 3 propulsion engine.

g(O'fl[t)=nl°gfi-nfil°gO+(fl-1)Zl°gti-

ö

"

i=1

Differentiating g(O, filt) with respect to 0 and fl, and setting the results equal to zero, yields the two equations 0 - ~0 -

0 F

0--Sfi8B--nfi

,

(4)

nlogO+~-~logti-i=] ~õ) log õ

.

(5)

The first equation can be solved for 0 (in terms of fl), yielding tn

0 - n]/~

(6)

Substituting this back into (5) yields 0 = ~ - n log n~ß/~+

~=~

log ti -

This simplifies to n

0 = ~ - n log tù +

log ti i~l

Solving for fl gives n

ß = Ei~=] log(tù/ti)

"

•

\tù)

\tù)

The Weibull nonhomogeneous Poisson process

51

log(t,/t,) = log 1 = 0, the M L E of 3 can be

Since the last term in the sum is written as n

B = ~i'=-]] log(t,/ti)

(7)

Once the M L E of/3 is computed, it can be substituted back into (6), giving the M L E for 0 Ô=

tn

It can be shown (Rigdon and Basu, 2000) that

2n/3 has a chi-square distribution with 2(n - 1) degrees of freedom. This result can be used to prove that n

E(~) = ~ / 3 An unbiased estimator is therefore

B

-

n-2fi n

=

n-2

"-~ löl~ i = 1 og(l"/ti)

(9)

The result that 2n/3/fi has a chi-square distribution with 2 ( n - 1) degrees of freedom can be used to derive a confidence interval for/3 and to determine critical regions for hypothesis tests regarding/3. To obtain a 1 0 0 ( 1 - e)% two-sided confidence interval for /3 we look up X~_«/2(2(n - 1)) and Z2/2(2(n - 1)), the points that leave an area of ~/2 in the left and right tails, respectively, of the chi-square distribution with 2(n - 1) degrees of freedom. Since these values satisfy P

Z2_c~/2(2(n - 1)) < - = - < Z ~ / 2 ( 2 ( n - 1))

= 1 - 0~

/3 the confidence interval for/3 is

Z2_«/2(2(n- 1))/) 2n

Z2/2(2(n- 1))/} 2n

(10)

When/3 = 1 the intensity function reduces to 2(t) = (1/O)(t/O) 1-1 = 1/0, which means that the process is a homogeneous Poisson process. It is often of interest to see whether the confidence interval includes or excludes the value fi = 1. The hypothesis Ho:/3 = 1 can be tested against one of the alternatives H1 : /3 • 1,//1 :/3 < 1, or H1 :/3 > 1. For the two-sided alternative Ho :/3 ¢ 1, the rule is to reject Ho if

A. P. Basu and S. E. Rigdon

52

2nfl/fi < Z~_«/2(2(n- 1))

2 2nfi/fi > Z~/2(2(n1)) .

or

This is equivalent to

B<

2n

or /?>

Z2/2(2(n- 1))

2n

(11)

Z2 ~ / 2 ( 2 ( n - 1))

F o r the one-sided test H0 : fi _> 1 versus//1 :/3 < 1, the rule is to reject H0 if 2n B ~" Z2(2(n - 1)) " F o r testing H0 :/3 _< 1 versus H1 :/3 > 1, that is, a test of whether the system is remaining stable or possibly improving against the alternative that it is deteriorating, the rule is to reject H0 if 2n

B> ;g2_~(2(n-

1))

The statistic W = (Ô/O)~ is a pivotal quantity. Finkelstein (1976) used simulation to obtain percentage points of the distribution of W for various values of n. Tables in Finkelstein's p a p e r can be used to c o m p u t e confidence intervals for 0.

4.2. Time truncation W h e n the data are time truncated, the n u m b e r of failures N is r a n d o m and the time that testing stops is fixed. (The reverse is true for the failure truncated case.) Let t denote the truncation time, and let T1 < T2 < ... < TN < t denote the observed event times before time t. Since the n u m b e r of failures N is r a n d o m , we observe a r a n d o m n u m b e r of r a n d o m variables. Thus, the likelihood taust take into account the r a n d o m n e s s of N as well as the r a n d o m n e s s of the failure times tl < t2 < .. • < tN. (It is possible that we observe no failures before time t, in which case N = 0 and the M L E s of fi and 0 do not exist.) The joint density of (N, T l , . . . ~TN) is

f ( n , tl,..,

th) = f f N ( n ) f ( t l , . . . ,tnln), '

[fN(0),

n > 1, n =

0

.

The r a n d o m variable N has a Poisson distribution with m e a n (t/O) #, so

t/O) ô e x p - ( t / O ) ~ fN(n)

n!

,

n = O, 1 » . . . .

The conditional distribution of T1 < / ' 2 < • • • < TN given N = n is that of n order statistics f r o m the distribution with cdf

G(y) =

{

0,

y
A(y)/A(t), 1,

0 <_y < t, y>t .

53

The Weibull nonhomogeneous Poissonprocess

F o r the power law process

(Y/O)~ C(y)

-

A(t) = J~ 2(x)dx = (t/O) p, so (Y)~,

0<

(,7õ7-

-

y -

The density corresponding to G is then

B(y) fl-1 g(Y)

=

0 ~y(f.

7

-

"

The density o f T1 < T2 < ... < T~ given N = n is therefore

f(t,,

t2,..

, t nln)

= n! I In C'(ti) = n! ii~l B (~) fi_ 1 0 < t l < t 2 < ' " < t n < t .

i=l

The joint density o f N and T1 < T2 < ... <

f(n, tl, t2, .., th) = =-

TN is thus

n[

0~~ i=1

ti

n[ i~l exp

-(t/O) ~

\tl

,

n > 1, _

0 < t l

and

f(O) =exp[-(tlO) B]

n=0

This is nearly identical to the likelihood function from the failure truncated case, a remarkable fact given the completely different approaches to their derivations. The log-likelihood function is thus 11

«(0,filn, t ) =nlog fi-nfilog 0+ (il-l)~--~ log tii=1

(~ä)P.

\w,J/

The M L E s are obtained by differentiating 6(0,/~ln, t) with respect to 0 and fi, setting the results equal to zero, and solving. The results are N

(12)

B -- ENi_l log(t/ti) and Ô=

t N~/~ "

(13)

The M L E s o f fi and 0 are similar to the failure truncated case, the m o s t obvious differences being the replacement o f N for n, and t for t». The sum in the den o m i n a t o r for D goes up to i = N, rather than just up to i = n - 1.

54

A. P. Basu and S. E. Rigdon

It can be shown that if n _> 1 then

Because of the condition n _> 1, the estimator B

N-lB_ N

N-

1

~N=I log(t/ti)

is called the conditionally unbiased estimator for fl. Confidence intervals and tests of hypotheses for/? are based on the result that conditioned on N = n, the quantity 2nf/fi has a chi-square distribution with 2n degrees of freedom. Following the same procedure as the failure truncated case, we find that a 100(1 - «)% confidence interval for fi is given by

X2_«/2(2n)/}

Z2~/2(2n)/~

< fl <

2n

(14)

2n

Since the power law process becomes a homogeneous Poisson process when B = 1, it is often of interest to see whether the confidence interval contains the value fi = 1. The result that 2nf/fl has a chi-square distribution with 2n degrees of freedom can also be used to construct a hypothesis test of Ho:fl=flo

vs

Hl:fißfio

-

The rule is to reject Ho whenever

2nfio/fl

< Z2_«/2

or

2nflo/fi > Z2c~/2

or equivalently

2nfi° or ~ > ~2nil° B<..mZ~/2

(15)

Zl-«/2

Again, it is often useful to test H0 : fi = 1 versus H1 : fi # 1 since the power law process reduces to the homogeneous Poisson process when fi = 1. One-sided tests, that is, tests with a one-sided alternative such as H1 : fl < 1 or H1 : fl > 1 can also be constructed. There appears to be no method of obtaining exact confidence intervals for 0 when the data are time truncated. Bain and Engelhardt (1980) suggested an approximate method that uses the tables of Finkelstein (1976).

4.3. Examples EXAMPLE 1 (Coal mining disasters). This is an example of a failure truncated data set because observation stopped at the time of the last event. Applying the power law process, we find that the MLEs of the parameters fl and 0 are

The Weibull nonhomogeneous Poisson process

B - - N-~189 Z-~i=I

190 log 40549/ti

55

0.670819

and Ô

-

40549 ^ ~ 16.2557 . 1901/~

A value ofD less than one was expected because the coal mining disasters seem to be getting less frequent. The estimate of the mean function A(t) is ^

Ä(t) =

(~) /~~ ~\ ~ ~/ ~0~70~~~~ 0.15404 t067082 .

The smooth curve in Figure 2 is the estimated mean function Ä(t). EXAMPLE 2 (Failures of copy machine). The copier failure data set is also failure truncated. The MLEs are 18 B = ~ ] 7 log

1.08055

196877/ti

and Ô _ 196877 - - ~ 181~

13567.6 .

We expected a value of/~ near one because the system was neither improving nor deteriorating. The smooth curve in Figure 3 is the estimated mean function ^

Ä(t) = (0)/~~ ( l ' 0 8t 0 s s )~

~ 0.00003425 t 1-0a055

EXAMPLE 3 (Executions in the US). The execution data are time truncated because observation stopped on 7 March 2000, four days after the last execution. For this data set, the MLEs are 617 B ~- N-~617 log Z..ai= 1

8450/ti

3.05149

and Ô _ 8450 ~1029.09 . 6171/8 The estimate of the mean function, which is shown as the smooth curve in Figure 4, is

56

A. P. Basu and S. E. Rigdon

( t Ä(t) ~ \ ~ j

,]3.05149 ~ (6.41991 × 10 -1°) t 3-°»149

EXAMPLE 4 (USS Halfbeak). For the failure truncated data from the number 3 main propulsion engine of the USS Halfbeak, we have B=

71 ~ 2.76034 ~~__°1 log 25518/ti

Ô

25518 ~ ~ 5447.3 . 711/fl

and -

The estimated mean function, ( l Ä(t) ~ \ ~ ,

,]2.76034 10 1It276034 ~ 4.8614 x

is shown as the smooth curve in Figure 5. The fit is not particularly good, however. For this data set, a piecewise linear function seems to give a better fit. A piecewise linear mean function corresponds to a flat intensity function that takes a single jump. For the USS Halfbeak data, this change point seems to be about t -- 19,000 h. Before this time, the intensity is roughly 0.0009 failures per hour, whereas after t = 19,000, the intensity is roughly 0.0082 failures per hour. Figure 6 shows the data along with this piecewise linear mean function. This example shows that the power law process is not always a good fit to failure data.

Bt) r~

8o ù-m

0

6O

4O cD

20 co »

5000

I0000

15000

20000

25000

t

Cumulative operating time Fig. 6. A linear change-point model applied to the USS Halfbeak data.

The Weibull nonhomogeneous Poisson process

57

5. Goodness-of-fittests There are two transformations that lead to goodness-of-fit tests for the power law process when the data are failure truncated. The more common is called the ratiopower transformation, Ri : (ti/tn)fl~

i:

1~2~...

(16)

~/~/- 1 ~

where/~ is the unbiased estimator B:n-2/}_

n-2

/,/

n-1

y~~log(tù/t~)

i=1

If

A(t)

is the mean function then

A(ti) Ri--A(tn),

i= l,2, . . , n - 1

are distributed as n - 1 order statistics from a uniform distribution over the interval [0, 1]. Since A(t) = (t/O)~ for the power law process, we have

A(«,) Iti/O? (ti~ ~, \~j

Ri-~(t~)-- ~ -

i= 1,2, . . , n -

1 .

(17)

If fi were known, then any goodness-of-fit test for the uniform distribution could be applied to the Ri's. In most cases, however, the parameter fi is unknown and it taust be estimated from the observed failure data. With the unbiased estimator used in place of il, the Ri from Eq. (17) becomes the Æi defined in (16). The Ri's are distributed, approximately, as order statistics from a uniform distribution on the interval [0, 1]. The quantity 2i- 1 2 ( n - 1) is the expectation of t h e / t h order statistic from a uniform distribution on [0, 1]. The test statistic for the Cramér-von Mises test involves the squared difference between k; and ( 2 i - 1 ) / [ 2 ( n - 1)]; specifically

1 C2 - 12(n-- 1) +

nB( i--1 ",

2i~_1 ~2 Æi

2 ( n - 1)J

(18)

The null hypothesis that the process is a power-law process is rejected for large values of C2. Tables for obtaining critical values can be found in Crow (1990) and Rigdon and Basu (2000). EXAMPLE 4 (USS Halfbeak). We apply the Cramer-von Mises test to the failure data from the USS Halfbeak. As mentioned in the previous section, a better

A. P. Basu and S. E. Rigdon

58

model seems to be the piecewise linear mean function shown in Figure 6. The test statistic is 1

2i-1

n-1

~2

{-Z(Æi 2-~--1)'// ~ 0 . 7 5 8

C2--12(71-1)

.

i=1 ' ,

For c~= 0.05, the critical value obtained form Rigdon and Basu (2000) is 0.390. Since 0.758 > 0.390 we would reject the null hypothesis that the power law process is an adequate model. Since the percentage point corresponding to = 0.001 is 0.512, the P-value for this test is less than 0.001. Rigdon (1989) suggested a second transformation that leads to a goodnessof-fit test for the power law process. The log-ratio transformation is Ui=log(tn/&-i),

i= l,2,

..,n-1

.

If the null hypothesis of the power law process is true, then given T, = &, the random variables Tl, T 2 , . . , Tn_l are distributed as n - 1 order statistics from the distribution with cdf

{

oB/ y < 0, G(y)= th)~, 0_

tn .

It then follows that the random variables:

(~q~V2~~, (~~~~~ ä/

\~/

'"\

to /

are distributed as n - 1 order statistics from the UNIF(0,1) distribution. By L e m m a 30 of Rigdon and Basu (2000, p. 119) we can conclude that the n - 1 r a n d o m variables - log

= -filog--

- log

- f l l o g T1

are distributed as n - 1 order statistics flora the EXP(1) distribution. If we divide each r a n d o m variable by fi, we conclude that U~ < U2 < --- < U~_~ are distributed as n - 1 order statistics from an exponential distribution with mean 1/fl. After this transformation is made, any goodness-of-fit test for the exponential distribution with unknown mean can be used to test the adequacy of the power law process. Rigdon and Basu (2000) discuss the use of Lilliefors' (1969) test for exponentiality, but a number of other tests for exponentiality can be applied.

6. E s t i m a t i o n of the intensity function

It is often of interest to estimate the intensity with which events are occurring when the data collection stops. For the failure truncated case this is at time &, that

The Weibull nonhomogeneous Poisson process

59

is, the time of the nth failure. For the time truncated case this is at time t, the truncation time. The MLEs are ^

B B(tn) = õ

(o)

fl-1

fi tn = -(tn/rtl/~)fi tn

~'

(failure truncated case),

^

B@)/~-l__

~(')=õ

/}t/}-I

ND

(t/~x,~/=T

(timetruncatedcase)

Rigdon and Basu (1988, 1989, 1990) studied classes of estimators of 2(t,) and 2(t) of the form

e~t}

(failure truncated case),

6,

cN/)

(time truncated case) .

t

Consider the failure truncated case first. The bias of the estimator c,B/tn is

B F ( n - 1/fl) [ n& ( n - 1/fl)] Bias - Ö -F-(n) Ln - 2 and the mean squared error (MSE) is MSE=

(fl)2. F(n_-_2/fi)_ \ O J F(n)(n - 2)(n - 3)

x [(n + 1 - 2/fi)(n - 2/fi)(n - 2)(n - 3) - 2c,(n - 2/fi)(n - 3) + c2ùn2] . For a fixed fl (n -

1/fi)(n

- 2)

Cn

makes the bias equal to zero, whereas

(n - 2/fl)(n - 3) C n

minimizes the MSE. Since a good estimator should perform well for both reliability improvement (fi > 1) and deterioration (fl < 1), Rigdon and Basu (1988, 1989, 1990) suggested using fl = 1 in the expressions above, yielding the estimators 2UB=(n--1)(n--2)fi ntn

60

A. P. Basu and S. E. Rigdon

and BMMSE = (n -- 2)(n -- 3)]~ ntù The first estimator is unbiased when fi = 1, and the second has m i n i m u m m e a n squared error within the class {c~ß/tn} when fi = 1. The estimator )~MMSE is a good estimator, in the sense of having a small MSE, for a wide range of il's. F o r the time truncated case, Rigdon and Basu (1990) studied the estimators BUB = (N - 1)(N - 2)fl Nt and ,~MMSE = (N - 2)(N - 3)fl Nt The estimator )~MMSE had a high efficiency relative to the M L E for a wide range of il's. Rigdon and Basu (2000) give tables for calculating confidence intervals for 2(tù) (failure truncated case) or 2(t) (time truncated case). These tables give values col and co2 (failure truncated case) which yield the confidence interval

o~l~(tù) < ,z(to) < ~o2~(tù) and Pl and P2 (time truncated case) which yield the confidence interval

pl'(t) < x(t) < p2~(t) . EXANPLE 2 (Failures of copy machine). There were n = 18 failures for this machine, and the process was failure truncated. We found previously that B ~ 1.08055 and tù---196877. The M L E ,~(190877) and the M M S E estimator 2MMSE(196877) are )~(196877) - 18 x 1.08055 ~ 0.000099 196877 and )~MMSE(196877) = (18 -- 2)(18 -- 3)1.08055 18 X 196877 ~ 0.000073 . Units for these estimates are "failures per copy".

7. Multiple systems modeled with the power law process So far, we have assumed that we observe a single system. In practice, however, we often observe a n u m b e r of repairable systems. F o r example, a rental car c o m p a n y

The Weibull nonhomogeneous Poisson process

61

may keep repair records on each car in its fleet. In a case like this, we observe multiple failures on multiple systems. H o w we handle the data analysis for multiple systems depends on what assumptions we are willing to make regarding the systems. Consider the following: 1. All systems are identical, in the sense that the failure process for each system is the power law process with parameters/~ and 0. (/3 and 0 are the same for all systems.) 2. All systems are different and each is modeled by its own power law process. (The ith system is modeled by a power law process with parameters/3i and 3. All systems have the same growth parameter/3, but possibly different 0's. 4. All systems are different, but they are similar enough to assume that the parameters of the power law process are drawn from some prior distribution.

Oi.)

Under the second assumption, each process is estimated separately using the methods of the previous section. We now discuss separately the other three cases.

7.1. Identical power law processes N o w we assume that all k systems are identical and that they are modeled by a power law process with parameters 0 and /3. This is a strong assumption and should not be made without justification. Let denote the time of the j t h failure on the/th system. Also, let ni denote the number of observed failures for system i, and N = nl + n2 + • • • + nk. We assume that system i is observed until time T~, where T, = ti,~i (failure truncated case) or T~ > (time truncated case). Because the failures on separate systems are independent, the likelihood function is

tij

ti,ni

= /3N o-ô~i=In~ l~j~_ltij ) g(O,/3)

k exp

L(O,/3)

Differentiating the log-likelihood = log with respect to 0 and with respect to/3, setting the results equal to zero, and simplifying, yields

O= ( ~~=l liß ) 1/~ N

-

(19)

and B = O-fl

N ~~=1 T/Blog T,, - ~k=l Ejril log t/j

(20)

The expression for 0 in (19) can be substituted into (20) yielding one nonlinear equation in one unknown. Once the estimate/) is obtained, it can be substituted

62

A. P. Basu and S. E. Rigdon

into (19) to obtain the MLE of 0. For the special case when all k systems are time truncated at the same time T, the MLEs have the closed form expressions N

B

:

Ek=l ~~/1 log(T/tij)

and

ô_

kl/~T N1/~

If we condition on the time of the nth failure (failure truncated case) or on the number of observed failures (time truncated case) we obtain a conditional likelihood function. This conditional likelihood can be used to find the conditional maximum likelihood estimators (CMLEs) for ig and 0 (Crow, 1974)

ß

a4 log(Ti/@

(21)

= 2~=a 2>,

and

õ=(

~~~~)l~~

(22)

where

mi =

{

ni - 1 if data are failure truncated, ni if data are time truncated

and

k M= Zmi i=i

.

Since the estimator ]~ has the property that 2nfl/fi ,--, X2 (2M), we have

\

2M

< ig <

~-~

-] = l - Œ

.

Thus, a 100(1 - Œ)% confidence interval for ig is

XI_Œ/2(2M) ~ 2 (2M) ig Z~/2 2M < ig < 2M Suppose we want to test whether the growth parameters for the k systems are the same; that is, we wish to test the hypothesis

H0: & = ig2 . . . . .

&

The Weibullnonhomogeneous Poissonprocess

63

against the alternative that at least two of the 3's are different. For an arbitrary k, this requires an approximation based on the chi-square distribution, but for the special case of k = 2, an exact procedure is available. When k = 2, we can find independent estimators/)1 and/)2 of/71 and /72 that satisfy 2nl/?1 N )~2(2ml) and

2n2/72 ~ )~2(2m2) ,

B1

/)2

where mi = ni - 1 if system i was failure truncated and mi = ni if system i was time truncated. The statistic

2nlfll/ 2 m l F_31/

~~~~/~m2

therefore has an F distribution with 2ml and 2m2 degrees of freedom. When the null hypothesis that/71 =/?2 is true, we have

A size ~ test of H0 "/71 =/?2 against the two-sided alternative Ha •/71 ¢/?2 rejects H0 whenever or

F < Fl_~/2(2ml,2m2)

F > F~/2(2ml,2m2) .

Similar tests can be constructed for one-sided alternatives. For testing the equality of growth parameters for k systems, that is, a test of

g0 :/71 =/?2 . . . . .

/?k

against the alternative that at least two of the/?'s are different, we taust resort to a likelihood ratio test. In this situation, the likelihood ratio is k

L R = M log/?* - ~

mi log/)i ,

i=1

whereßi is the C M L E for t h e / t h system from Eq. (21) and/?* is M m ~i=1(i//?,)

B*--

k

Using Bartlett's (1937) approximation, we have 2LR~z2(k_I) a

,

A. P. Basu and S. E. Rigdon

64

where

1 (~1 1) a=l-~ 6(k- 1) \i=l"~~ For

an

approximate

size e test,

we reject

Ho:fil

= fi2 . . . . .

fix when

2 ù2LR/a > ~«(k1).

7.2. Power law process for non-identical systems I f we assume that all systems have the same growth p a r a m e t e r 8, but possibly different scale parameters, 0 1 , 0 2 , . . , 0k, then the likelihood function becomes

L(O'fi)= kiHl{[~fl(tij~fi-1]exp[--(Ti/Oi)Bl}\OJiJ =

I~°~~~~)C=~

i~=itiJ) exp -i~=l(Ti/Oi)~] ~ '~-~

~~

Differentiating the log-likelihood function, and setting the results equal to zero gives

50i- Oi Oi \Öi/ =0' i= 1,2, ..,k êg N k k ,~ k /T~\ ô i=1

i=1 j = l

T~

i=1

F r o m the first equation we have Ôi =

T/ n]/~

(23)

and f r o m the second equation we have N

B = 2ki_l E~~~_Ilog(Ti/ti«) Once we c o m p u t e /~, we can substitute i = 1 , 2 , . . , k . The C M L E of fi is

its value into (23) to obtain

Ôi,

M

B = ~ik-1E~Z, log(Ti/tij) The estimator /~ has the p r o p e r t y that 2Mfi/fi~Z2(2M) where M = m l + m2 + --- + mk. This result leads to the 100(1 - a ) % confidence interval

65

The WeibullnonhomogeneousPoissonprocess Z12_~/2(2M)/~

Z;~/2(2M)/~

2M

2M

7.3. Parametric empirical Bayes models for the PLP Engelhardt and Bain (1987) suggested a parametric empirical Bayes (PEB) model based on the parametrization ,l(t) = ,Zfit ~-1,

t > 0 .

PEB procedures are easier to work with if the intensity function is parametrized in this way. Suppose that the k systems have the same growth parameter fi, but the parameters 21,22». • • »•k are drawn from a gamma prior distribution with pdf

ba2b-1 F(a~eXp(-b,t),

p ( 2 l a , b) -

2 > 0 .

The parameters a and b a r e called hyper-parameters and p is called the hyper-prior distribntion. We assume that a and b are fixed but unknown. Let ti denote the vector of failure times for system i and let T =

Itl,h,...,tk]

•

Because all k systems operate independently, the pdf of T given 2 is

D

i=l

i=1

Thus, the pdf of T given a, b, and fi is

p(Tla,b, fi) . . . .

/ 0 ~ / 0 ~ p(T, Äla,b, fi)d21.., d;~~

. . .j[i . °°

=

fo°°P(Tl 2 , fl)P(ÄIa, b)d2 1... d2k

flN B

(ii~l ]~it tij )fl 1 d21.., d/k. i=I

N bak

/ k ,j

\/~-1 k eoo

kl=l j=l

/

=~ ~ [ n n , , , ) / k

n / Än~+a-'exp[-(Tf+b))°i] d)~i i=l a0

,7i \fi-1 k

:flNbak~ \'=(iI~lil~lltij/

IIi=l(Tl. fl/'(n~b)a!+a "

66

A. P. Basu and S. E. Rigdon

The log-likelihood function of (a, b, fi) is therefore

g(a,b, fi) = N l o g fi + aklog b - k l o g F(a) k

+ ~

ni

k

Zlog

k

tij+ Z l o g F(ni+a)- Z(ni+a)log(Tp + b).

i=1 j = l

i=l

i=1

In general, we must use an iterative procedure to approximate the MLEs of a, b, and fi. Engelhardt and Bain (1987) showed that in the special case where all k systems are time truncated at the same time T, we have

g(a, b, fl) = Nlog fl + aklog b - klog r(a) k

+ ~

ni

~

k

log tij + Z 1 o g F(ni + a) - (N + ka)log(T/~ + b) ,

i=1 j : l

i=i

so the MLE of fi is N

(24)

B : 2ki:i 2j~l log(T/tij) and the MLE ä of the parameter a satisfies

k ni

1

~:__~~a+7- 1

--klog(1 +N)

(25)

The solution & to this equation can be approximated by a numerical method. Once ä and fi have been determined, the MLE of b can be found from B--

käT~ N

(26)

Estimates of 2~, 2 2 , . . , 2~ can be obtained once we ger point estimates of a, b, and fi. If we take these estimates as the true values of the parameter s (a, b, and fl), we find that the posterior distribution of 2i is GAM(ni + ä,/; + t~). A point estimate for 2; is therefore

Bi - ä + ni

+ t/B

(27)

8. An extension of the power law process

The minimal repair assumption, which implies the nonhomogeneous Poisson process, may be unreasonable in many situations. Often a system is repaired to a condition better than just before the failure, but not quite as good as new. Lakey and Rigdon (1992) suggested a model called the modulated power law proeess that

The Weibull nonhomogeneous Poisson process

67

contains as special cases the power law process and a gamma renewal process. The model can be described as follows. Suppose that (unobservable) shocks occur according to a power law process with parameters 0 and fl. A failure occurs not at every shock, but rather at every ~cth shock, where ~cis a positive integer. If ~c= 1, then a failure occurs at every shock and the model becomes a power law process. On the other hand, if Ô = 1, then the shock process is a homogeneous Poisson process, and since a failure occurs at every ~cth shock, the times between failure are independent gamma random variables. Black and Rigdon (1996) showed that the joint density of q, t 2 , . . , t,, is

f(tl, t2,..., tù10, ~, ~) = e(0, ~, ~) =

2(ti)[A(ti) - A(ti-1)] ~-1

exp[-A(&)l

O

{r(~)]"

' ,

where 2(t) = (fl/O)(t/O) ~-1 and A(t) = fö2(x)dx = (t/O) p are the intensity function and mean function for the process of shocks, respectively. Since this joint density is a density for every t¢ > 0, it is not necessary to assume that ~c is a positive integer. The MLEs of 0, fi, and ~ccan be obtained by applying an iterative method to solve the likelihood equations. The log-likelihood is

g(O, fl,~c) = - ( ~ ) ~ + n l o g f i - n l o g F(t¢)- nfi~clog 0 n

+ (fl-1)Zlog

n

ti-{-(t¢- 1 ) ~ l o g ( t / B -t/B ') .

i=1

i=1

Differentiating with respect to O,/~, and x gives ao

o O

o ' n

Oft--

~ö) log\ J ö + ~ - m c l o g 0 + Z l o g t i i=1

+ ôg ~~c-

B , ~ lo6 ~~- 4-1 log ~, 1 i=, ~( -t~l '

1) Z_.,

nO(~c)- nÔlog 0-- ~ l o g ( t e -t~_l) , i=1

where O denotes the digamma function O(x) = F'(x)/F(x) and to is defined to be 0. The MLEs of 0, fl, and ~: are the solutions to the non-linear equation

~J

68

A. P. Basu and S. E. Rigdon

T h i s s y s t e m o f e q u a t i o n s c a n be s o l v e d b y a n i t e r a t i v e m e t h o d s u c h as N e w t o n R a p h s o n . B l a c k a n d R i g d o n (1996) suggest strategies f o r o b t a i n i n g a n initial e s t i m a t e . T h e y also discuss a s y m p t o t i c c o n f i d e n c e i n t e r v a l s a n d tests o f h y p o t h e s e s f o r the p a r a m e t e r s o f the m o d u l a t e d p o w e r l a w process.

References Ascher, H. (1968). Evaluation of repairable systems reliability using the 'bad-as-old' concept. IEEE Trans. Reliab. R-17, 103-110. Bain, L. J. and M. Engelhardt (1980). Inferences on the parameters and current system reliability for a time truncated Weibull process. Technometrics 22, 421M26. Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proc. R. Soc. Lond. A 160, 268~82. Black, S. E. and S. E. Rigdon (1996). Statistical inference for a modulated power-law process. J. Qual. Technol. 28, 81-90. Crow, L R. (1974). Reliability analysis for complex systems. In Reliability and Biometry, pp. 379-410 (Eds. F. Proschan and R. J. Serfting). Crow, L. R. (1990). Evaluating the reliability of repairable systems. In 1990 Annual Reliability and Maintainability Symposium, pp. 275-279. Crowder, M. J., A. C. Kimber, R. L. Smith and T. J. Sweeting (1991). Statistical Analysis of Reliability Data. Chapman & Hall, London. Engelhardt, M. and L. J. Bain (1987). Statistical analysis of a compound power law model for repairable systems. IEEE Trans. Reliab. R-36, 39~396. Finkelstein, J. M. (1976). Confidence bounds on the parameters of the Weibull process. Technometrics 18, 115-117. Jarrett, R. G. (1979). A note on the intervals between coal-mining disasters. Biometrika 66, 191 193. Lakey, M. J. and S. E. Rigdon (1992). The modulated power law process. In Transactions of the 46th Annual Quality Congress, pp. 559 563. Lilliefors, H. W. (1969). On the Kolmogorov-Smirnov test for the exponential distribution with mean unknown. J. Amer. Stat. Assoc. 64, 387-389. Maguire, B. A., E. S. Pearson and A. H. A. Wynn (i952). The time intervals between industrial accidents. Biometrika 39, 168-180. Rigdon, S. E. (1989). Testing goodness-of-fit for the power law process. Commun. Stat. A-18, 46654676. Rigdon, S. E. and A. P. Basu (1988). Estimating the intensity function of a Weibull Poisson process at the current time: failure truncated case. J. Stat. Comput. Simul. 30, 17-38. Rigdon, S. E. and A. P. Basu (1989). Mean squared errors of estimators of the intensity function of a nonhomogeneous Poisson process. Statist. Prob. Lett. 8, 445-449. Rigdon, S. E. and A. P. Basu (1990). Estimating the intensity function of a power law process at the current time: time truncated case. Commun. Stat. Comput. Simul. B-19, 1079-1104. Rigdon, S. E. and A. P. Basu (2000). StatisticaI Methods for the Reliability of Repairable Systems. Wiley, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook ofStatistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserve&

"~ ù.1

Bathtub-Shaped Failure Rate Life Distributions

C. D. Lai, M . X i e a n d D. N. P. M u r t h y

Notation and acronym

T f ( t ) , F(t), F(t)

lifetime random variable pdf, cdf, survival function of T failure rate function

h(t) I~

ET

#(t)

mean residual life at t

tl(t)

-S(t)/f(t)

HF 1(t)

T T T transform scaled T T T transform set of burn-in times discrete analog of f a n d h test statistic the empirical cdf

ÓF(t) = HFI(t)/HF1(1) B*

B., hi Tn Fn

Other abbreviations are given in Table l(a) and (b).

1. Introduction

The concept of aging is very important in reliability theory. ' N o aging' means the age of a component has no effect on the distribution of residual lifetime. 'Positive aging' describes the situation where residual lifetime tends to decrease, in some probabilistic sense, with increasing age of component. On the other hand, 'negative aging' has an opposite effect on the residual lifetime. In reliability, we often characterize a lifetime distribution through the following three functions: • Reliability or survival function F(t) = 1 - F(t). • Failure rate function defined by

h(t) - f(t)

(2.1)

e(t)

The failure rate is also often known as the hazard rate 69

70

C. D. Lai, M. Xie and D. N. P. Murthy

• Mean residual life (MRL) is defined by #(t) =

x)dx

t) ,

(2.2)

All these three functions provide probabilistic information on the residual lifetime and hence aging classes may be formed according to the behaviour of the aging effect on a component. h (t)dt may be interpreted as the probability of an item will fail in (t, t + dt] given that it has survived time t. We m a y interpret/~(t) in this way: given an item is o f a g e t, the expected value of the random remaining life is called the M R L at age t. We may loosely classify the failure rates into three groups: • Monotonic failure rates: h(t) is either increasing or decreasing. • Bathtub failure rates: h(t) has a bathtub or U-shape. • Generalized bathtub failure rates: h(t) is a polynomial, or has a roller-coaster shape or some generalization. M a n y notions ofaging have been defined in reliability literature. In Section 2, we will provide a list of aging notions and a chain of implications and the link between failure rate and the MRL. The list is certainly not meant to be exhaustive. Following this we devote our attention to the bathtub failure rate distributions. A review of this class of life distributions was given by Rajarshi and Rajarshi (1988). However, a significant amount of literature on this subject has appeared over the last decade. Thus there is a need for another thorough study. In this chapter, we give a detailed review of m a n y different facets and issues and finally conclude with a brief discussion of some models extending the traditional bathtub failure distributions. The present chapter comprises the following sections: • • • • • • • • • •

Introduction Notions of aging Bathtub-shaped failure rate life distributions Families of bathtub-shaped failure rate distributions Construction techniques for B F R distributions Tests of exponentiality versus bathtub distributions Change points estimations for B F R distributions Mean residual life and bathtub-shaped life distributions Optimal burn-in time for bathtub distributions Models extending the traditional bathtub distributions

2. Notions of aging In recent years, m a n y statistical aging properties have been defined through one of these three functions (i.e., failure rate, survival and mean residual life). The best known are the nine classes listed in Table l(a).

Bathtub-shaped failure rate life distributions

71

They are connected by the chain of implications (see, for example, Deshpande et al., 1986; Kochar and Wiens, 1987) as indicated below IFR

~

DMRL

IFRA ~

~

NBU

~

NBUFR

~

NBUFRA

NBUE

~

HNBUE

~

£

Other known classes are now listed according to alphabetical order in Table l(b). In addition to the one given above, other chains of relationships are also given in Joe and Proschan (1983), for example NBUq=~NBUP-«

for a l l 0 < c ~ < 1 ~ N B U E

Table 1 Abbreviation (a) IFR IFRA DMRL NBU NBUE NBUFR NBUFRA HNBUE £ (b) BFR BMRL-t0 DIMRL DPRL-c¢ DVRL IDMRL NWBUE NBAFR NBUC NBUFR NBU-t0 NBUP-~ SIFR SNBU UBFR

Name of the class

References

Increasing failure rate Increasing failure rate average Decreasing mean residual life New better than used New better than used in expectation New better than used in failure rate New better than used in failure rate average Harmonically new better than used in expectation Class with aging property based on Laplace transform

Barlow and Proschan (1981) Barlow and Proschan (1981) Bryson and Siddiqui (1969) Barlow and Proschan (1981) Barlow and Proschan (1981) Deshpande et al. (1986) Lob (1984)

Bathtub-shaped failure rate Better mean residual life at toa Decreasing initiatty then increasing mean residual life Decreasing 100c~ percentile residual life Decreasing variance of residual life Increasing initially then decreasing mean residual life New worse then better than used in expectation New better than average failure rate New better than used in convex ordering New bettet than used in failure rate New better than used of age to New better than used with respect to 100c~ percentile Stochastically increasing failure rate Stochastically new better than used Upside-down bathtub-shaped failure rate

Barlow and Proschan (1981) Kulasekera and Park (1987) Guess et al. (1986)

Rolski (1975) Klefsjö (1983)

Joe and Proschan (1984) Launer (1984) Guess et al. (1986) Mitra and Basu (1994) Loh (1984) Cao and Wang (1991) Deshpande et al. (1986) Hollander et al. (1986) Joe and Proschan (1984) Singh and Deshpande (1985) Singh and Deshpande (1985) Sweet (1990)

Residual life declines during the time 0 to to, and thereafter is no longer greater than what it was at to.

72

c. D. Lai, M. Xie and D. N. P. Murthy

IFR¢=~DPRL-e

for a l l 0 < e <

DPRL~DPRL-e~NBUP-~

1 forany0
1 .

There are other more chains of relationship but we will not elaborate on them here. All the classes in Table l(a) as well as some others in Table l(b) describe an item whose reliability is getting worse, to a greater or lesser extent as it ages, there are classes describing items whose reliability is improving with age - DFR, DFRA, etc. These are the dual classes of IFR, IFRA, etc. The first group is also described as having positive aging whereas the second as having negative aging. It is also possible to place distributions in order with respect to the extent to which they possess (for instance) the I F R property - see Kochar and Wiens (1987). Both the positive aging and the negative aging share a common characteristic, namely, the residual lifetime tends to decrease (or increase), in some probabilistic sense, with increasing age of the component. In other words, the residual lifetime is monotonic with respect to age. However, in many practical applications, the effect of age is initially beneficial (a burn-in phase where negative aging takes place), but after a certain period, it is age adverse indicating a 'wear-out' phase where aging is positive. This kind of non-monotonic aging phenomenon is offen modelled using life distributions that display bathtub-shaped failure rates. Nearly all the bathtub failure rate distributions have an upside-down bathtub shape MRL. For this reason, the present chapter also includes a discussion on the relationship between BFR and other non-monotonic M R L classes.

3. Bathtub-shaped failure rate life distributions A class of life distributions which has received considerable attention is the class of bathtub-shaped failure rate life distributions. A systematic account of such distributions was given by Rajarshi and Rajarshi (1988). We say that F is bathtub-shaped failure rate (BFR) if its failure rate function decreases at first and then remains constant for a period and finally it increases with time. In other words, the failure rate function has a bathtub shape. This corresponds to the three distinct phases of a component or a system: early life, useful life and wear out as shown in Figure 1. During the early life period, failures tend to be caused by manufacturing defects or birth defects in the case of human beings. Failures in the useful life period can be called chance failures. The wearout region has an increasing failure rate with time because older the unit the more likely it is to fail. The class of lifetime distribution having a bathtub shape failure rate function is very important because the lifetime of electronic, electromechanical, and mechanical products are often modeled with this feature. In survival analysis, the lifetime of human beings exhibits this pattern. Kao (1959), Lieberman (1969),

Bathtub-shapedfailure rate life distributions

h(t)

2

73

I I I I I I I

early life

, I I

useful life

wear out

>t

Fig. 1. A bathtub failure rate. Glaser (1980), and Lawless (1982) give m a n y examples of bathtub shape life distributions. There are critiques of bathtub shaped failure rate distributions, see for example, K r o h n (1969), Talbot (1977) and Sherwin (1997).

3.1. Acronyms for bathtub shaped failure rate life distributions The bathtub shaped failure rate life distributions, offen known simply as bathtub distributions, have a failure rate curve that resembles a bathtub shape. Such failure rate curves are also known as U-shaped or J-shaped curve and it is unknown to us who first coined the phrase " b a t h t u b " for these life distributions. Unlike other class notions of aging, there appears no consistent acronyms or abbreviations. Some of these are: BT, BTD, BFR, BTF, BTFR, B T F R D , BTR, DI, DIB and, etc. It seems to us the preferred one would be B F R which we now adopt.

3.2. Definitions and basic results DEFINITION 1 (Glaser, 1980). Let F be a cdfwith a failure rate function h(t) which is continuous. Then F is B F R if there exists a to such that: (a) h(t) is decreasing for t < to, (b) h(t) is increasing for t > to. i.e., h'(t) < 0 for t < to, h'(to) =- 0 and h'(t) > 0 for t > to. ttere the strict monotonicity is implied. For example, when h(t) is increasing, it is strictly increasing. The bathtub curves given in this definition would probably represent some U-shaped tubs rather bathtubs as there is no interval for which h(t) is a constant. If F is not absolutely continuous, we may define B F R through the conditional reliability function F(x]t)-Æ(t+x)= ,

F(t)

F(t)=l-F(t)>O

.

(3.1)

c. D. Lai, M. Xie and D. N. P. Murthy

74

DEFINITION 2 (Haupt and Scabe, 1997). F is BFR if there exists a to such that • /~(xlt) is increasing in t for 0 _< t < to, 0 _< x _< to - t, • F(xlt ) is decreasing in t for to _< t < oc,x >_ 0. Definition 1 may be modified to allow for a more 'comfortable' bathtub shape shown in Figure 1. DEFINITION 3 (Mitra and Basu, 1995). A life distribution F which is absolutely continuous and having support [0, oc) is said to be a BFR distribution if there exists a to _> 0 such that h(t) is non-increasing for [0, to) and non-decreasing on

[to, oo). In this definition, the 'flat' part of the bathtub is allowed though not explicitly. to is refereed to as a change point of the distribution F by the above-named authors. A more explicit definition that gives rise to curves having definite 'bathtub shapes' is as follows. DEFINITION 4 (Mi, 1995). A distribution F is a bathtub shape life distribution if there exists 0 _< tl _< t2 < oc such that: (a) h(t) is strictly increasing if 0 < t < tl; (b) h(t) is a constant if tl _< t _< t2; and (c) strictly increasing if t _> t2. Several comments are in order: 1. In Definition 4, Mi called the points tl and t2 as the change points of h(t). If tl = t2 = 0 , then a BFR becomes an IFR; and if h = t2 ---+oo, then h(t) is strictly decreasing so becoming a DFR. In general, if q = t2, then the interval for which h(t) is a constant degenerates to a single point. In other words, the strict monotonic failure rate distributions I F R and D F R may be treated as the special cases of BFR in this definitions. Park (1985) also used the same definition. 2. In Definition 4, at most two change points are allowed. In other words, the points in the interval (tl, t2) are not change points according to Mi (1995) but would have been called the change points according to Mitra and Basu (1995). 3. Definition 4 may be rewritten as

h(t) =

{

hl(t) 2

for t_< h, for tl _< t _< t2,

h2(t)

for t _> t2 ,

(3.2)

where hl (t) is strictly decreasing in [0, t] and h2(t) is strictly increasing for t _> t2. We do not know of many parametric distributions that possess this property. However, Jiang and Murthy (1997c) found a distribution while studying sectional models involving three Weibull distributions. More details will be given in Section 4.1.

Bathtub-shaped failure rate life distributions

75

4. We may differentiate the types of bathtub failure rates based on the asymptotic nature of h(t) as t approaches 0 and infinity, h(t) may be finite or infinite at these asymptotes. Following Mitra and Basu (1996a), we let {BFRs} denote the family of the bathtub distributions covered by Definition 1 with the subscript s standing for "strict". We use {BFR} to denote the family of bathtub distributions covered by Definitions 3 and 4. Clearly, {BFR} = {BFRs} U {IFR} U {DFR}. Another definition of a bathtub shaped failure rate distribution may be defined through logff(t). DEFINITION 5 (Deshpande and Suresh, 1990). A life distribution F having support on [0, oc) is said to be a bathtub shaped failure rate distribution if there exists a point to such that - log/~(t) is concave in [0, to) and convex in It0, oc). It is to be noted that the above definition of a BFR distribution is quite general and extends the idea of distributions possessing a bathtub shaped failure rate to situations where the failure rate itself does not exist. In studying the property of B F R class, in particular its relationship with other classes, one has to be aware which definition of BFR is used.

3.3. Some basic properties Mitra and Basu (1996a) present some basic properties concerning the bound for the survival functions and moments of a BFR distribution. Closure properties of the BFR class under the formation of coherent systems, convolutions and mixtures are also dealt with. • Su~pose F is BFR, then Æ(t) <_ G(t) where G is exponential with mean {h(t0)}- . Here to is a change point at which h is minimum. • ET ~ < r-~,k - h(to)

.

> O.

• A life distributlon F in (BFR} with mean equal to e:rk - r ( ~ + 1)

(h(t0)} k is necessarily an exponential. • Convolution of distributions in {BFRs} is not necessarily in {BFRs}. In fact, even the {BFR} class is not closed under convolution. This is seen in the following example: B(t)=½(e -t+e-t/2),

t>0;

G ( t ) = e -¢ .

Then the failure rate function of the convolution H = F , G is given by 0.5, h(4) mm 0.5533 and h(t) ----+ 0.5 - - (t_l)e-t+e te-t+2e t/~2t/2. , accordingly, h(0) mm 0, h(2) Z as t --, oc which does not have a bathtub shape. • The mixture of BFR distributions need not be BFR.

h(t)

C. D. Lai, M, Xie and D. N. P. Murthy

76

• Suppose we have a competing risk model: Æ(t) =/~1 (t)f'2(t) where the lifetime of each component is BFR with a common turning point to. Then the lifetime of the system again has a BFR distribution with to as one of its turning points. (Note: the turning point here is defined as that given in Definition 3. Also a competing risk model is simply a series system). • A parallel system of two independent components from {BFR} need not be BFR. 4. Families of bathtub-shaped failure rate distributions

Many parametric families of bathtub-shaped life distributions have been constructed over the last two decades. Ideally, we should classify them into groups or strata according to some comlnon characteristics. However, this exercise seems untenable. Instead, we summarize them into two categories: (a) lifetime distributions that have explicit expressions for failure rates and (b) distributions whose failure rate functions are unwieldy or unknown. For the latter, we give only either the probability density function or the distribution function, whichever is more convenient. Behaviour of h(t) at the origin or infinity is given whenever is possible. For Sections 4.1 and 4.1.1, we endeavour to list the failure rates in the increasing order of sophistication.

4.1. Bathtub distributions with expIicit failure rate functions Quadratic and generalization. • Bain (1974, 1978), Gore et al. (1986) considered quadratic failure rate:

h(t) = o:+flt+Tt2;~ > 0, fl < 0,7 > 0 , which has a bathtub shape, h(0) = c~, h(t) ~ oc as t --+ ec. • It is easy to verify that h(t)= exp{h(t)} has a bathtub shape if h(t) has a bathtub shape. For example, BT 9 of Rajarshi and Rajarshi (1988) listed h(t) = 1 9 exp{c~ +/~t + 7t2}, c~,7 -> O, 0 > fl _> -2(c~7) /- as havlng a bathtub shape. • In fact, any increasing function of a bathtub failure rate in itself has a bathtub shape failure rate.

A competing risk model. k(t)=l+fl~t+?~t~-I ,

c~,fi,? > 0; c ~ > 2 .

(Murthy et al. 1973). h(0) - cqh(t) ~ oc as t --+ oc. • This may be considered as a competing risk model involving a Lomax distribution (the Pareto distribution of the second kind) and a Weibull distribution. • The case when c~ = 2 is considered in Hjorth (1980): c~ ~ß

h(t) = 1 +/~---5÷ 27t,

A flexible family.

7 _< 5 -

77

Bathtub-shaped faiIure rate life distributions

• Gaver and Acar (1979) have proposed a model that has a bathtub-shaped failure rate given by h(t) = ~ + 9(t) + k(t) where 9(t) > 0 is a decreasing function t such that limt-~,o 9(t) --+ O, k(t) is a decreasing function of t such that k(0) = 0 and limt~~ k(t) --+ ec and 2 is any real number such that h(t) > O. • For example, 0 h(t)=2+t~~+etP

,

e, 0_>0; t , q ) , p > O ,

which is an extension of Murthy et al. (1973) and studied by Jaisingh et al. (1987). If both 9(t) and k(t) are failure rate functions, then this model is simply a competing model involving three distributions. • Other special cases are: Canfield and Borgman (1975): h(t) = 01cqt «~-~ + 02 + 03~3/«3-1, 0~3 > 2, ~1 < 1 h(t) ---+oc as t ~ 0 or oc. Hjorth (1980): h ( t ) = c ~ / ( l + f i t ) + T t ; 0 < 7 _ < « / ? , h(0)=c~, h(t)--+oc, as t ---+oo.

Makeham's curve. • This is listed as No. 15 in Section 5 of Rajarshi and Rajarshi (1988): h(t)=3exp(#t)+e/?(l+fit) -j,

#6<e/?2,

t>0

,

h(O) = 6 + «/?, h(t) ---, oc as t ---+oc . Reliability models involving two or more Weibull distributions. Competing risk model. • Xie and Lai (1995) and Jiang and Murthy (1997c) considered a competing risk model involving two Weibull distributions resulting in a failure rate:

h(t)=/?1tll~1t/h-l+/?2q2132t#2

1 t]i>0,

/?i>0,

i=1,2.

h(t) has a bathtub shape when/71 < 1 and/?2 > 1. The turning point to given by

tO = ~/?1(1 --/?1)(t/2)/~2~I/(Ô2-Pl) (&(1--~ß2)~J

"

Also h(0) = h(oc) = oc. Sectional model with two WeibuIl distributions. • Murthy and Jiang (1997) consider two sectional models involving two Weibull distributions having failure rate functions given by

{ (/71/~1)(t/~1)~1-1, o < t < So, satisfying two conditions: to = [/?@/r/2B2]1/(/~~-&), 7 = ( 1 - / ? ) t o so that h(t) is continuous at to.

C. D. Lai, M. Xie and D. N. P. Murthy

78

• The second model is the same as above except that y = 0. Now the shift parameter y has no influence on the shape type of the failure rate. Thus the two models are essentially the same. For//1 1. • Jiang and Murthy (1997b) extended their results to give two sectional models involving three Weibull distributions. Four types of bathtub shapes are possible for both models.

Exponen tial power. • Smith and Bain (1975, 1976), Dhillon (1981), Paranjpe et al. (1985), Paranjpe and Rarjasi (1986) and Leemis (1986) studied h(t) = b//(//t)b-le (~t)~, / / > 0, b = 1/2 . h(t) --+ oc when t --+ 0 o r t --+ e « F o r / / = 1, the middle part of the bath is quite flat. B(t) has a rather simple expression, i.e., F(t) = e x p { - ( e (~t)b - 1)}. Double exponential power. h(t) =//c~t ~-1 exp(//t ~) exp[exp(//t~) - 1],c~ < 1 (Paranjpe et al. 1985; Paranjpe et al. 1986). Clearly, h ( t ) ~ t -+ ec. The survival function Æ(t) is quite straightforward,

ec as t - + 0 or

4.1.1. Finite fange distribution families Power-function distribution. • Mukherjee and Islam (1983) (see also Lai and Mukherjee, 1986) proposed a finite fange distribution with failure rate: h ( t ) - ~ptp-1 Ztp ,

O<_t
p
h(t) ~ oo when t ---+0 o r t --+ 0 . Beta failure rate distribution. • Moore and Lai (1994) proposed another finite fange distribution having h(t) = c(t +p)a-1 (q t)ó-1; 0 < a < 1, b < - 1 , 0 < t < q, c > 0,p > 0. _

h(O) = c p a - l q b-l,

h(t)--+oo

ast---+q .

Integrated beta failure rate distribution. • Lai et al. (1998) considered a lifetime distribution with which the cumulative failure function is a beta function, but the failure rate is given by h(t)=t ~ l(a-t)b-l{a-(a+b)t},

0
a>0,

b<0

It is obvious that h(t) ---+oo as t ~ 0 or 1. A minor extension yields H(t) =

h(x)dx

/0 ~

= c t ~ ( 1 - d t ) b,

b<0,

0
O
.

.

79

Bathtub-shaped failure rate life distributions

Govindarajula distribution. • Govindarajula

(1977)

considered

a

family

of

distributions

having

h ( t ) = [ 2 ( 2 + l ) t ~ ° - l ( 1 - t ) 2 ] -1, 2>1, 0
h(t) =

a, a > 0; Æ real. a a [ l - 2(t/a)'/«] '

• The fange of this generalized Weibull r a n d o m variable is (0, oo) for 2 < 0 and (0, a/2 ~) for 2 > 0. h(t) has a b a t h t u b shape for e > 1 and 2 > 0.

h(t) --+ oo as t--+ 0

or a / U

.

The distribution function is

~~~/_-I, 0 ~~t~«~~~~)l~~ 1 • Chen (2000) considered a generalized Weibull distribution F(t) = 1 - e J~(l-J) where 2, fi > 0 with h(t) = 2Bt~-le t~, t > O.

Haupt and Schabe distribution. • H a u p t and Schabe (1992, 1994) p r o p o s e d a b a t h t u b distribution having

h(t)

=

{

l+2fl

2T~/B2+(l+2fi)t/T(l+fl-~/B2+(l+2fl)t/T) '

0 < t < T, - 1 / 3 < fl < 1,

0,

otherwise

with

h(O) -

1+2/~

2Tfl '

h(t) -+ oc as t-+ oc .

{~,

T h e cdf is given by

F(t)

_/~ + x//~2+

t>T, (1 +

O,

2p)t/T,

0
• Schabe (1994b) showed that the m e a n time between unscheduled removals has an upside d o w n b a t h t u b shape.

4.2. Bathtub distributions with unknown or unwieldy failure rates Exponentiated Weibull family. • M u d h o l k a r and Srivastava (1993) considered

F(t)=[1-exp(-(t/a)«)]

°,

O<_t < oo

C. D. Lai, M. Xie and D. N. P. Murthy

80

and the quantile function

Q(u)=F-l(u)=«

-log(1-u

U°)

0<

< 1 .

Gamma mixture family. • Glaser (1980), Kunitz and P a m m e (1993), and Pamme and Kunitz (1993) all considered: f ( t ) = Pf1 (t) + qf2(t), p + q = 1, t > O, B(t) = ~~Jt~J-le ~t/F(Tj) , for either 71 > 2, 72 = 1 or 71 > 1,72 < 1. Generalized gamma. • Glaser (1980), M c D o n a l d and Richards (1987a, b) and Richards and McD o n a l d (1987) considered a generalized gamma distribution having probability density: f ( t ) = «t ~v-l exp[-(t/fi)'l],

Œ> O, i l > O ,

v > 1, uc~ < 1 .

Generalized exponential distributions. • Cubic exponential family. Glaser (1980), Cobb et al. (1983) considered: f(t) =«expl-st-

flt 2 - 7 t 3 ] ,

«<~

.

B F R distributions arise if: (i) « and/~ real, 7 > 0, or (ii) ~ real, fl > 0, 7 = 0 or (iii) ~ > 0,/~ = 7 = 0. • Exponential family of densities. Glaser (1980), Cobb (1981), Cobb et al. (1983) and P h a m - G i a (1994) all considered

f(t) = ùexp[-~t-

fit 2 + 71nt] = ùt~ e x p [ - o ~ t - fit2j,

7< 0 .

BFR distributions occur for: (i) 7 real, fi > 0,7 > - 1 or (ii) > 0, fl = 0,7 > - 1 . Piecewise exponential family. • Kunitz (1989) considers a family o f extremal class of distributions based on the T T T transform OF that changes once from convex to concave in (0, 1) and that possess the K o l m o g o r o v distance max{A,e} represented by the following graph (Figure 2). • The corresponding life distributions are part of the family of piecewise exponential distributions.

4.3. A mistaken identity." The mixed Weibull family K a o (1959) considered a mixture o f t w o Weibull distributions in the following form:

F(t) = pF~ (t) + qF2(t),p + q = 1, Fl(t)= 1-exp{-t~l/Œ1} , Q(t)

1

-exp{-(,-

~1 > 0 ,

~2)~2/~2~,

0 < f i l < 1, '

> ~2, ~2 > o, e2 > 1

81

Bathtub-shapedfailure rate life distributions

1.0

/° OF(t) ///

/'/

/

!

i/ 1-v

0.(

El.0

Fig. 2. TTT-transform of a piecewiseexponentialdistribution. He claimed that this family has a bathtub-shaped but several authors contradicted his result, for example, Glaser (1980), Pamme and Kunitz (1993), and Jiang and Murthy (1998) all showed that F defined above can never be a BFR distribution. Unfortunately, several other books cited the result of Kao (1959). 4.4. S o m e comments on the bathtub shapes

Almost all the parametric distributions given above do not have an interval for which h(t) is a constant. However, many of these would have a nearly flat middle part if the parameters are chosen properly. It appears that only a sectional model can achieve a constant to describe the 'useful' life sector. It is pointed out by Haupt and Schabe (1997) that for many of these BFR distributions, estimations of parameters often resort to extensive iterative procedures. Furthermore, the main characteristics of these distributions such as moments and quantiles are not available in closed forms. For these reasons, we have not discussed the subject of estimations for BFR distributions in this review.

5. Construction techniques for BFR distributions There are many ways of constructing BFR distributions. The following may not be an exhaustive list.

82

C. D. Lai, M. Xie and D. N. P. Murthy

Glaser's technique.

• Glaser (1980) chose a function ~/(t) which fulfils the following criteria: (a) ~/(t) -- - f ' ( t ) / f ( t ) and f ( t ) is a density function; (b) there exists a to > 0 such that ~/'(t) < 0 for all t c (0, to), ~/'(t0) = 0 and ~/'(t) > 0 for all t > to; (c) there exists a Y0 > 0 such that Jyõ [f(y)/f(yo)]q(yo)dy - 1 = O. Convex function.

• W i t h the definition of a BFR distribution, we can define a BFR by choosing a positive convex function h(t) over (0, oe) such that f ö h(t)dt = ce (Rajarshi and Rajarshi, 1988). Function o f random variables.

• This procedure is due to Griffith (1982). • Let X have an exponential distribution with mean 1, and let ~p(-) be a strictly increasing differentiable (except perhaps at ml and rn2) function on I0, ce). Further, if ~(.) is convex on [0, tal), linear on (ml, m2) and concave on (m2, ce) (where possibly ml -- m2), then ~p(X) has a BFR. Reliability and stochastic mechanisms.

• Series system (competing risk model). Suppose we have a series system of two independent components. It is well known that the failure rate of the system is simply the sum of the two component failure rates. If one of them has an IFR distribution and the other has a D F R distribution, then the system lifetime may have a bathtub-shaped failure rate. Models obtained by Murthy et al. (1973), Canfield and Borgman (1975), Gaver and Acar (1979) are of this type. • Stochastic failure models. Consider the life distribution of a device subject to a sequence of shocks occurring randomly in time according to a homogeneous Poisson process, under appropriate conditions on the probability of surviving a given number of shocks, Mitra and Basu (1996b) have shown that the lifetime of the device has a BFR. I D M R L classes.

Also known as upside-down bathtub mean residual life distributions. A bathtub shaped failure rate distribution may arise via an I D M R L model. • Stochastic differential equation models and population abundance distributions. • Mixture. gamma mixture, Weibull mixture (different location parameters), mixture of three distributions (Krohn, 1969). Sectional models.

• Piecewise linear in three regions (Shooman, 1969), Colvert and Boardman (1976), Jaisingh et al. (1987). For example,

{

sl - qit,

h(t) =

0 < t _< tl,

~2, g 2 -}-

tl < t < t2,

q2(t

--

t2),

t > t2 ,

Bathtub-shaped failure rate life distributions

83

subject to the conditions tl = (el - e 2 ) / t / l , el > e2 > 0, t/1 , g/2 > 0. Other sectional models that give rise to BFR distributions are Jiang and Murthy (1997b), Murthy and Jiang (1997). Polynomial o f finite order. • Jaisingh et al. (1987) and Shooman (1968) suggested a polynomial of finiteorder failure model: h(t) = ao + alt + • •. + ant n. As the constants ai, i = 0 , . . . , n may be positive or negative, bathtub shapes can be achieved. T T T transformation. • See Kunitz (1989) and Haupt and Schabe (1997) for details. Truncation o f D F R distribution. • Schabe (1994) has constructed bathtub shaped failure rate distributions from decreasing failure rate distributions by truncations.

6. Tests of exponentiality versus bathtub distribution 6.1. Test based on total time on test ( T T T ) transform

Bergman (1979) suggested a test based on the total time on test (TTTT) transform for testing exponentiality against bathtub shaped distributions. Aarset (1985) derived the exact distribution of this test under the null hypothesis of exponentiality. Aarset (1987) also derived another test that is based on T T T plot which is equivalent to the well-known Cramer-von-Mises test statistic. A Monte Carlo power comparison of the two tests are performed by Kunitz 1989). See also Xie (1988). More recently, see Haupt and Schabe (1997). Formally, the T T T transform is defined as: HF1(t) =

B

F-l(t)

F(u)du, and

(6.1)

d0

Scaled T T T transform: OF(t) = H ; 1(t)/HF 1(1) .

(6.2)

Now d 1 1 ~ttH~ (t) -- h(F_l(t) )

Since F is increasing in t, it follows that F -1 (t) is also increasing. We note that as 1 OF(t) = h(F I(t))HFI(1) '

it is obvious that OF is concave for F being IFR and convex for F being DFR. For a bathtub shape distribution, we expect the T T T transform curve to have an s-shape, that is we anticipate the T T T plot to lie below the 45°-line in its leftmost part and above the line in its rightmost part (Aarset, 1987; Kunitz, 1989). The curves in Figure 3 summarize the situations.

84

C. D. Lai, M. Xie and D. N. P. Murthy

Fig. 3. Plot of ~ße-against time t.

As pointed out by Haupt and Schabe (1997), such an s-shaped curve need not cross the diagonal line. An example can be given by the following function: for 0 < u < ¼, { 1~(12u HFI(U) =

~6(28u

10)

for¼_<ù_<½, forl_<ù<~,

~6(20u

4)

for3
-

2)

1 .

The above authors also proved the following theorem which formally verifies the observations made earlier. THEOREM 1. F is BFR if and only if the scaled TTT transform ~bF(U) has one reflection point u0 such that 0 < u0 < 1 and 4)F(U) is convex on [0, u0] and concave on [u0, 1]. T e s t statistics. Apart from tests based on TTT transforms, there are other methods available for testing constant failure rate against bathtub shape. For example, Park (1988) used the following statistic:

• Ler X(1) < X(2).-- < X(n) denote the order statistics corresponding to the random sample from F and ler G be the empirical cdf. The test statistic is

Bathtub-shaped failure rate life distributions

k-1 Tn = Z n-2{4i- (2 i=1

-

85

p)n]X(i)

n

+ Z n - 2 [ - - 4 i + (3 -[-p)n]X(i ) - En l(p)(1 - p ) ( 1 - 2p) i=k

where k - [np]+, F~-1 (p) = X(k). • Note: [x]+ is the smallest integer greater than or equal to x, p = F(to) = the proportion of population that dies at or before the change point to. Instead of using Th, Park (1988) uses the following scale invariant test T* =

rùlxn, xn = ~i:~ x,.In. • The test rejects H0 in favour of H1 : F being BFR at the approximate « level if

B T 2 / a o >_z~, where z~ is the upper Œ-quantile of the standard normal distribution. We note in passing that a0 = 1/3 - p +p2. • Although our main aim in this section is to study the testing procedures for {BFR}, we note that there are several test procedures for the {IDMRL}. Guess et al. (1986) developed two testing procedures for H0 : F is exponential versus HI: F is IDMRL; one when the change point is known and the other with unknown change point but known p, the proportion of the population that die before the change point to. For other test statistics, see Lim and Park (1995a, b, 1998). • Xie (1987) considered testing constant failure rate against some partially non-monotone alternatives. Details are omitted here.

6.2. Other tests Apart from the TTT-plot mentioned earlier, there are other graphical tools for identifying bathtub-shaped life distributions. For example, Kunitz and Pamme (1991, 1993) proposed a plotting technique named by the "aging plot" which is a generalization of the TTT-plot. Kunitz and Pamme (1991) consider another plotting technique which is well known in reliability literature, namely, the socalled Weibull probability plot (WPP). These authors claimed that "Convex shapes in Weibull probability plot might indicate an underlying distribution with a trend change from DFR- to IFR-behaviour (bathtub-shaped hazard rates)". Pamme and Kunitz (1993) discussed the combined application of graphical tools and parametric estimation to identify a bathtub shaped failure distribution.

7. Change point estimations for BFR distributions Estimation of change points for BFR (and others) is relevant particularly in the context of maintenance policies, since, as is natural, one would hardly think of replacing a component of having such a life distribution before the 'threshold' unknown age of to is achieved. Kulasekera and Lal Saxena (1991) have considered this problem. They first constructed kernel density estimators and empirical cdf to estimate f and F,

86

C. D. Lai, M. Xie and D. N. P. Murthy

respectively. These are then used to estimate h and its change point. Asymptotic properties of the change point estimators are also considered. The estimators proposed by them seem complicated. Mitra and Basu (1995) considered the problem of estimating change points in various non-monotonic aging models. In the case of BFR distributions with a unique change point to, they proposed the following procedure: • Let B be an upper bound for the unknown change point, i.e., to _< B < oe,

F(B) < 1 • Let fn be a continuous kernel estimate o f f . We then estimate the failure rate function by hn(t)

-

fn(t) _

en(t)

'

where Bù(t) = 1 - n / ( n + 1) f0 t A ( y ) d y

The factor n/(n + 1) is introduced in the definition of Æn simply to ensure that hn(t) is well defined, as f ö f~(x)cbc _< 1. It can be deduced that Ihn(t) - h(t)] -+ 0 a.s. as /'t --+ OC.

Define: An = {0 < t < B : hn(t) is minimum}. The required estimator is now given by ton ~ infAn ~ min An. Suresh (1992) also obtained two estimates of the change point; one by using the definition of BFR distribution, another by a characterization of BFR distribution in terms of T T T transform.

8. Applieations Failure times of jet engine starters. • Kamins (1962) synopsized a very large set of failure times of jet engines starters in the form of histogram representation of hazard rate Barlow and Proschan (1981, p. 55) used this histogram as an example of real data which can be adequately modeled by a bathtub failure rate. • Siddiqui and Kumar (1991) used the finite fange model of Mukherjee and Islam (1983) to fit the failure data of V600 indicator tubes used in aircraft radar sets collected by Davis (1952). Car failures. • Xie and Lai (1995) used an additive Weibull model to analyze an actual set of car failure times data collected during unit test (name of the brand is suppressed for the sake of confidentiality). The failure rate has a bathtub shape. Bus motor failures. • Davis (1952) obtained a large number of datasets on bus motor failures. Two of these exhibited bathtub shaped failure rates. Bain (1974) observed that the

Bathtub-shaped failure rate life distributions

87

quadratic failure model fits well to one of these whereas Smith and Bain (1975) found the exponential power model fits well with the other set.

Electronic tubes failures. • Kao (1959) used a mixed Weibull distribution to fit the actual life testing data gathered by a small-scaled life test of some 800 6AQ5A's conducted at Cornell University.

Electricity generators' failures. • Failure data of 500 M W generators were collected over 6-year period and Dhillon (1981) found the exponential power model fits well with this dataset.

Biological and ecological appIications. Birds. Paranjpe and Rajarshi (1986) fit the exponential model and the double exponential power model to survival data of birds in Deevey (1947) and Pinder et al. (1978).

Deer. Gore et al. (1986) used the quadratic failure rate model to study the decomposition rates of heaps of pellet of Axis axis (a deer species) in an animal reserve.

Halley's mortality life data. • Halley obtained the survival data of 1000 children born in the city of Breslau in Germany (now Wroclaw in Poland). The failure rate has a bathtub shape. Jaisingh et al. (1987) and Moore and Lai (1994) found some models that would fit the dataset well.

Load-capacity (stress-strength) interference. • Lewis and Chen (1994) showed that the infant mortality, constant failure rate (Poisson failures), and aging are associated with capacity variability, load variability and capacity deterioration, respectively. Bathtub-shaped failure rate curves are obtained when all three failure types are present.

Non-Hodgkin's L ymphoma survival. • Alidrisi et al. (1991) obtained survival data of 989 patients treated for nonHodgkin's Lymphoma from the King Faisal Specialist Hospital and Research Center in Riyadh, Saudi Arabia. The failure rate function has a clear bathtub shape.

Preventive maintenance schemes. • A preventive scheine with periodic checkup has been found to be an important device to avoid catastrophic system failure. Of late, the same scheme has been found to be useful in the clinical follow-up studies of patients suffering from chronic diseases such as hypertension, diabetes, cancers, etc. • Gross and Clark (1975) have found that the bathtub failure rates are highly applicable for preventive maintenance schemes. • Biswas and Abid (1991) studied the optimal time for checkup in a maintenance scheme under bathtub and Weibull type failure rates.

Centrifugal water pumps. • Pamme and Kunitz (1993) found the mixed gamma model (which has a bathtub shaped failure) gives a good fit to the Centrifugal Water Pumps data.

88

C. D. Lai, M. Xie and D. N. P. Murthy

9. Mean residual life and bathtub shaped life distributions 9.1. M e a n residual life

Recall, the mean residual life is defined as #(t) = E { X - tIX > t} =

F(x)dx

F(t) .

(9.1)

Although numerous studies have been conducted on the MRL distributions and its applications, few of them involve BFR. It is weil known that the class of IFR is contained in the class of decreasing mean residual lifetime (DMRL). A DMRL distribution need not be an IFR. One may conjecture that BFR may be related to a class of life distributions whose mean residual life #(t) is increasing and then decreasing (that is, the mean residual life has an upside down bathtub shape). Indeed this relationship has been observed empirically as pointed by Rajarshi and Rajarshi (1988) who said in their review article "It can be observed from the life-tables of human and animal populations that the shape of the empirical MRL function is upside-down bathtub". DEFINITION 6. A life distribution having an upside-down bathtub MRL is often known as IDMRL. Again, care needs to be taken as some authors, e.g, Mi (1995) included the flat part in his definition of a bathtub or upside-down bathtub shape whereas others, like Gupta and Akman (1995a, b), do not allow a flat part to be included in their definitions. Let {IDMRLs} be the family of distributions having upside-down bathtub MRL but excluding DMRL and IMRL. Let us also define: {IDMRL} = {IDMRLs} tO{IMRL} tO {DMRL}. In other words, {IDMRLs} is a strict {IDMRL} which does not contain {IMRL} or {DMRL}. Guess and Proschan (1988) have postulated two possible situations whereby IDMRL model may be applicable: (i) Length of time employees stay with certain companies: An employee with a company for four years has more time and career invested in the company than an employee of only two months. The MRL of the four-year employee is likely to be longer than the MRL of the two-month employee. After this initial IMRL (this is called 'inertia' by social scientists), the process of aging and retirement yield DMRL period. (ii) Life length of human: High infant mortality explains the initial IMRL. Deterioration and aging explains the DMRL stage. Muth (1977) described the relationship between h(t) and #(t) h(t) = [#'(t) + 1]/#(t) indicating that #'(t) _> -1.

(9.2)

89

Bathtub-shaped failure rate life distributions

Mi (1995) has proved the following theorem which generalizes the result of Park (1985). THEOREM 2. If the failure rate function h(t) has a bathtub shape, then the associated M R L has an upside-down bathtub shape. That is {BFR} c {IDMRL}. The converse is not necessarily true as demonstrated by the following example given by Mi (1995). Example: #(t)=

{

t2+l 2t 4 e x p ( - ¼ ( t - 2))

if0 2 .

It is easy to verify that/~(t) has an upside-down mean residual life. It follows from the above equation that:

{ (2t-I- 1)/(/2 -]- l) h(t)

il0

3/(20

1/4exp(1/4(t-

2) - 1))

< t < 1, if 1 < t < 2, t> 2 ,

which does not have a bathtub shape. Theorem 1 above indicates that {BFR} C {IDMRL} and the inclusion is in a strict sense. However, the classes {BFRs} defined according to Definition 1 may not be included in {IDMRLs} i.e., {BFRs} g {IDMRLs}. The following example given by Muth (1980) will demonstrate the inclusion is invalid: 1

#(t)-l+2.3t

2'

t_>0 ,

which is a decreasing function. It can be shown that

h(t) = (1 + 2.3t2) 2 - 4.6t 1 + 2.3t 2

,

t> 0

has bathtub shape. However, the inclusion is possible when an additional condition is imposed upon the BFR class. THEOREM 3 (Gupta and Akam, 1995b). I f F c {BFRs} and that h(0) > 1/#, then F c {IDMRLs}. That is, {BFRs} - {h(0) < 1/#} C {IDMRLs}. (Note: {h(0) < 1//~} = {all the failure rate functionsthat have h(0) < 1//@)

9.2. Bathtub-shaped failure rate and decreasing percentile residual life function The "c~-percentile residual life function" (c~-percentile RLF) was first defined by Haines and Singpurwalla (1974). Joe and Proschan (1984) show that this function may be expressed as q~,~(t) = F 1(1 - (1 - ~)ff(t)) .

(9.3)

90

C. D. Lai, M. Xie and D. N. P. Murthy

A distribution is DFRL-~, if and only if for some c~, 0 < c~ < 1,q~,F(t) decreases in t. Launer (1993) has shown that a distribution with a bathtub-shaped failure rate, h(t), is DPRL-c~ for all ~0 < Œ< 1 for some e0 > 0, provided there exists a to with h(to) >_ h(O). 9.3. Relationships among N W B U E ,

B F R and I D M R L

classes

DEFINITION 7 (Mitra and Basu, 1994). A life distribution F having finite mean # is said to be 'new worse then better than used in expectation' (NWBUE) if there exists a point to such that _> # #(t)],_<#

if t < to, ift_>t0 .

(9 ,4)

THEOREM 4 (Mitra and Basu, 1994). Let F be a continuous and strictly increasing life distribution. I f F is B F R with mean #, then it is N W B U E . Notationally, {BFR} C {NWBUE}. The converse of the above theorem is false as can be seen from the following example: B(t) =

{

e -t, e - l ~ ta,

((ù - g + 1)),

t/8exp

0 _ < t < 1, 1 <_ t < 2, 2_
.

The corresponding M R L is given by

~(t)=

{ {

1,

0_
t,

0_
4/t,

2 <_t < oc ,

so F is N W B U E with to = 4. The failure rate function is given h(t):

h(t) =

1, 2/t, (t/4-1/t),

0_
reveals that F is not BFR. TnEOREM 5. (Mitra and Basu, 1994) I f F is IDMRL(t0), then F is NWBUE(tó) with tó _> to, i.e., { I D M R L } C {NWBUE}. (Note: to tó are the change points of the failure rate and mean residual lifetime, respectively.) Mitra and Basu (1994) provide an example to show that a life distribution can be N W B U E without being I D M R L , i.e., the inclusion is strict.

Bathtub-shapedfailure rate life distributions

91

Earlier, Mi (1995) shows that {BFR} c {IDMRL}. Also Mitra and Basu (1994) have shown that {BFR} c {NWBUE}, and {IDMRL} c {NWBUE}. Combining these relationships, we now have a chain of inclusions: {BFR} C {IDMRL} C {NWBUE} . 9.4. Non-monotonic aging classes and their duals In this chapter, we deal with three non-monotonic aging classes, namely, {BFR}, {IDMRL} and {NWBUE}(we will no longer consider DPRL-e for the rest of the chapter). We now consider the dual classes of these aging families. DEFINITION 8. A life distribution F is UBFR if it has an upside-down bathtub failure rate function h(t). For example, Sweet (1990) shows that the log normal belongs to this class. The other one is the mixtures Weibull by exponential (Gurland and Sethuraman, 1994). For other examples, see Gupta (1995) and Ghitany (1998). DEFINITION 9. A life distribution F is DIMRL if its mean residual life has a bathtub shape. Ghitany (1998) showed that a generalized gamma distribution developed by Agarwal and Kalla (1996) has a bathtub-shaped mean residual life if the parameters satisfy certain constraints. The dual of {NWBUE} does not appear to have been considered. 10. Optimal burn-in time for bathtub distributions

10.1. Concepts of burn-in Due to the high failure rate in the early stages of component life (most notably, silicon and integrated circuits), burn-in has been widely accepted as a method of screening out failures before these components are shipped to customers or put into the field operations. That is, before delivery to the customers, the components are tested under electrical or thermal conditions that approximate the working conditions in field operation. Those components which fail during the burn-in procedure will be scrapped or repaired and only those which survive the burn-in procedure will be considered to be of good quality. These are then shipped to customers or pur into field operation. With insufficient burn-in, high initial failure rates cause high repair costs. On the other hand, with excessive burn-in, the reduced failure rate will be at the cost of increased capital and recurring costs. A major problem is to decide how long the procedure should continue. The best time to stop the burn-in process for a given criterion to be optimized is called the optimal burn-in time. A general background on burn-in can be found in Kuo and Kuo (1983), and Kuo (1984).

92

C. D. Lai, M. Xie and D. N. P. Murthy

10.2. Burn-in and bathtub distributions We now consider the optimal burn-in time for BFR distributions under different criteria

10.2.1. Maximizing the reliability for a given mission time Let the lifetime T of a component have a continuous bathtub shape failure rate h(t). This component is required to accomplish a mission which lasts for time 7. The reliability of completing the mission is thus/~(z). If we burn-in the component for a time b and if the component survives the burn-in, then the conditional reliability of accomplishing the mission is given by B(b + v) _ exp ( -

~(b)

(10.1)

fö+z \ Jb h(t)dt)

We want to determine the optimal burn-in time such that the survival probability (10.1) above will be maximized. The set of burn-in times is defined as B*=

{

F(»+¢)

b _ > 0 : - F(b-=~-maXt>o

P(t+¢)~ F(t)

J "

(10.2)

Alternatively,

B* =

{/7

b >_ 0 :

h(s)ds = min

;+~ }

t>_O at

h(s)ds

.

(10.3)

Mi (1994b) characterizes the structure of the set B*. THEOREM 6. (Mi, 1994a) Let the continuous failure rate h(t) have a bathtub shape with change points tl and t2, and z > 0 be a given mission time. • If z < t2 - q, then the optimal burn-in occurs at each point of [tl, t2 - 7], i.e., B* • ItI, t2 - 7]. • If z > t2 - tl, then the optimal burn-in occurs on or before the first change point tl, i.e., B* = {b*} and b* c [0, tl] where B* is defined as above.

10.2.2. Maximizing the mean residual lifetime M R L Often the quality of products is taken to be the length of time they give satisfactory service, i.e., their lifetime. After products are manufactured, the only way to improve this äspect of their quality is to operate them for a fixed period of time, say b. Suppose cost is not to be considered, it is reasonable to set out goal for the longest mean lifetime. Accordingly, we need to determine b such that M R L is maximized, as only those items that snrvive the fixed burn-in time are placed in service. In other words, we want to find b* such that #(b*) = maó~{P(b)} .

(10.4)

Bathtub-shapedfailure rate life distributions

93

THEOREM 7 (Mi, 1995). Assume h(t) has a bathtub shape and is differentiable. Then: • tl = 0: there is no need to burn-in, that is, b* = 0. • t2 = oc and tl > 0; then we can always choose b* = tl. • tl = t2 = oo (F is actually DFR): the cost should be considered. • 0 < tl <_ t2 < co, b* equals the unique change t* of #(t).

Accordingly, we never need to burn-in products longer than the first change point tl unless F is DFR.

10.2.3. Optimal burn-in time under age replacement with complete repair policy We consider the age replacement policy described in Barlow and Proschan (1965). Let cf denote the cost incurred for each failure in field operation and ca, satisfying 0 < ca < cf, the cost incurred for each non-failed item which is replaced at age T > 0 in field operation. THEOREM 8 (Mi, 1994b). Suppose the failure rate function h(t) is differentiable and has a bathtub shape. Then under the age replacement policy with complete repair at failure, the optimal burn-in time b* and the corresponding optimal age T* = T*(b*) satisfy 0 _< b* < tl and b* + T* = b* + T*(b*) > t2, where T*(b*) is either the unique solution of the equation given below

h(b+ T) f r F ( b + t) dt F(b + T) _ c / + k(b) «i - c~ Jo F(b) + B(b)

(10.5)

or equal to oc depending on whether (10.3) has a solution or not. Here k(b) is the expected cost of burn-in per item: k(b) = Eh(b) = c 0

f0b/~(t)dt c~F(b) FT~ ~

(10.6)

10.2.4. Burn-in time under block replacement and minimal repair policy Mi (1995) has also obtained the optimal burn-in time under the block replacement and minimal repair policy. The result, as given in Theorem 2 of Mi (1995), is rather complicated.

10.3. Burn-in time for BFR lifetime under warranty policies Warranty for durable goods provides a peace of mind for the consumers. It is a written guarantee that states the manufacturer of that good will provide labour and patts to fix or replace the defective good when necessary. Consumers will feel safe when they purchase a good that comes with warranty. However, the warranty cost may drastically reduce profitability.

C. D. Lai, M. Xie and D. N. P. Murthy

94

Burn-in is a common procedure to improve the quality of the products after they have been produced, but it is also costly. Following Nguyen and Murthy (1982) and Chou and Tang (1992), we assume that cost is additive and has the following elements: co: c1: c2: c3: c4:

the the the the the

manufacturing cost per unit without burn-in; fixed setup cost of burn-in per unit; cost per unit time of burn-in per unit; shop repair or replacement cost per failure; extra repair or replacement cost per failure during the warranty period.

Nguyen and Murthy (1982) proposed a model to determine the optimal burn-in time for products sold with warranty. Let Tw > 0 be the length of the warranty period. They considered two types of warranty policies. One is the failure-free policy where all failed products are repaired or replaced by independent and identically distributed one during the warranty period. The other is the rebate policy where the consumer is refunded some amount Ro(t) if the product fails at time t during the warranty period [0, Tw]. These authors then considered the total cost as the sum of burn-in cost and the warranty cost. By assuming that the failure rate of products has a bathtub shape with change points tl = t2 (i.e., having a unique change point), they obtain several results regarding optimal burn-in time minimizing the total mean cost function. Their conclusion is that the optimal burn-in time is no later than the unique change point. In Nguyen and Murthy's model, the refund amount function Ro(t) is assumed to be a decreasing linear function of t in [0, Tw]. Mi (1997) also consider the same problem but relaxed the assumptions in two ways: (1) h(t) may have two change points according to Definition 4, (2) Ro(t) is an arbitrary decreasing function of t. Mi discussed different product type-warranty policy combinations and concluded that in each case optimal burn-in time b* that minimizes the total mean cost function c(b) never exceeds the first change point t 1. It is believed that the replacement-free policy favours the consumers at the expense of the manufacturer, and the rebate (pro-rata) policy favours the manufacturer at the expense of the consumers. Because of this, Nguyen and Murthy (1984) introduced a mixed warranty policy that initially starts with a replacementfree period followed by a pro-rata period. This will be more reasonable from both the manufacturers' and the consumers' point of view. In addition to the above classification, warranties can also be renewable or non-renewable. Mi (1999) considers the burn-in problem under renewable mixed warranty policy. THEOREM 9 (Mi, 1999). Suppose the lifetime distribution F has a bathtubshaped failure rate function h(t) with change points 0 < tl _< t2 < oc. Then for the renewable mixed warranty with decreasing rebate function Ro(t) >_ O, the optimal burn-in time b* that minimizes the mean warranty cost C(b) must satisfy b* _< q.

Bathtub-shaped failure rate liJb distributions

95

11. Models extending the traditional BFR distributions

11.1. Discrete bathtub shape failure rate life distributions Most of the lifetimes are continuous in nature and hence many continuous life distributions have been proposed in literature. On the other hand, discrete data may arise from various common situations, for example: (a) A device is monitored or inspected for failures only once per time period (e.g., an hour, a day), and the observation is the number of time periods successfully completed prior to failure of device. In fact, lifetime data are often discrete even when the underlying process is continuous. (b) Some equipment operates in cycles. A frequently referred example is a copier whose life length would be the total number of copies it produces. Another example is the number of on/oft cycles of a switch before failure occurs. (c) An experimenter often discretizes or groups continuous data. Interests in discrete failure data came relatively late in comparison to its continuous analog. Earlier works on discrete lifetime distributions, see Ebrahimi (1986), Padgett and Spurrier (1985), Salvia and Bollinger (1982) and Xekalaki (1983). Suppose T is a discrete random variable with distribution function F and without loss of generality we can assume the support set of F is {0, 1 , 2 , . . , }. The failure rate of T is defined as

j5

hi-p{v>

f,

i)-Ri

'

(11.1)

where Ri = Pr{T _> i) is the survival probability, R0 = 1. Unlike the continuous but case, the survival function Ri hefe is not equal to 1 - F ( i ) , Ri+l = 1 - F(i) = fr(i) instead. The mean residual life function is defined as/~i = E{T - iIT > i):

~Rj+I

(11.2)

#i = 4"~ Ri j=t

so the mean/~ is given by # = kt0 = R1 + R 2

(11.3)

+'"

Salvia and Bollinger (1982) developed some results for discrete models analogous to continuous models. Some of their results are given as below:

fo=ho,fk-hk(1-ho)(1-hl)...(1-hk-1),k= Rk = (1 -- ho)(1 - h l ) . . . (1 - hk-x),k = 1 , 2 , . . oo

,

,

(11.4)

(11.»)

oe k - 1

E(T/= ~0 = ~ R~ = ~ 1-I(1 - hj/ k=l

1,2,..

k-1 j=0

(11.6)

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C. D. Lai, M . Xie and D. N. P. M u r t h y

We first need to define the concepts of bathtub shape and upside-down bathtub shape in the discrete case: DEFINITION 10 (Mi, 1993). A sequence { a i , i > 0} of real numbers is said to have a bathtub shape or an upside-down bathtub shape if there exits integers 1 < nl _< n2 < oc such that ao

> a l > a2 > • •. an

1 ~ anl

ao < a l < a2 < . . . a n - i

" " " ~- an2 ~ an2+l <~ " " "or~

:

< an1 =- "'" ~- an2 > anz+l > "'"

DEFINITION 11. A discrete distribution function F is said to have a bathtub shape if the sequence {bi, i > 0} has a bathtub shape. There are not too many discrete lifetime models having bathtub shapes. Lai and Wang (1995) provided a parametric finite range model defined as: B=P{r=i}-

~v

,

i=0,1,2,..,N,

~x=0 X

hi = hi

~ ~ - i x~

i = 0, 1 , 2 , . . , N

which is a discrete analogue to the finite range distribution proposed by Mukherjee and Islam (1983) . It was shown that for c~ < 0, F has a bathtub shape. Figure 4 shows the plot of the failure rates versus time for the above model with N=20. Mi (1993) established the relationship between the bathtub-shaped distributions and the upside down mean residual life (IDMRL).

1.0

•

0.8 F a

i

0,6

I u

x

0,4

x

R 0,2 a t -0.0 e

÷ •

÷

•

x *

x +

x •

x ÷

x ÷

x ÷

x ÷

x ÷

o o

o

o

o

o

o

o

o

o

o

o

,x

¥

o

o

,x

o

o

o

I

I

I

I

I

4

8

12

16

20

Time Fig. 4. Plots of failure rate hi vs time for the discrete model with N = 20. Caption: o,+, × correspond to « = 2, - I and - 2 , respectively.

Bathtub-shapedfailure rate life distributions

97

THEOREM 10. Let F be a lifetime distribution having support { 1 , 2 , . . } . Suppose (i) hi has a bathtub shape with change points nl and n2 ~ ex) with, (ii) f l > p+l" 1 Then {#i, i _> 1} has an upside-down shape with a unique change point, denoted by k0, and k0 _< nl, or two change points k0 - 1 and k0 _< nl. Notes: • It seems to us that Mi's definition of the mean p given by # _= .]'~/~(t)dt is incorrect. From his definition, fl = F(1) + F(2) +/~(3) + . . . . R2 + R3 + . . . , which is not the same as (11.3) so we think bis condition (ii) should be fl > 1/# instead o f f l > 1/(p + 1). • If the support for F were {0, 1 , 2 , . . } instead of { 1 , 2 , . . } then fl would be replaced by f0 = h0 whereby the condition (ii) would reduce to h0 > 1/#. This seems to be consistent with the result of Gupta and Akman (1995b). Using the example we have in this section, the mean residual life plot is given as (see Fig. 5): REMARK. Suppose we consider the optimal burn-in time for obtaining the longest mean residual life in field operation. From the above theorem we see that if the failure rate sequence has a bathtub shape with change points n~and n2, then the optimal burn-in time need not exceed the first change point nl. The converse of the above theorem is not necessarily true. Guess and Park (1988) give a counter example. Mi (1993) shows that under an additional condition the converse is true.

11.2. Generalized bathtub curves Some researchers (see, for example, Kuo and Kuo, 1983) collecting failure statistical data found that component failure rates offen follow a more complex

6.0

4.5 3.0 1.5 -0.0

I

I

I

I

5

10

15

2O

Time

Fig. 5. Plot of/~i, for ~

1,N = 20.

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C. D. Lai, M. Xie and D. N. P. Murthy

pattern than those described the bathtub curves. The modifications of this curve are known as generalized/modified bathtub curves. The major difl'erence between the traditional and the generalized curves is the behaviour during their infant mortality part. Jensen and Patersen (1982) suggested a two-stage model of infant mortality: the first stage has an increasing failure rate, indicating failures that rise from comparatively coarse defects, such as those from imperfect manufacturing, improper handling, or defective control processes. Such a failure rate in this first stage peaks quickly and is followed by a period of decreasing failure rate. Jensen and Patersen (1982) applied this model to the failures of modern electronic devices. K o g a n (1988) introduced a set of "Polynomial models" of failure rates that generalized bathtub curves. One of his models has a shape like the following (see Fig. 6). This curve has a minimum at tl and a m a x i m u m at to. The failure rate function h(t) is quite flat in the interval ( q - a , q + b). K o g a n (1988) constructed a polynomial model in which a = b such that h(h - a) = h(tl + a). To begin with, he assumes that the failure rate function has a derivative h'(t)=A(t-to)2r-l(t-tl)

2m-1,

t>O,

A>0,

tl_>to>0,

where r and m are positive integers such that A(tl -}- a - to)2r-la 2m-1 < e. For the special case with r = m = 1, it is found the failure rate equation is h(t) = At3/3 - A(to 4- tl)t2/2 + Atotlt + h(O)

subject to A ( q - a - to)a < e .

/ tl-a

tl

Fig. 6. Generalized bathtub curve.

tl+b

Time

Bathtub-shaped failure rate lfe distributions

99

N o tes

• In the above curve we have a bathtub shape for t > to. • A different form is one where it has a bathtub shape for t < to. • Kogan (1988) gave another generalized curve which has two bathtubs jointed by a hump. • For a mixture involving two Weibull distributions, Jiang and Murthy (1998) have shown that we can have both of the above two shapes. They have also shown that in this case, the shape for small t is the same as that for large t so that if it is increasing (decreasing) for small t then it is also increasing (decreasing) for large t. 11.3. R o l l e r - c o a s t e r curves

Wong (1988, 1989, 1990, 1991) as well as Wong and Lindstrom (1988) considered a generalization of the bathtub failure curve, called the roller-coaster curve. Essentially, the name suggests that the failure rate has a roller-coaster shape, a bathtub with one or more humps. Wong (1991) suggested some plausible physical reasons for the formation of the roller-coaster shape. The generation of the shape starts with the basic failure mechanisms, which lead to the generally decreasing failure rate. The humps could be caused by changing hazard conditions, wear-out failure distribution of flawed items, distribution of flaw sizes or residual small size flaws left in the equipment because of test and inspection limitations. Jiang and Murthy (1997b) considered two sectional models involving three Weibull distributions. By considering different constraints on the parameters, various shapes of failure rate functions result. Two of these have a 'roller~zoaster' shapes. The same authors, Jiang and Murthy (1998), also considered a mixture involving two 2-parameter Weibull distributions. They concluded that the failure rate shape can be one of eight different types. Two of these are monotonic whereas the rest can be viewed as 'roller-coaster' shaped. We also note in passing that in addition to the two models mentioned here, these two authors also considered a competing risk model (Jiang and Muthy, 1997c), and a multiplicative model (Jiang and Murthy, 1997a) involving two Weibull distributions. Bathtubshaped failure rates can all arise in these models except the mixture model. References Aarset, M. V. (1985). The null distribution of a test of constant versus bathtub-failure rate. Scand. J. Star. 12, 55-62. Aarset, M. V. (1987). How to identify a bathtub hazard rate. IEEE Trans. Reliab. R-36, 106-108. Agarwal, S. K. and S. T. Kalla (1996). A generalized gamma distribution and its application in reliability. Commun. Stat. - Theory Meth. 25(1), 201-210. Alidrisi, M. S., S. Abad and O. Ozkul (1991). Regression models for estimating survival of patients with non-Hodgkin's lymphoma. Microelectron. Reliab. 31, 473-480. Bain, L. J. (1974). Analysis of linear failure rate life-testing distributions. Technometrics 16, 551 560. Bain, L. J. (1978). Statistical Analysis of Reliability and Life-Testing Models. Marcel Dekker, New York.

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Barlow, R. E. and F. Proschan (1965). Mathematical Theory ofReliability. Wiley, New York. Barlow, R. E. and F. Proschan (1981). Statistical Theory of Reliaóility and Life Testing: Probability Models. To begin with, Silver Spring, MD. Bergman, B. (1979). On age replacement and the total time on test concept. Scandi. 91 Stat. 6, 161 168. Biswas, S. and M. A. Abid (1991). Optimal time of a periodic check-up preventive maintenance scheine under bath-tub and Weibull type failure rates. Internat. J. Syst. Sci. 22(12), 2651-2661. Bryson, M. C. and M. M. Siddiqui (1969). Some criteria for aging. J. Amer. Star. Assoc. 64, 14721483. Canfield, R. V. and L. E. Borgman (1975). Some distributions of time to failure for reliability applications. Technometrics 17(2), 263-268. Cao, J. and Y. Wang (1991). The NBUC and NWUC classes of life distributions. J. Appl. Prob. 28, 473-479. Chen, Z. (2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Stat. Probab. Lett. 49, 155-161. Chou, K. and K. Tang (1992). Burn-in time and estimation of change point with Weibull-exponential mixture distributions. Dec. Sci. 23, 973-990. Cobb, L. (1981). The multimodal exponential families of statistical catastrophe theory. In Statistical Distributions in Scientißc Work (Eds. G. P. Taillie, B. Patil). Reidel Press. Baldessari, Dordrecht, Holland. Cobb, L., P. Koppstein and N. H. Chen (1983). Estimation and moment recursion relations for multimodal distributions of the exponential families. J. Arner. Stat. Assoc. 78, 124~130. Colvert, R. E. and T. J. Boardman (1976). Estimation in the piece-wise constant hazard rate model. Commun. Stat. - Theory Meth. 1, i013-1029. Davis, D. J. (1952). An analysis of some failure data. J. Amer. Stat. Assoc. 47, 113-150. Deevey, E. S. Jr. (1947). Life-tables for natural populations. Q. J. Biol. 283 314. Deshpande, J. V., S. C. Kochar and H. Singh (1986). Aspects of positive aging. J. Appl. Prob. 23, 748-758. Deshpande, J. V. and R. P. Suresh (1990). Non-monotonic aging. Scandinavian J. Stat. 17, 257-262. Dhillon, B. S. (1981). Life distributions. IEEE Trans. Reliab. R-30(4), 457M60. Ebrahimi, N. (1986). Classes of discrete decreasing and increasing mean-residual-life distributions. IEEE Trans. Reliab. R-35(5), 403405. Gaver, P. D. and M. Acar (1979). Analytical hazard representation for use in reliability, mortality and simulation studies. Commun. Stat. - Simulat. Comput. 8(2), 91 111. Ghitany, M. E. (1998). On a recent generalization of gamma distribution. Commun. Stat. Theory Meth. 27(1), 223-233. Glaser, R. E. (1980). Bathtub and related failure rate characterizations. J. Amer. Stat. Assoc. 75, 667-672. Gore, A. P., S. A. Paranjpe, M. B. Rajarshi and M. Gadgil (1986). Some methods of summarizing survivalship in nonstandard situations. Biometr. J. 28, 577-586. Govindarajula, Z. (1977). A Class of distributions useful in life-testing and reliability. IEEE Trans. Reliab. R-26(1), 67-09. Griffith, W. S. (1982). Representation of distributions having monotone or bathtub-shaped failure rates. IEEE Trans. Reliab. R-31(1), 95-96. Gross, A. J. and V. A. Clark (1975). Survival Distributions; Reliability Applications in Biomedical Sciences. Wiley, New York. Guess, F. M. and D. H. Park (1988). Modelling discrete bathtub and upside-down bathtub mean residual-life functions. IEEE Trans. Reliab. R-37(5), 545-549. Guess, F., M. Hollander and F. Proschan (1986). Testing exponentiality versus a trend change in mean residual life. Ann. Stat. 14, 1388 1398. Guess, F. and F. Proschan (1988). Mean residual life: theory and applications. In Handbook o f Statistics, pp. 215-224 (Eds. P. R. Krishnaiah and C. R. Rao). Vol. 7: Quality Control and Reliability, Elsevier, Amsterdam. Gupta P. L. (1995). Aging characteristics of Weibull mixtures. Bullet. Internat. Statist. Institute: Proceedings of the 50th Session, pp. 435436. 2 1 ~ 9 August, Beijing.

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Gupta, R. C. and H. O. Akman (1995a). Non-monotonic failure rates and mean residual life functions. Bulletin of the International Statistical Institute." Proceedings of 50th Session, pp. 437-438. 2139 August, Beijing. Gupta, R. C. and H. O. Akman (1995b). Mean residual life functions for certain types of nonmonotonic aging. Commun. Stat.: Stochastic Models. 11(1), 219-225. Gurland, J. and J. Sethuraman (1994). Reversal of increasing failure rates when pooling failure data. Technometrics 36(4), 416-418. Haines, L. and N. D. Singpurwalla (1974). Some contributions to stochastic characterizations of wear. In Reliability and Biometry: Statistical Analysis of Lifelength, pp. 46-80 (Eds. Proschan and Serfling). Society for Industrial and Applied Mathematics. Haupt, E. and H. Schabe (1992). A new model for a lifetime distribution with bathtub shaped failure rate. Microelectron. Reliab. 32(5), 633 639. Haupt, E. and H. Schabe (1994). Constructing lifetime distributions with bathtub shaped failure rate from DFR distributions. Microelectron. Reliab. 34(9), 1501 1508. Haupt, E. and H. Schabe (1997). The TTT transformation and a new bathtub distribution model. J. Stat. Plan. Infer. 60(2), 229-240. Hjorth, U. (1980). A reliability distribution with increasing, decreasing, constant and bathtub-shaped failure rates. Technometrics 22(1), 99-107. Hollander, M., H. D. Park and F. Proschan (1986). A class of life distributions for aging. J. Amer. Stat. Assoc. 81, 9145. Jaisingh, L. R., W. J. Kolarik, and D. K. Dey (1987). A flexible bathtub hazard model for nonrepairable systems with uncensored data. Microelectron. Reliab. 27(1), 87 103. Jensen, F. and N. E. Petersen (1982). Burn-in: An Engineering Approach to the Design and Analysis of Burn-in Proeedures. Wiley, New York. Jiang R. and D. N. P. Murthy (1997a). Parametric study of multiplicative model involving two Weibull distributions. Reliab. Eng. Syst. Safety 55, 217-226. Jiang R. and D. N. P. Murthy (1997b). Two sectional models involving three Weibull distributions. Qual. Reliab. Eng. Internat. 13, 83 96. Jiang R. and D. N. P. Murthy (1997c). Parametric study of competing risk model involving two Weibu11 distributions. Internat. J. Reliab. Qual. Safety Eng. 4(1), 17-34. Jiang R. and D. N. P. Murthy (1998). Mixture of Weibull distributions - Parametric characterization of failure rate function. Appl. Stochastic Models Data Anal. 14, 47-65. Joe, H and F. Proschan (1983). Tests for properties of the percentile residual life function. Commun. Stat. - Theory Meth. 12, 1087-1119. Joe, H and Proschan, F (1984). Percentile residual life, Oper. Res. 32, 668-677. Kamins, M (1962). Rules for planned replacement of aircraft and missile parts. RAND Memo RM-2810-PR; RAND Corp, Santa Monica, CA, USA. Kao, J. H. K. (1959). A graphical estimation of mixed Weibull parameters in life testing of electronic tubes. Technometrics 1, 389-407. Klefsjö, B. (1983). A useful ageing property based on the Laplace transform. J. Appl. Probab. 20, 615-626. Kochar, S. C. and D. D. Wiens (1987). Partial orderings of life distributions with respect to ageing properties, Nav. Res. Logist. 34, 823-829. Kogan, V. I. (1988). Polynomial models of generalized bathtub curves and related moments of the order statistics. S I A M J. Appl. Math. 48(2), 416-424. Krohn C. A. (1969). Hazard versus renewal rate of electronic items. IEEE Trans. Reliab. R-18(2), 64~73. Kulaseker, K. B. and H. D. Park (1987). The class of bettet mean residual life at age to. Mieroeleetron. Reliab. 27, 725 735. Kuo, W. (1984). Reliability enhancement through optimal burn-in. IEEE Trans. Reliab. R-33(2), 145-156. Kuo, W. and Y. Kuo (1983). Facing optimal burn-in. IIIE Trans. Reliab. R-32(2), 145 156. Kunitz, H. (1989). A new class of bathtub-shaped hazard rates and its application in a comparison of two test-statistics. IEEE Trans. Reliab. 38(3), 351-353.

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Knnitz, H. and H. Pamme (1991). Graphical tools for life time analysis. Stat. Papers 32, 85-113. Kunitz, H. and Pamme, H. (1993). The mixed gamma ageing model in life data analysis. Stat. Papers 34, 303-318. Kulasekera, K. B. and K. M. La1 Saxena (1991). Estimation of change point in failure rate models. J. Stat. Plan. Infer. 29, 111-124. Lai, C. D. and D. Q. Wang (1995). A finite range discrete life distribution, International Journal of Reliability, Qual. Safety Eng. 2(2), 147-160. Lai, C. D., T. Moore and M. Xie (1998). The beta integrated failure rate model. In Proceedings of the International workshop on Reliability Modelling and Analysis-from Theory to Praetiee, pp. 153-159. November 1998, National University of Singapore, Singapore. Lai, C. D. and S. P. Mukherjee (1986). A note on 'A finite range of Failure Times'. Microeleetron. Reliab. 26, 183-189. Launer, R. L. (1984). Inequalities for NBUE and NWUE life distributions. Oper. Res. 32, 660-667. Launer, R. L. (1993). Graphical techniques for analyzing failure data with the percentile residual-life function. IEEE Trans. Reliab. 42(1), 71-75. Lawless, J. F. (1982). Statistieal Models and Methods for Life Time Data. Wiley, New York. Leemis, L. M. (1986). Lifetime distribution identities. IEEE Trans. Reliab. R-35, 170-174. Lewis, E. E. and H. C. Chen (1994). Load-capacity interference and the bathtub curve. IEEE Trans. Reliab. 43(3), 470~475. Lieberman, G. J. (1969). The status and impact of reliability methodology, Naval Res. Logis. Q. 14, 17-35. Lim J. and. D. H. Park (1995a). Trend change in mean residual life. IEEE Trans. Reliab. 44(2), 291-296. Lim J. and D. H. Park (1995b). A family of tests for trend change in mean residual life. Commun. Stat. Theory Meth. 27(5), 1163-1179. Lob, W. Y. (1984). A new generalization of NBU distributions. IEEE Trans. Reliab. R-33, 419-422. McDonald, J. B. and D. O. Richards (1987a). Model selection: Some generalized distributions. Commun. Stat. - Theory Meth. 16(4), 1049. McDonald, J. B. and D. O. Richards (1987b). Hazard rates and generalized beta distributions. 1EEE Trans. Reliab. R-36(4), 463-466. Mi, J. (1993). Discrete bathtub failure rate and upside-down bathtub mean residual life. Nav. Res. Logist. 40, 361-371. Mi, J. (1994a). Maximization of a survival probability and its application. J. Appl. Prob. 31, 10261033. Mi, J. (1994b). Bnrn-in and maintenance policies. Adv. Appl. Prob. 26, 207 221. Mi, J. (1995). Bathtub failure rate and upside-down bathtnb mean residual life. IEEE Trans. Reliab. 44(3), 388-391. Mi, J. (1997). Warranty policies and burn-in. Nav. Res. Logist. (44), 199-209. Mi, J. (1999). Comparisons of renewable warranties. Nav. Res. Logist. 16, 91-106. Mitra, M. and S. K. Basu (1994). On a nonparametric family of life distributions and its dual. aT. Stat. Plan. Infer. 39, 385 397. Mitra, M. and S. K. Basu (1995). Change point estimation in non-monotonic aging models. Arm. Inst. Stat. Math. 47, 483-491. Mitra, M. and S. K. Basu (1996a). On some properties of the bathtub failure rate family of life distributions. Microelectron. Reliab. 36(5), 679-684. Mitra, M. and S. K. Basu (1996b). Shock models leading to non-monotonic ageing classes of life distributions. J. Stat. Plan. Infer. 55, 131-138. Moore, T. and C. D. Lai (i994). The beta failure rate distribution. In Proceedings of 30th Annual Conference of Operational Research Society of NZ/45th Annual Conference of New Zealand Statistical Association, pp. 339 344. Mudholkar, G. S. and D. K. Srivastava (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Trans. Reliab. 42(2), 299-302.

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Mudholkar, G. S., D. K. Srivastava and G. D. Kollia (1996). A generalization of the Weibull distribution with application to the analysis of survival data. J. Amer. Stat. Assoc. 91, 1575 1583. Mukherjee, S. P. and A. Islam (1983). A finite-range distribution of failure times. Nav. Res. Logist. Q. 30, 487-491. Murthy, D. N. P. and R. Jiang (1997). Parametric study of sectional models involving two Weibull distributions. Reliab. Eng. Syst. Sqfety 56, 151-159. Murthy, V. K., G. Swartz and K. Yuen (1973). Realistic models for mortality rates and their estimation. Technical Reports I and Il, Department of Biostatistics, University of California at Los Angeles. Muth, E. J. (1997). Reliability models with positive memory derived from the mean residual life function. In Theory andApplications ofReliability, Vol. 2, pp. 401~435 (Eds. C. P. Tsokos and I. N. Shimi). Academic Press, New York. Nguyen, D. G. and D. N. P. Murthy (1982). Optimal burn-in time to minimize cost for products sold under warranty. IIE Trans. 14, 167 174. Nguyen, D. G. and D. N. P. Murthy (1984). Cost analysis of warranty policies. Nav. Res. Logist. Q. 31, 525-541. Padgett W. J. and J. D. Spurrier (1985). On discrete failure models. IEEE Trans. Reliab. R-34(3), 253-255. Pamme, H. and H. Kunitz (1993). Detection and modeling of aging properties in lifetime data. In Advances in Reliability, pp. 291 302 (Ed. A. P. Basu). Elsevier, Amsterdam. Paranjpe, S. A., M. B. Rajarshi and A. P. Gore (1985). On a model for failure rates. Biom. J. 27, 913-917. Paranjpe, S. A., M. B. Rajarshi (1986). Modelling non-monotonic survivorship data with bathtub distributions. Ecology 67, 1693-1695. Park, K. S. (1985). Effect of burn-in on mean residual life. IEEE Trans. Reliab. R-34(5), 522-523. Park, D. H. (1988). Testing whether failure rate changes its trend. IEEE Trans. Reliab. 37(4), 375-378. Pham-Gia, T. (1994). The hazard rate of the power-quadratic exponential family of distributions. Stat. Prob. Lett. 20(5), 375 382. Pinder, J. E., J. G. Wiener and M. H. Smith (1978). The Weibui1 distribution: a new method of summarizing survivorship data. Ecology 59, 175-179. Rajarshi, S. and M. B. Rajarshi (1988). Bathtub distributions: a review. Commun. Star. - Theory Meth. 17(8), 2597-2621. Richards, D. O. and J. B. McDonald (1987). A general methodology for determining distributional forms with applications in reliability. J. Stat. Plan. Infer. 16, 365-376. Rolski, T. (1975). Mean residual life. Bull. Int. Stat. Inst. 46, 266-270. Salvia, A. A. and R. C. Bollinger (1982). On discrete hazard functions. IEEE Trans. Reliab. R-31, 458-459. Schabe, H. (1994a). Constmcting lifetime distributions with bathtub shaped failure rate from DFR distributions. Microelectron. Reliab. 34(9), 1501 1508. Schabe, H. (1994b). Time between unscheduled removals. Microelectron. Reliab. 34(11), 1787 1794. Sherwin, D. J. (1997). Concerning bathtubs, maintained systems, and human frailty. IEEE Trans. Reliab. 46(2), 162. Shooman, M. L. (1969). Probabilistic Reliability." An Engineering Approach. McGraw Hill Electrical and Electronic Engineering Series, McGraw Hill, New York. Siddiqui, S. A. and S. Kumar (1991). Bayesian estimation of reliability function and hazard rate. Microelectron. Reliab. 31(1), 53-57. Singh, H. and J. V. Deshpande (1985). On some new ageing properties. Scand. J. Stat. 12, 213 220. Smith, R. M. and L. J. Bain (1975). An exponential power life-testing distribution. Comrnun. Stat. Theot7 Meth. 4, 469-481. Smith, R. M. and L. J. Bain (1976). Correlation-type goodness of fit statistics with censored sampling, life-testing distribution. Commun. Stat. - Theory Meth. 5, 119-132.

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Sweet, A. L. (1990). On the hazard rate of the lognormat distribution. IEEE Trans. Reliab. 39(3), 325-328. Suresh, R. P. (1992). On estimating change-point in a bathtub failure rate model. In Proceedings of the Symposium on Statistical Inference (Trivandrum), Publication, 24, Centre Math. Sci., Trivandrum. pp. 103-113. Talbot, J. P. P. (1977). The Bathtub myth. Qual. Assur. 3, 107-108. Wong, K. L. (1988). The bathtub does not hold water any more. Qual. Reliab. Eng. Int. 4, 279282. Wong, K. L. (1989). Roller--coaster curve is in. Qual. Reliab. Eng. Int. 5(1), 29 36. Wong, K. L. (1990). Demonstrating reliability and reliability growth with environmental stress screening data. In Proceedings of the Annum Reliability and Maintainability Symposium, Piscataway, NJ, USA, pp. 47-52. Wong, K. L. (1991). The physical basis for the roller-coaster hazard rate curve for electronics. Qual. Reliab. Eng. Int. 7(6), 489~495. Wong, K. L. and D. L. Lindstrom (1988). Offthe bathtub onto the roller--coaster curve. In Proceedings of the Annual Reliability and Maintainability Symposium, pp. 356-363. Piscataway, NJ, USA. Xekalaki, E. (1983). Hazard functions and life distributions in discrete time. Commun. Stat. - Theory Meth. 12, 2503 2509. Xie, M. (1987). Testing constant failure rate against some partially monotone alternatives. Microelectron. Reliab. 27(3), 557-565. Xie, M. (1988). Some total time on test quantiles useful for testing constant against bathtub-shaped failure rate distributions. Scand. J. Stat. 16, 13%144. Xie, M. and C. D. Lai (1995). Reliability analysis using an additive Weibull model with bathtubshaped fäilure rate. Reliab. Eng. Syst. Safety 52, 87-93.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

A

Equilibrium Distribution its Role in Reliability Theory

A. Chatterjee and S. P. Mukherjee

Equilibrium distribution has drawn considerable interest among applied probability and reliability theoreticians in recent years. Various generalized order relations and aging properties have been introduced in the literature based on equilibrium distributions of higher orders and in higher dimensions. An exhaustive account of such relations and aging classes has been tried to provide in this article along with their implications and a few preservation properties.

1. Introduction

Reliability and survival analysis have derived considerable strength from advances in applied probability and related mathematics. In turn, problems dealing with the dynamic behavior of various types of systems have motivated several meaningful extensions and generalizations of ideas and results in connection with probability distributions and stochastic processes. Thus, for example, notions of partial orders and of dominance relations among probability distributions have been used to derive interesting results in classification and characterizations of univariate as well as multivariate life (failure-time) distributions. Equilibrium distributions originated in the context of renewal processes, which are used in the study of repairable systems. Of course, such distributions can also be derived as particular cases of weighted distributions introduced earlier by Fisher (1934) in the context of ascertainment of gene frequencies. Again one can look upon the equilibrium distribution as the common limiting distribution of both used life and residual life in the context of reliability and survival analysis. Equilibrium distributions have been considered by the present authors along with several others for their relevance to studies of various properties of life distributions and the aging behavior of associated systems. The present article attempts to put together most of the important findings related to equilibrium distributions and their bearings on reliability analysis. In particular, dominance relations between original and equilibrium distributions and their aging implications have been mentioned in Section 3. Section 4 is de105

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voted to equilibrium distributions of higher orders and introduction of different generalized orderings and aging properties derived from such distributions. While Section 5 considers the preservation of some generalized order relations and a few aging properties under selected reliability operations, Section 6 deals with equilibrium distributions in higher dimensions and dominance relations involving them along with their interpretations in terms of multivariate aging properties. A few characterization of exponential distributions in terms of equilibrium distributions have been reported in Section 8 while Section 7 refers briefly about the uses of equilibrium distributions in the study of repairable systems. The paper ends up with certain concluding remarks and mentioning of several open problems.

2. History and genesis Weighted distributions arise in situations where the recorded observations cannot be considered to constitute a random sample from the original distribution. The reasons behind this, as observed by Rao (1965) are threefold. (i) Non-observation of some events. (ii) Damage caused to original observations. (iii) Adoption of unequal probability sampling. Consider a natural mechanism generating a random variable X with probability density function (probability mass function)f(x]O), OeO. For drawing a random sample of observations on X, one has to use a method of selection which gives the same chance to any observation (produced by the original mechanism) of being included into the sample. In practice it may so happen that the relative chances of inclusion oftwo observations x and y are w(x) : wO, ), where w(.) is a non-negative valued function. Then the recorded X, denoted as Xw has the probability density function (probability mass function) fW(xlO) = w(x)f(xlO)/w , where w = E(w(x)) with °E' for expectation. Define by Ut and Vt the backward and forward recurrence times of a stochastic process generated by X at age t. In reliability terminology they are known as used life and residual life at t, respectively. It follows from Cox (1962), pp. 61-66, that the limiting distribution (probability density function) of both Ut and Vt is given by the equilibrium distribution of X as R(2)(x,X) = R(x,X)/E(X), where R(x,X) is the reliability function of X at x. It is interesting to note that both equilibrium distribution and residual life distribution may be generated from the above-weighted distribution where the weight functions, respectively, are inverse of the failure rate function and I[x>tl function.

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The present article is too much constrained to describe all the stochastic-order relations (some authors prefer the phrases dominance relations among life distributions) and various aging properties as applicable in reliability analysis. However, for the sake of clarity reference may be made to Van Zwet (1964), Barlow and Proschan (1975), Ross (1983), Deshpande et al. (1986), Singh (1989), Deshpande et al. (1990), Chatterjee (1993), Nanda et al. (1996a, b), Gupta and Kirmani (1998) among many others. In what follows these definitions and concepts are assumed to be valid and accepted.

3. Equilibrium distributions and dominance relations Interest in the study of consequences of dominance relations originated from the works of Van Zwet (1964) who studied the consequences of convex ordering where the reference distribution is exponential distribution. With the introduction of various partial orderings and order (dominance) relations over the years the consequences (in terms of various aging properties) with respect to several forms of reference distributions became the point of interest. In continuation to those approaches Deshpande et al. (1990) and Chatterjee (1993) also investigated nonexponential referral cases. Consequences of the dominance relations where the reference distributions are either residual life or equilibrium distribution have been extensively studied there. (Some other interesting consequences for such dominance relations such as comparison of equilibrium distribution with referral exponential were also cited.) It is also interesting to study the consequences of various order relations among equilibrium distributions of two distinct random variables in terms of the corresponding relations between the original distributions of the same random variables. The proofs of all such results are somewhat routine and do not require any elegant or sophisticated mathematical techniques as demonstrated in Deshpande et al. (1990) or Chatterjee (1993). However, the consequences obtained therein may be of immense interest and of practical use particularly in the context of various inference problems. The following table gives a brief account of all such consequences. In the table 'ordered distribution pairs (F(x,X), G(x, Y)) where the latter is the reference distribution' means nonnegative random variables X and Y have reliability (distribution) functions respectively as F and G and X > • Y, where ' , ' stands for various order relations e.g. LR, FR, F R A or d, MRL, d(2), E, CVRL etc. (Table 1). It will be interesting to note that the consequences for residual life (Rt) is always one step ahead of the corresponding results for equilibrium distribution (R(2)). This may be possibly due to the fact that R(2) may be generated from Rt under a limiting argument as discussed in Section 2. Moreover order relations between two distinct random variables is always one step behind in terms of the order relations among the equilibrium distributions of them. It is generally observed, as mentioned in Deshpande et al. (1990), that the stronger the positive aging properties enjoyed by R ( x , X ) so is that enjoyed by R(2)(x,X) and vice versa. Towards this one can refer to the works of Whitt (1985) and Deshpande et al.

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Table 1 Consequences of dominance relations in terms of aging properties and order relations Ordered distribution pairs where the latter is the reference distribution

Additiorial requirement

Dominance relation

Consequence

LR FR FRA or d MRL d(2) d(2) E

R IFR R DMRL R NBUE R DVRL R(2) NBUE R DVRL c.v. of R < 1

--

LR FR FRA or d

X >FR }1 X > MRL Y X > HAMR Y

-

FR FRA or d MRL d(2) E d(2) CVRL

R R R R R R R

FR MRL

R IFR R DMRL

LR FR FRA or d E

X > FR Y R NWUE R HNWUE c.v. of R > 1

(R(x,X),R(2) (x,X)), R(2) being the equilibrium distribution of R

-

R IVRL R NWUE

(R(2)(x,X), R(2)(x, Y))

(i) (R(x,X),Rt(x,X)), Rt being the reliabiIity function of the residual random variable at 't' corresponding to R

R NWU

(ii) (Rtl (x,X), Rt2(x, X)), 0 < t~ < t2 (R(2)(x,X), Exp(x, Y)), Exp being the orte parameter exponential distribution having the same mean as R

-

IFR NBU DMRL generalized NBUE NBUE DMRL ICVRL

(1986) for further results. The dual of the consequences in terms of negative aging may be easily obtained by reversing the dominance relations and by considering t h e d u a l o f t h e a d d i t i o n a l a g i n g p r o p e r t i e s as a n d w h e n r e q u i r e d .

4. Equilibrium distributions of higher orders: Properties and generalized order relations (p+l) D e f i n e R(P): 1R1 ~ [0, oc) b y R 0 ) = R and R ( x , X ) =f[xoo)R(P)(t,X)dt, c o r r e s p o n d i n g t o t h e r a n d o m v a r i a b l e X . F o l l o w i n g R i e m a n n anc[ L i o u v i l l e (cf. Z y g m u n d , 1959, p. 133) it c a n b e s h o w n b y i n d u c t i o n a n d s u c c e s s i v e a p p l i c a t i o n of Fubini's Theorem that R(p+I)(x,Äz) = E x ( X - x)+P/p!,

w h e r e x+ = m a x ( x , 0) .

(4.1)

Equilibrium distribution - its role in reliability theory

109

For each p either R (p)(x, X) is non-negative and finite for all non-negative x or else it is infinite; depending only on the right tail of R(x,X). More specifically, from (4.1) it follows that R(P)(x,X) is finite if and only iflfo,~)y(P-1)dR(y,X)l < oc. Generate for all positive integer p the sequence of equilibrium distributions corresponding to R as

R(p+I)(X,X) = [ R(p)(t,X)dt/Tp(X), «[xùco) where 7p(X) =

Jo ,2) R(p)(t,X)dt

(4.2)

with R(l) = R, for every x. Trivially R (°)(x,X) = R(o)(x,X) =- f(x,X), the probability density function of X for every x. It is evident that the functions defined in (4.2) are proper reliability functions in the sense that they are non-increasing, left-continuous and satisfy the boundary conditions R(p)(co) = 0 and R(p)(0) = 1. Define the failure rate (FR) and mean remaining life (MRL) functions of R(p)(x,X), respectively as

2R~~(x, X) = -R~)(x, X)/R(p)(x, X)

and

rR~~(x,X) = fx,oo)R(p)(t,X)dt/R(p)(x,X)

J

Also define the reliability function of the residual life beyond 't' as for every non-negative x. The above-mentioned sequences viz. R (p)(x,X) and R(p)(x,X) were first studied by Mukherjee and Chatterjee (1992a) in the context of higher-order stochastic dominance. For two random variables X and Y Fishburn (1980) first introduced the concept of stochastic dominance of higher order. Afterwards O'Brien (1984) explored the idea and developed various moment inequalities based on them. Mukherjee and Chatterjee (1992a) extended the problem in terms of reliability function. For two random variables X and Y having reliability functions R(x,X) and R(x, Y) if R (p) (x, X) and R (p)(x, Y) exist and are finite the following definitions were proposed.

Rt(p)(x,X) = R(p)(t + x,X)/R(p)(t,X),

DEFINITION 4.1. For any positive integer p, X has pth-order stochastic dominance over Y, written as X >d(p) Y provided R (p)(x, X) >_ R (p)(x, Y), for all non-negative x. It is obvious that pth-order stochastic dominance implies that of (p + 1)thorder for all positive integer p, but the converse is not necessarily true. This in view of (4.1) may be taken to be an alternative proof of the first part of Proposition 1.7.1 in Stoyan (1983), p. 22 that second-order stochastic dominance implies pth-order stochastic dominance for all positive integer p. The second part also follows by considering the dual version as considered in O'Brien (1984) or Fishburn (1980). It may be pointed out that Definition 4.1 is the dual version of Fishburn's (op. cit.) definition ofpth-order stochastic dominance.

A. Chatterjee and S. P. Mukherjee

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Another definition of interest to study dominance relation between two random variables X and je provided E(X p) = ~ip(X) and E(Y p) = ~ip(Y), exist and are finite, may be stated as follows. DEFINITION 4.2. For any positive integer p, X has pth-order moment dominance over Y, written as X >M(p) Y provided ~ip(X) > #p(je), for all non-negative x. Afterwards R(p)(x,X) along with 2Re)(x,X), rR(~)(x,X) and Rt~)(x,X) were explored by Fagiuoli and Pellery (1993, 1994), Nanda et al. (1996a, b) to develop the concept of various stochastic order relations of pth-order and a few generalized aging classes based on them. For the sake of clarity some of them are quoted in the following. The order relations were proposed by Fagiuoli and Pellerey (1993) and are valid for two non-negative continuous random variables X and Y having reliability function corresponding to the pth-order equilibrium distribution as R(p)(x,X) and R(p)(x, Y). DEFINITION 4.3. X is said to have pth-order failure rate dominance over Y (written as X >p-FR Y) provided for all non-negative integer p, R(p)(x,X)/R(p)(x, Y) is non-decreasing in x. It is intuitive that X >p-FR Y ~

2Re/(x, Y) > 2R¢o/(x,X) for all x .

DEFINITION 4.4. X is said to have stochastic dominance based on the equilibrium distribution of order p over Y (written as X >p-ST Y) provided R(p)(x,X)/Roa)(x , Y) > R(p)(O,X)/R(p)(O,Y) which is equivalent to have R(p)(x,X) > R~)(x, Y) for all x. It is to be noted that d(p) is different from p-ST. DEFINITION 4.5. X is said to have pth-order mean remaining life dominance over Y (written as X >p-MRL Y) provided rRel (x,X) > re(r) (x, Y) for all x. DEFINITION 4.6. X is said to have pth-order convex ordering over Y (written as X >p-CX Y) provided fx,~)R(p)(t,X)dt >_ fx,~)R(p)(t, Y)dt for all x. DEFINITION 4.7. X is said to have pth-order concave ordering over Y (written as X >p cv Y) provided fox Y)dt for all x. [,] R(p)(t,X)dt -> fox)R(p)(t, [,The following definitions are involved in the generalized aging properties of a non-negative continuous random variable. -

DEFINITION 4.8. X is said to have increasing (decreasing) failure rate distribution ofpth-order (written as X is p-IFR (DFR)) provided 2R~/(x,X) is non-decreasing (non-increasing) in x. It should be noted that X is p-IFR holds provided Rt(e) (x,X) is non-increasing in t _> 0, which is equivalent to have RCp) (x,X)/R(p_1)(x,X) is non-increasing in x.

Equilibrium distribution - its role in reliability theory

111

DEFINITION 4.9. X is said to have increasing (decreasing) failure rate average distribution of pth-order (written as X is p - I F R A (DFRA)) provided f[0,~] 2R(~)(t,X)dt/x is non-decreasing (non-increasing) in x. DEFINITION 4.10. X is said to have new better (worse) than used distribution of pth-order (written as X is p - N B U (NWU)) provided Rt(p)(x,X) < (>_)R(p)(x,X) for all x, t. It is interesting to note the following implications of some higher-order dominance relations and aging classes. (i) 0 - F R ~=~ LR; 1 - F R ~ FR; 2 - F R e=~ MRL; 3 - F R e:~ VRL, 1 - ST ~, d and 2 - ST ~~, H A M R ; 1 - M R L ~ MRL; 1 - CX ~ CX and 1 - CV e:~ CV (with respect to several dominance relations along with their natural dual). (ii) 0 - I F R e=~ ILR; 1 - I F R +~, IFR; 2 - I F R e=~ D M R L ; 3 - I F R ~=~D V R L ; 1 I F R A e=~ I F R A ; 1 - N B U e=~ NBU; (with respect to various aging properties along with their natural dual). Certain characterizations of order relations between higher-order moments of two random variables (not necessarily non-negative) have been discussed in Fishburn (1980) and O'Brien (1984). Their results were in terms of distribution function of higher order. It has been observed by Mukherjee and Chatterjee (1992a) that similar relationship between higher order equilibrium distribution and stochastic dominance of higher order may be established if one views the stochastic dominance in terms of reliability function instead of distribution function. Mukherjee and Chatterjee (op. cit.) also established a few properties of equilibrium distribution while discussing on order relations between moments of two non-negative r a n d o m variables. In what follows some of the results and theorems of Mukherjee and Chatterjee (op. cit.) will be cited.

RESULT 4.1. Under the existence of density f = - R ' ; ~p+l (.X-) = f[0 oo) R(p+ l) ( x , X ) dx = #(p+l)(X)/(p+l)#p(X), for all non-negative integer p, ~here l~p(X)=

E(x~). PROOF. • The result will be established by inducting on positive integer p. • The result is well known for p = 0; as f[o,~)R(x,X)dx = E(X) = #I(X). • It can be shown after some algebraic manipulation that for p = 2; • Let it be assumed that the result is true for p = q - 1 (q > 2). t Then f[o,~) R ( q ) ( x , X ) d x -~ l~q/q#(q_l). • It is also evident that if the result is true for p = q - 1, it must be true for p = k, for all non-negative integer k for which q - 2 >_ k. Consider the case for p = q.

A. Chatte~jeeand S. P. Mukherjee

112

f[o,~) R(q+~) (x, X)dx

= (q#(q_l)(X)/#q(X)) f[o,~).fro,tJ dxR(¢)(t'X)dt'by

Fubini's theorem.

: (q(q _ 1)u(q_;)(x)/Uq(X)) ~ ,~) f[o ,~JtdtR(«-l)(z'X)dz' = (q(q- 1)#(q_;)(X)/l~q(X))~,co) (z2/2)R(q-1)(z'X)dz' = (q(q z

"''7

z

• • "7

= (q(q -

1)(q

- 2)#(q_3)(X)/2#q(X)) f[o ,oo)(w3/3)R(q-2)(w'X)dw'

1)(q -

2)...l#(q_q)(X)/2.3.4...

q,uq(X))

x ffo,co)(uq+l/(q + 1))f(u,X)du, under the existence of density and repeated applications of Fubini's theorem,

= (q[/(q + 1)!)#(q+l)(X)/gq(X), = ¢t(q+l)(X)/(q @ 1)#q(X) . THnOg~M 4.1. On x E [0, oc), for two random variables X and Y and for a given non-negative integer p, the following statements are equivalent. 2R,/(x,X ) < 2R~)(x, Y) ,

R(p)(x,X)/R(p)(X, Y)

is non-decreasing in x ,

Rt(p)(x,X) < Rt(p)(X, Y), Röp) (x,X)/Rt(p)(x, Y)

(4.3) (4.4)

for every non-negative t ,

is non-decreasing in x,

(4.5)

for every hOrt-negative t ,

(4.6) (Observe that (4.4) has been stated as the definition of p-FR as given in (4.3).) PROOF. The proof follows by observing the following facts: (a) For all non-negative x and integer p _> 2

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113

(d/dx) (Re)(x, X)/Re)(x, Y)) > 0

¢=~ ( p - 1)#e_2)(Y)Re)(x,X)Re_u(x , Y)/#e_l)(Y)

>_ Co- ~>e_2>(X)Re/(~, r)Re ~l(x,X)/~~_~l(x)

** 1/~~,~(x, Y) >_ 1/~~,~~(x,X) (b) For all non-negative x and integer p > 2

2RCoI(x,X) = 1/rRC~ ~I(x,X)

and

2R~I(x ,Y) = 1/rR~_li(X, Y) .

(C) For all hOrt-negative x and integer p > 2

R,~) (~,X) > <el (x, r) R e ) ( x , X ) / R e ) ( x , Y) is non-decreasing in x . (d) For all non-negative x and integer p > 2 (d/eLf) (R,e) (x, X)/Rt(p)(x, Y) ) 2 0

1/r~~_l~(X+ ~, v) >_ 1/~~~ ,~(x+t,x) For p = 1 the proof is straight forward. Observing that, for all non-negative x and t,

R,~) (x,X) > R,~)(x, Y)

@ f[x+t,oc)R(p-1)(x,X)/~,oc)R(p-1)(X'~z) Putting t = 0 and recalling Result 4.1, after certain routine calculations for all non-negative x the following can be obtained:

Be_l)(Y)//Ae_l)(X) ŒR e) (x, Y)/R @ (x,X)

.

With the above simple observations, the following theorem may be stated proofs of which is quite simple and is given in Mukherjee and Chatterjee (op. cit.). TnEOREM 4.2. (i) If X (ii) If X (iii) If X

any of the statements in Theorem 4.1 holds true and X >Me 1) Y, then >d(p) Y. any of the statements in Theorem 4.1 holds true and X >M0,-2) Y, then >M(p-1) Yany of the statements in Theorem 4.1 holds true and X >Me 2) Y, then >te) Y'

A strong bearing of second-order stochastic dominance on harmonic new better (worse) than used ( H N B U E / H N W U E ) class may be observed. In fact a life

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distribution with finite mean is H N B U E / H N W U E provided it is dominated by (dominates) an exponential density having the same mean in second order stochastic (d(2)) sense. An extension of this observation to the d(p) case may be of interest, for which the following definition is proposed. DEFINITION 4.11. A life distribution for the random variable X with reliability function R(x,X) is said to be generalized HNBUE (HNWUE) ofpth-order provided the harmonic mean of the MRL function over (0,x) of R(p)(x,X) is dominated by (dominates) the corresponding arithmetic mean of R(p)(x,X), for every hOrt-negative integer p. After certain routine calculations an alternative definition may be proposed as follows. DEFINITION 4.11 (a). R is generalized H N W U E of pth-order provided, for all positive integer p and non-negative x

R(P+I)(x,X) >_#p(X)exp(-pX#(p_l)(X)/#p(X))/p! (The dual version is obtained by reversing the inequality). It is obvious that putting p = 1 the usual H N B U E / H N W U E class may be generated. The strong bearing among generalized H N B U E / H N W U E class and d(p) dominance may be stated in form of the following theorem. THEOREM 4.3. R is generalized HNBUE (HNWUE) of pth-order provided it is dominated by (dominates) an exponential density having the same mean in (p + 1)th-order stochastic sense. PROOF. The proof follows by considering the expression for R(p+I)(x,X) when X has exponential density having the same mean as R and applying Definition 4.1 to generate the condition given in Definition 4.11 (a). 5. Closure of generalized order relations

Preservation of various dominance relations under several reliability operations has been examined by Mukherjee and Chatterjee (1991a, b), Boland et al. (1994), Shaked and Shantikumar (1994) among others. As mentioned in Stoyan (1983) the following reliability operations are of wide interest in the context of dominance relations or aging properties: (i) (ii) (iii) (iv) (v) (vi) (vii)

Convolution. Mixture distributions. Formation of coherent system. Poisson shock model. Laplace transformation. Weak convergence. Monotonic transformation.

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In what follows, few such reliability operations will be considered to establish the closure properties of the generalized dominance relations. The rest will be taken up in later communications.

5.1. Preservation under mixture of distributions The preservation of likelihood ratio (LR), failure rate (FR) and stochastic (d) dominances under mixtures has been considered by Mukherjee and Chatterjee (1991a). Later an elegant proof with respect to FR - case has been provided by Boland et al. (1994). A detailed discussion in this regard may be available in Shaked and Shantikumar (1994). In continuation Nanda et al. (1996a) considered the closure of the generalized dominance relations under mixture distributions. The basic problem along with the results will be discussed now. Let the non-negative continuous random variables Xo and Yo respectively have reliability functions R(x,Xo) and R(x, Yo) where 0 is guided by a random mechanism having distribution function G(O). Let the compound or mixture random variables which are again non-negative and continuous based on Xo and Yo be denoted as X and Y having reliability functions R(x,X) and R(x, Y), where

R(x,X) = I R(x'X°)dG(O)' R(x, Y) = / R(x, Yo)dG(O) Consider the reliability functions of the pth-order equilibrium distribution corresponding to Xo and Yo as Rco)(x,Xo), and R(p)(X, Yo), respectively. The question that will be addressed may be stated in terms of the following theorem. THEOREM 5.1. (i) IfXol >p-FR YO2,for all 01 and 02 in the support of 0; then under the above set up X >p-FR Y. (il) IfXol >p-sT Yo2,for all 01 and 02 in the support of 0; then under the above set up X >p-ST Y. (iii) IfX01 >p-CXX02, for all 01 and 02 in the support of 0; then under the above set up X >p-CX Y. (iv) IfX01 >p-CV Yo2,for all 01 and 02 in the support of 0; then under the above set up X >p-CV Y. (Although Nanda et al. (op. cit.) considered two different 01 and 02 having same probability model G(O), for all practical purposes they may be taken to be equal.) Before proving the theorem an important lemma has been proved by Nanda et al. (1996a) which is as follows. LEMMA 5.1. For non-negative integer p, (i) R(p)(x,X) = f(-oo,oo)tip-1 (O)R(p)(x,Xo)dG(O),

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A. ChatteJjeeandS. P. Mukherjee

7p(X) = f(-~,o~)tip-l(O)'/p(Xo)dG(O), where tip(O)= HP=oTj(Xo)/IIP_oTj(X) with by convention/3_ 1(0) = 1.

PROOF. The proof follows by induction on p, starting with p = 1 and invoking Fubini's theorem for changing the order of integration. Proof of the theorem.

(i)

~01>p-FR1102 e=~ for

(d/dx)(R(p)(x,Xol)/R(p)(x, Y02)) ~" 0 <=> [R(p)(x,Xol)R(p U(x, YO2))]/~/(p_I(YO2) all 01 and 02,

>- [Æ~-ll(x,Xol)R~l(x, Y02))]/~p ~(x0~) .

FIP=oTj(Yo)/IIP_oTj(y)

Define tip(0) : with by convention ti*l (0) = 1. Multiplication of both sides of the inequality by tip-1 (01)tip_l (02) and then application of expectation operator chronologically with respect to 01 and 02 yields

[/_oo,~)[tip-l(O)R(p-1)(x,Yo)/Tp I(Yo)]dG(O)]

~I/~ ~,~/~; l(0/~~~(~,~0/d~C0/J_>0 Observe that the left side of the above inequality is the numerator of the derivative of

[a~(-~,~)tiP-l(O)R(p)(X'X°)dG(O)]/[/( oc,oo)fl;-'(Õ)R(p)(X'Y°)dG(Õ)1 = R~I (x,X)/R~~(x, Y) ,

(by Lemma 5.1) with respect to x indicating that it is non-decreasing in x, which proves that X >p-FR Y.

Equilibrium distribution - its role in reliability theory

(ii)

Xol >p-ST Yo2

~

for all 0 and 02,

117

R(p)(x,Xol)/R(p)(x,Iio2)>_1

~:} ])p_l(Y02).f[x,e~)R(pl)(t,Xol)dt1 ~ 'p l(XO1)

f[x,~)R(Pl)(t'Y°2)dtI "

Multiplication of both sides of the inequality by ~p_2(01) • ~p_2(02) and then application of expectation operator chronologically with respect to 01 and 02 yields

>-I.f_~,~)flp2(O)Tp-l(X°)dG(O)l/[f(_~,o~)~*p-2(O)Tp-~(Y°)dG(O)1 " Interchanging the order of integration both in numerator and denominator of the left side of the inequality and invoking Lemma 5.1

ùf[x,oo)R(p1)(t,X)dt/.~~,oo)R(p_l)(t,Y)dt >_Ts_l(X)/Ts_l(Y), assuringX >p_sTY . For CX and CV orderings the basic idea of the proofs and the mode of derivation are almost similar and hence are omitted. For details reference may be made to Nanda et al. (op. cit.).

5.2. Characterization through Laplace transform The characterization of the usual dominance relations (particularly LR, FR and d) under Laplace transformation has been extensively studied by Kebir (1994), Shaked and Shantikumar (1994). Vinogradov (1973) has first studied characterization of the aging properties (more specifically IFR class) through Laplace Transformation. Later Block and Savits (1980) extended the theory to DMRL, IFRA, NBU, and NBUE classes. An obvious generalization of these results to characterize the p-FR and p-ST orderings as well as p-IFR, p-IFRA and p-NBU classes has been attempted and solved by Nanda (1995). A few definitions of discrete generalized orderings and generalized aging properties have been introduced in Fagiuoli and Pellerey (1993). Ler X and Y be two discrete non-negative random variables having probability mass functions {Pk} and {q»}, respectively, for non-negative integer k. Define for positive integer p, the discrete version of the equilibrium distributions as:

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S(p)(k,X) =

S(p 1)(i,X)

S(p_l)(i»X )

with S(0)(k, X) =

Pk,

i=k+ I

S(p)(k, Y) =

S(p_0 (i , Y) i~k+ 1

/#_o

Se_l)(/, Y)

with S(0)(k, Y) = qk •

DEFINITION 5.1. (i) For two non-negative discrete random variables X and F, X is said to have pth-order failure rate dominance over Y (written X >p-VR I1) provided S(p)(k,X)/S(p)(k, Y) is non-decreasing in k. (ii) For two non-negative discrete random variables X and F, X is said to have stochastic dominance based on equilibrium distribution of order p over F (written X >p-ST Y) provided S(p)(k,X) > (S(p)(k, Y). Let for a non-negative random variable X with reliability function R(x,X), define the Laplace transform of X as ~(2,x)

=-

c¢ù(2,X) =

B0 ,~) exp(-Lv)

dR(x,X), 2 > 0 and denote

(-l)"(d'/(d2)")[(1 - +(2,X))/2]/n!, with

C~o(2,X) = (1 - 0(2,X))/2 and ~ù(2,x) = 2%

,(2,x)

.

Similar definition for another non-negative random variable Y having reliability function R(x, Y) is assumed to hold and is denoted by c¢,(4, Y), with the respective boundary condition and recurrence relation. Write for positive integer n, nonnegative x and any Bord measurable function defined on [0, oc), Fx (n, x) = 2 exp ( - ) « ) (2x)" -I / (n - 1)!

F;.g(n) = B ~) F2(n,x)g(x)dx with F~g(O) = 1

I

Kebir (1994) and Shaked and Shantikumar (1994) considered the following results proofs of which will be omitted as they are simple. LEMMA 5.2 (Kebir, 1994).

c¢ù(2,X) = [ F;~(n,x)R(x,X) JE0ùoc)

dx, for positive integer n

m

LEMMA 5.3 (Shaked and Shantikumar, 1994). ~, (2,X) and e, (2, Y) are proper discrete reliability functions .

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Let N~x and N[ be two discrete random variables having respective reliability functions ~n(2,X) and ~~(2, Y). Invoking Lemma 5.3 it can be shown that:

F~R(p)(n,X) = f

F~(n,x)R(p)(x,X)dx and

r~~R(p)(n,r) = f

r~(n,x)Re)(x, r)dx

J[0ùoc) J[o,oo)

are also proper discrete reliability functions. The following lemmas will be used in the sequel and are proved in Nanda (1995). LEMMA 5.4. For all non-negative integer p,

(i) ~ rjR(p)(k,X) = 2~p(X)r~R(p+l)(n,X), k=n+l

(ii) ~ F~R(p)(k,X) = 1 + 2~p(X). k=0

Taking n = 0 a n d p = 1 in (i), orte can have E(Nf) = )LE(X) and E(N~) = 2E(Y). LEMMA 5.5. For all non-negative integers n and p, (i) S(p)(n, Nf) = FxR(p)(n + 1,X) and (ii) S(p)(n,N[) = F;R(p)(n + 1, Y). Three more lemmas will be required for convenience in discussion, which are: LEMMA 5.6 (Block and Savits, 1980). Ler x > 0 be a continuity point of any distribution function F. Let 2 = 2 ( x , n ) satisfy limn~~(n/2)=x and limn-~oo(1/2) = 0. Write dGn(t)= {2n+lt"exp(-2t)}dt/n!. Then Gù converges weakly to G, where G is degenerate at x. LEMMA 5.7 (Block and Savits, 1980). Let x > 0 be a continuity point ofR(p)(x,X). Let ,l = ,~(x, n) satisfy limn-~oo(n/2) = x and limn_~o~(1/2) = 0. Then for any nonnegative integer p, limù~oo F ~R(p)(n, X) = R(p) (x, X). LEMMA 5.8 (Block and Savits, 1980).

(d/d)OF;~R(p) (n,X) = (n/2) [F,~R(p)(n,X) - FxR(p)(n + 1, X)] . Block and Savits (1980) argued that for each 2 > 0, c~n(2,X) specifies a discrete reliability function corresponding to a specific cumulative damage model where X is the random threshold of the system and n being the number of shocks absorbed. It is to be noted that in a Poisson shock model the system is subject to a sequence of shocks occurring randomly as events in a Poisson process with intensity 2; each shock causes a random amount of damage to the system and the damages accumulate successively.

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Consider two systems which are subjected to sequence of shocks and the nature of shock be as above. Suppose X and Y be two random thresholds of the systems, respectively. Then the following theorem indicates that the necessary and sufficient condition that the thresholds are ordered according to p-FR (p-ST) sense is that the discrete lifetimes of the systems experienced by a sequence of exponentially spanned shocks are also ordered in p-FR (p-ST) sense. THEOREM 5.2. (i) (il)

X >p-VR Y ¢# Nf >p-VRN;Y for X >p-SW Y ¢:>Nf >p-sT Nf for

all positive 2, all positive 2.

PROOF. (i) ( ~ ) X >p-VR Y ¢V R@ (x,X)/R(p) (x, Y) is non-decreasing in x > 0, which means for some c > 0, R(p)(x,X) - cRco)(x, y) has atmost one change of sign and if it does occur, it occurs from below. Also it is well known that Poisson mass function P(2x, n - 1) is TP2 in (n,x), for x > 0. Thus by variation diminishing property of TP2 function (Karlin, 1968) B0,oo)Iexp (-2x)()«)"-~/(n - 1)!] [R(p)(x,X) - cR(p)(x, Y)] dx has atmost one change of sign and if it does occur, it occurs from below. Hence, for all positive integer n,

ff[o,oo)[exp(-2x)(2x)"-' / (n -1)'] R(p)(x,X)dx/ ~o

[eXp(-2x)()~x)" l/(n-1)']R(p)(x,Y)]dx

~CX3) L

is non-decreasing in n. Thus F2R(p)(n,X)/F~R(p)(n,Y) is non-decreasing in n and hence by Lemma 5.5 S(p)(n,N[r)/S(p)(n, Nf) is non-decreasing in n yielding N x >p-eR Nr, for all positive 2. ( ~ ) Note that the numerator of (d/d2)IraR~,)(~,x)/r~R~)(n, by invoking Lemma 5.8 can be written as

~)],

which reduces to

+ 1, Y) - r;R(p)(n, Y)r~R(p)(n + 1,X)] . (5.1) By virtue of IF,R~)(n,X)/F;,R(p)(n, Y)I being non-decreasing in n; (5.1) beeomes non-positive. Thus considering (5.1) as a funetion of 2, F~R(p)(n,X)/F;tR(p)(n,Y) is non-increasing in 2. Hence F,/xR¢o)(n,X)/F,/xR¢~)(n, Y) is non-decreasing in x. (n/,Z) [r)Rc~ ) (n,X)r)R(p)(n

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121

Let 0 < xl < xa be two continuity points of both R(p)(,X) and R(p)(*, Y). Then

r,/xaR(p) (n,X)/rn/xaR(p)(n, Y) >_F~/x1R(p)(n,X)/Fn/«lR(p)(n, Y), for positive integer n. Taking limits as n approaches infinity and using Lemma 4.7,

R(p)(x2,X)/R(p)(X2, Y) > R(p)(Xl,X)/R(p)(X~, Y), for 0 < xl < x» As the set of discontinuity points of a reliability function is at most countable which means the set of continuity points is dense, R(p)(x,X)/R(p)(x, Y) is nondecreasing in x yielding X >p-FR Y. (il) ( ~ ) X >p-ST Y ~:* R(p)(X,X) > R(p)(x, Y) for all x _> 0. Multiplication of both sides of the above expression by F;.(n,x) and integration over x ~ I0, oc) yields F;R(p)(n,X) >_F~R(p)(n,Y), which by Lemma 5.5 reduces to S(p)(n, N x) >_ S(p)(n, NX), for all non-negative integer n, Thus Nf >p-ST N;Y, for all positive 2. ( ~ ) N~ >p-ST Nr, for all positive 2 ~ F~R(p)(n,X) > F~R(p)(n, Y) for all positive integer n. Let x (> 0) be continuity point of both R(p)(.X) and R(p)(*, Y). Then for each positive integer n, F~/~R(p)(n,X) > F~/~R(p)(n, Y). Taking limit as n approaches infinity on both sides of the above expression and using Lemma 5.7, R(p) (x,X) > R(p)(x, Y), for all continuity points x. Thus by the similar argument as of p-FR case, X >p-ST Y follows. Notes: (i) The above characterization is a generalization of the characterization of usual FR - dominance under Laplace Transform as has been stated in Kebir (1994) and Shaked and Shantikumar (1994). However, for p = 0, 2 and 3 the results corresponding to LR, M R L and VRL - dominance easily follow. (ii) The above characterization is a generalization of the characterization of usual d - dominance under Laplace Transform as has been stated in Kebir (1994) and Shaked and Shantikumar (1994). However, for p = 2 the result corresponding to H A M R - dominance easily follows. The characterization of the generalized aging classes in terms of Laplace transform may be stated in terms of the following theorem. THEOREM 5.3. (i) A random variable X is p-IFR iff F;R(p)(n,X)/F;R(p 1)(n,X) is non-increasing in n i.e. - l o g [F;~R~,~(n,X)] is convex in n, for positive integer n. (ii) A random variable" X lS'Lp-IFRA~/lff"~F,tR(p)(n,X)] 1/n.ls non-increasing in n i.e. - l o g IC;oR(p)(n,X)l is star-shaped in n, for positive integer n. (iii) A random variable X is p-NBU iff F~R(p)(m,X)FxR(p)(n,X) >_F~R(p) (m+ n,X) i.e. - l o g [F~R(p)(n,X)l is super-additive in n, for positive integer n. PROOF.

(i) ( ~ ) Consider x (> 0) and x + y (> 0) be two continuity points of both and F~R(p_I)(n,X). Define 2 = n i x and k=[ny/x]. Then lim,,_+o~(n + k)/2 = x +y. By Lemma 5.6, one can show that Gn+k-1 converges

F~R(p)(n,X)

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122

weakly to a degenerate distribution with point of degeneracy x + y. Thus by Lemma 5.7, limn-~o~FxR(p)(n + k,X) = R(p)(X + y,X). As F;~R(pI (n,X)/rx R(p_l)(n,X) is non-increasing in n for positive integer n, by Lemma 5.8, F~R(p) (n,X)F~R(p_I)(n + k,X) - FxR(p)(n + k,X)F~R(p_I)(n,X) »_ O. Taking limit as n approaches infinity R(p) (x,X)R(p_l)(x q- y , X ) - R(p)(x q- y , X ) R ( p _ l ) ( x , X ) > 0 .

As the continuity points of reliäbility function are dense, the result follows immediately by noting that the above condition is equivalent to R(p)(x,X)/R(p_i)(x,X) is non-increasing in x i.e. -logR(p)(x,X) is convex in x. Thus X is p-IFR. ( ~ ) X is p-IFR ~ For any given c (> 0), R(p)(x,X) - cR(p_l)(x,X) has at most one change of sign and if a change does occur it occurs from above. As Poisson mass function P(2x, n - 1) is TP2 in (n,x), for positive integer n and x > 0, by variation diminishing property of TP2 function (Karlin, 1968)

B

,~)[exp(-)zc)(2x)n-l /(n - 1)!][R(p)(x,X) - cR(p_l)(x,X)]dcr

has atmost one change of sign and if it does occur, it occurs from above. This means FxR(p)(n,X) - cF~R(p_l)(n,X) has at most one change of sign and if it does occur, it occurs from above. Thus F~R(p)(n,X)/FxR(p_I)(n,X) is non-increasing in positive integer n which means -log[F~R(p)(n,X)] is convex in n, for positive integer n. (ii) Define c~,(2,X(p)) = f[0,~)F,~(n,x)R(x,X(p))dx, for positive integer n. Then

c~H(2,X(p) ) = J[Of,o~)[un exp(-2u) /n!]R(p) (u,X) du = 2([o [uH/n!]R(P)(u, Y) du, ùoo)

where R~)(u, Y) = exp(-)~u)R(p)(u,X)

= E[Y~p~l]/(n + l) ! • ( 3 ) Given that for positive integers n and k for all positive 2,

Taking x (> 0) and x + y (> 0) as the continuity points of R(p)(n,X) and using the same type of argument as used in (i), as n approaches infinity; the above expression reduces to

R~)(x,X) > [Re)(z +y,X)] ~/(~+~) .

Equilibrium distribution - its role in reliability theory

123

Since this is true for all continuity points x of R(p)(x,X) and the continuity points are dense, the above expression reducës to [R(p)(x,X)l l/x is non-increasing in x, which means - l o g R(p)(x,X) is star-shaped in x. Thus X is p-IFRA. ( ~ ) Observe that R(p)(, Y) is I F R A iff R(p)(,X) is I F R A and X is p-IFRA means X(p) is IFRA. Thus by using Corollary 6.5 of Barlow and Proschan (1975), X is p-IFRA implies: 1)!} l/(n+l)

is non-increasing in n, whichmeans

{Œù(2,X(p))} 1/("+1) is non-increasing in n, which further gives

[F;,R(p)(n,X)I 1/, is non-increasing in n, which is equivalent to - log[F2R(p)(n,X)l is star-shaped in n for positive integer n (iii) ( 3 ) Given that for positive integers n and k, F2R(p)(n,X)F;~R(p) (k,X) >_F~R(p)(n + k,X). Taking limit as n approaches infinity and by similar argument as in (i) or (ii), it follows that: [R(p)(x,X)l [R(p)(y,X)] >_ R(p)(x +y,X), for positive x and y, which means -log[R(p)(x,X)] is super-additive in x. Thus X is p-NBU. ( ~ ) As in (ii) Re)(,, Y) is NBU iff R(p)(, X) is NBU and X is p-NBU means X(p) is NBU. Thus X is p-NBU implies {E[X;~l]/(m@

l),}{E[X;~l]/(n-~ -

m+n+l]/( m -~-n Jc- 1)!} 1)'} > { E XI (p)

This yields -log[F;R(p)(n,X)l is super-additive n, for positive integer n. Notes:

(i) (A) For p = 1, the characterization of I F R class as stated in Vinogradov (1973) follows. The condition is - log F;R(n,X) is convex in n (positive), for all positive 2. (B) For p = 2, the characterization of D M R L class as stated in Block and Savits (1980) follows. The condition is ~~=n FxR(k,X)/FxR(n, X) is nonincreasing in n (positive), for all positive 2. (c) For p = 3, the characterization of DVRL class follows. The condition is ~k~=n F;R(2)(k,X)/F;~R(2)(n,X) is non-increasing in n (positive), for all positive 2. (ii) For p = 1, the characterization of I F R A class as stated in Block and Savits (1980) follows. The condition is -log[FxR(n,X)] is star-shaped in n (positive), for all positive 2. (iii) For p = 1, the characterization of NBU class as stated in Block and Savits (1980) follows. The condition is -log[FxR(n,X)] is super-additive in n (positive), for all positive 2.

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6. Equilibrinm distribution in higher dimension So far the discussion is concerned with a non-negative random variable X and its equilibrium distribution of higher orders. In this section the multivariate version of equilibrium distribution will be proposed with regard to the n-dimensional random vector X = ( X l , ~ , . . ,Xn) I defined over the non-negative orthant of ndimensional real space denoted as IP,+. The extension of the univariate concept to multivariate case was first introduced by Mukherjee and Chatterjee (1991b) while establishing certain closure properties of two multivariate aging classes proposed by Johnson and Kotz (1975), Marshall (1975), Zahedi (1985), and Arnold and Zahedi (1988). For the sake of completeness the definitions are cited below. DEFINITION 6.1. Let R(x) = P[X > x] be the reliability function of the n-dimensional random vector X defined over IR+. The extended real vector b = (bi, b 2 , . . , b,,)' is called the supremum of the support (sos) of R provided bj = inf{xc[0,cm)}{x : Rj(x) = 1},

j = l(1)n ,

where Rj(x) is the j t h univariate marginal of R. It is apparent that some or all of the components of b may be ec. The following two multivariate ageing classes are proposed for X, a random vector defined over IR,+, having reliability function R. Also let b denote the sos of R. DEFINITION 6.2. Define the Borel measurable function on IP,+ as A(x, X) = ()Vl (x, X), )v2(x, X ) , . . , ~.n(x, X)', where for all x c IR+ and x x / » ) / R & A x / j ) > x/») , where f j (xjIXü)> x0) ) is the conditional density of Xj given X 0 ) > x0) and

Rj(xjIX(i ) > x0) ) = P[Xj > xjIX(i ) > x(j)] with x(j) = (Xl,X2,.. ,xj_~,xj+l,... ,x,,)'. Johnson and Kotz (1975) as well as Marshall (1975) called A(x,X) as the hazard gradient vector. It is to be noted that on the set {x: P[X > x] = 0} which is equivalent to the set {x:R(x, X) = 0}, the function may be defined in an arbitrary fashion. DEFINITION 6.3. Define the Borel measurable function on IR+ as r(x, X) = (rl (x, X), r2(x, X ) , . . ,

r,(x, X))',

where for all x E IR+ and x x)

=[

«[,j ùcm)

R(tj, x(i), X)dtj/R(x, X),

where (tj, x0) ) means (Xl,X2,.. ,xj 1, tj,xj+l,...

,xn)' •

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Zahedi (1985) and Arnold and Zahedi (1988) called r(x, X) as the multivariate mean remaining life function. Here also it is to be noted that on the set {x:P[X > x] = 0} which is equivalent to the set {x:R(x,X) = 0}, the function may be defined in an arbitrary fashion. With the above definition of A(x, X) and r(x, X) the following two multivariate aging classes may be proposed. DEFINITION 6.4 (Johnson and Kotz, 1975; Marshall, 1975). A reliability function R (x, X) or the corresponding random vector X is said to be vector multivariate increasing (decreasing) hazard rate (VMIHR (VMDHR)) distribution provided for all x E IP,+ and x < b (lexicographically) ~q(x) is non-decreasing (non-increasing) in xj. DEFINITION 6.5 (Zahedi, 1985; Arnold and Zahedi, 1988). A reliability function R (x, X) or the corresponding random vector X is said to be decreasing (increasing) multivariate mean remaining life of type 2 (DMMRL (IMMRL)-2) distribution provided for all x ~ IR+ and x < b (lexicographically) rj (x) is nondecreasing (non-increasing) in xj. Mukherjee and Chatterjee (1991b) have considered the multivariate reliability function according to the equilibrium distribution of R(x, X) as

R(2)(x, X) = f~;,~) R( tj, x(i), X)dtj / fo,~) R( tj, x(i), X)dtj ,

(6.1)

An interpretation of R(2)(x,X ) may be available in Gupta and Sankaran (1998). However in line of Section 5 the ¢oncept of higher order equilibrium distribution in the multivariate set up may be introduced by generating a sequence of equilibrium distributions corresponding to R(x, X) for all positive integer p as follows:

R(p+~)(x,X)= /xj,~)R(p)(tj, x(j),X)dtj/ f[o,~)R(p)(tj, x(j),X)dtj .

(6.2)

It is interesting to note that as in the univariate case here also R(p+l)(x, X) represents a proper multivariate reliability function in the usual sense. With the above definition of R(2)(x, X) Mukherjee and Chatterjee (op. cit.) proved a theorem establishing a relationship between V M I H R (VMDHR)-ness of R(2) and D M M R L (IMMRL)-2-ness of R. The same result with few extensions, only in bivariate set up has been cited in Gupta and Sankaran (1998). However, a general version of those results involving R(p+I)(x, X) may be cited in the following theorem. THEOREM 6.1. R(p+l)(x, X) is V M I H R (VMDHR) iff R(p)(x,X) is D M M R L (IMMRL)-2.

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PROOF. Observe that for j = l(1)n, 2jR~+~)(x, X) = jth component of the hazard gradient of R(p+l) (x, X).

= -((~/~xj)R~+~l (,,, x))/R¢o+l/(x, x)

=-{-IR(p)(Xj,X(j),X)/f[o,~)R(p)(tj,x(j),X)dtjl}/ Ifxj,oo)R(p)(tj,x(j),X)dtj/~,~)R(p)(tj,x(j)'X)dtj1 = R(p)(x, X)/flxj,oo)R(p)(tj, x(j),X)dtj = 1/rjR(p)(x,X), rjR

where• (x, X) is the jth component of the multivariate mean remaining life (p functlon o~ R(p)(x, X). Hence, for j = l(1)n, on IP,+, 2jRe+~/(x, X) non-increasing (non-decreasing) in (x, X) non-decreasing (non-increasing) in The concept of hazard gradient dominance (a generalization of failure rate ordering as stated in Ross, 1983) and multivariate mean remaining life dominance (a generalization of mean remaining life ordering as stated in Gupta and Kirmani, 1987) between two non-negative random vectors X and Y has been introduced by Mukherjee and Chatterjee (1993). Moreover, the concept of multivariate stochastic dominance based on reliability function was discussed in Marshall and Olkin (1979). The definitions involving two non-negative random vectors X and Y are restated for the sake of continuity in discussion.

xj e=~rjR~,I

xj.

DEFINITION 6.6 (Mukherjee and Chatterjee, 1993). X is said to have hazard gradient dominance over Y (written as X >HG Y) provided for all x ~ IR+ and x A(x, X), where the inequality is valid lexicographically . DEFINITION 6.7 (Mukherjee and Chatterjee, 1993). X is said to have multivariate mean remaining life dominance over Y (written as X >MMRL Y) provided for all x E IR+ and x r(x, Y), where the inequality is valid lexicographically . DEFINITION 6.8 (Marshall and Olkin, 1979). X is said to have multivariate stochastic dominance over Y (written as X >x Y) provided for all x c IR+ and x R(,,, Y)

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With the introduction of higher-order equilibrium distribution in multivariate set up as stated in 6.2 the notions of hazard gradient vector and multivariare mean remaining life vector may be extended as follows, while X and b are as before. DEFINITION 6.9. Define the Borel measurable function on IR+ as Aco+l)(x , X) = (2(p+X)l(x,X), 2(p+l)2(x,X),.. ,2(p+un(x,X))', where for all x c IR+ and x x] = O} which is equivalent to the set {x : R(p) (x, X) = 0}, the function may be defined in an arbitrary fashion, DEFINITION 6.10. Define the Borel measurable function on IR+ as

r~+~)(, x) = (~~+1)~(", x), ~~+~)~(x, x ) , . . , r~+~)~(x, X))' for all x q IP.+ and x < b (lexicographically) , r(p+l)j(x, X) = ff[i.j,oc)R(p)(tj, x(j), X)dtj/R(p)(x, X) r(p+l) (x, X) may be called as the multivariate mean remaining life vector ofpthorder. Here also it is to be noted that on the set {x : P[X(p) > x] = 0} which is equivalent to the set {x:R(p)(X, X) = 0}, the function may be defined in an arbitrary fashion. With the above definition of R(p) (x, X), A(p)(x, X) and r(p) (x, X) in line of definitions of Section 4 the following multivariate dominance relations of pth-order may be proposed between two random vectors X and Y defined over IR+. Let b denote the common sos of the reliability functions corresponding to X and Y. DEFINITION 6.1 1. X is said to have pth-order hazard gradient dominance over Y (written as X >p-HG Y) provided for all x E IR+ and x A(p)(x, X), where the inequality is valid lexicographically .

DEFINITION 6.12. X is said to have pth-order multivariate mean remaining life dominance over Y (written as X >p-MMRL Y) provided for all x C IR+ and x _ r(p)(x, Y), where the inequality is valid lexicographically .

DEFINITION 6.13. X is said to have pth-order multivariate stochastic dominance over Y (written as X >p-x Y) provided for all x E IR+ and x (R(~)(x,Y)

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Before elaborating certain issues concerning the higher order multivariate equilibrium distribution, consider the random vector of the residual life beyond t as Xt(p) corresponding to X(p) the random vector having reliability function R(p) (x, X). Ler its reliability function be denoted by Rt(p)(x, X). Then

Rt(p>(x, x) -- P[x,(p) > x] = P[X(p) > x + tIX(p) > t] = R(p)(x + t, X)/R(p)(t, X) . Moreover higher-order representations of various aging properties as introduced by Buchanan and Singpurwalla (1977) may be defined as follows. DEFINITION 6.14. X is said to have pth-order multivariate increasing (decreasing) failure rate distribution - very strong (written X is p-MIFR (MDFR)-VS) provided for all x, t E IP,+ and x, t < b (lexicographically) R(p) (x + t, X)/R(P) (x, X) is non-increasing (non-decreasing) in x . DEFINITION 6.15. X is said to have pth-order multivariate new better (worse) than used distribution - very strong (written X is p-MNBU (MNWU)-VS) provided for all x, t E IP,+ and x, t R(p/(x + t, x) . The following theorems connect p-MIFR-VS and p-MNBU-VS with p-K in one hand and establish interrelations between p-HG, p - M M R L and p-K on the other. All the results as stated are generalizations of the results obtained in Mukherjee and Chatterjee (1993). THEOREM 6.2. (i) X is p-MIFR-VS provided for all x, y E IR+ and x < y ; « Xy(p) and vice versa. (ii) X is p-MNBU-VS provided for all y E IP,+ and y p-x Xy(p) and vice versa.

Proof. (i) Observe that for all x, y ~ IR,+ and x < y p-K Xy(p) ~ R(p)(x -~- t, X)/R(p)(x, X) > R(p)(y + t, X)/R(p)(y, X) ¢=~R(p)(x + t,X)/R(p)(x, X) is non-increasing in x ¢=~X is p-MIFR-VS (il) Observe that for all x, y E IR+ and x < y p-K Xy(p) ~ R(p)(x,X) ~_ R(p)(y ~- x, X)/R(p) (y, X)

R(p) (x, X)R(p)(y, X) > R(p)(y + x, X) ~=~X is p-MNBU-VS . Consider the residual life of X(p) beyond a random cut-off point Y having distribution function G(y, Y) and denoting it by Xv(p) with the corresponding reliability function Ry(p)(x,X) = P[X(p) > x + YIX(p) > Y]

= f(y

R@(x + y, X)dG(y, Y ) / E N+ and y
.~y

R(p) (y, X) dG(y, Y) . c IP-+ and y
THEOREM 6.3. X(p) >p K Xy(p) provided X is p-MIFR-VS. PROOF. Define the generic random variable Xv+t(p) with the reliability function Ry+t(p) (x, X) given by Ry+t(p) (x, X) = P[X(p) > x _u y + t]X(p) > Y + t]

= P[X(P) > x + Y + t]/P[X(p) > Y + t] .

(6.3)

It will be shown that if X is p-MIFR VS then Xt(p) >p-X Xy+t(p)Corlsider Rt(p)(x, X) - Ry+t(p)(x, X) = R(p)(x -~- t, X)/R(p)(t, X) - R(p)(Y -- x -It, X)/R(P)(Y + t, X) which has the same sign as

f,

[R(p)(x + t, X)R(P)(y + t, X) ' E N + and y
- R(p)(t, X)R(p)(y + x + t, X)]dG(y, Y)

(6.4)

assuming R(p)(t, X)R(p)(Y -4- t, X) = R(p)(t, X) f(y ~ N+ and y_ 0 , which means the integrand in (6.4) is non-negative. Thus Rt(p) (x» X) > Ry+t(p) (x, X) ensuring (6.3). Putting t = 0 in (6.3), it is quite natural that X(p) >p-K Xy(p). Notes: (i) The converse of the Theorem 6.3 is not true, although that of Theorem 6.2 is evidently valid.

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(ii) By reversing the dominance relations and inequalities and by replacing nonincreasing by non-decreasing the results corresponding to the dual versions of Theorems 6.2 and 6.3 may be derived. (iii) By substituting x* = x.1, y* = y.1 and t* = t.1 where 1 is an n-dimensional unit vector in place of x, y and t in the respective cases, the extended results concerning 'S', 'W' and 'VW' versions as referred in Buchanan and Singpurwalla (1977) can be easily obtained. THEOREM 6.4. For two random vectors X, Y the following statements are equivalent for x, t c IP-+ and x, t e-rm Y. (il) R(p)(x, X)/R(p)(x, Y) is non-decreasing in every component of x. (iii) Xt(p) >p-X Yt(p). PROOF. Observe that for x, t E IP,+ and x, t < b (lexicographically) and for all integer j = 1(1)n,

(6~fixe) ln(R(p)(x, X)/R(P)(x, Y)) = (R(p)(x, Y)/R(p)(x, X)R~p)(x, Y)) [R(p)(x, Y)(6/&•)R(p)(x, X)

- e(p / (x, x)(a/ax:)e(p I (,,, Y)]

= -~~(p/+(x, x) - ,~(p>+(x,Y) Thus (ii) ~

(3/&/) ln(R(p)(x, X)/R(P)(x, Y)) > O,

for all integer j = l(1)n ¢~ 2(p)/(x, Y) _> 2(p)/(x, X), for x e IP-+ and x R(p)(x + t, X)/R(p)(x + t, Y) _> R(p)(t, X)/R(p)(t, Y) (ii). THEOREM 6.5. (i) X >p-HG Y ~ X >v-K Y and the converse is not generally true, (ii) X >p-H~ Y ~ X >p-MMRL Y and the converse is not generally true. PROOF.

O) X >p-HG Y <=~for all x E IR+ and x A+)(~, X), where the inequality is valid lexicographically.

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Integration of both sides over the piecewise smooth path in [0, t] (cfs. Block, 1977; Galambos and Kotz, 1978; Mukherjee and Chatterjee, 1988) yields - lnR(p)(t, Y) _> - lnR(p)(t, X), assuring X >p-K Y •

(ii) By virtue of Theorem 6.4, for all x, t E IR+ and x, t p-HG Y ¢:> R(p)(X + t,X)/R(p)(x,X) > R(p)(X + t,Y)/R(p)(x,Y) .

(6.») Taking t = (tj, 0.t(j)) in particular, 6.5 reduces to

,e~/(xj + t» x~» x)/R~/(x, x) >_R~/% + t» x~»,v)/R~/(x, Y) Integrating both sides with respect to

/xj

tj over the

range [0, oc) yields

R(p)(yj'x(j)'X)dyj/R(p)(x'X) >- ~xj R(p)(yj, x(j),Y)dyj/R(p)(x,Y) ** r(p)j(x,X) >_ r(p)j(x,Y)

for all integer j = l(1)n .

The reverse implications are not valid since in even for n = 1 and p = 1, the corresponding implications are not true. For counter examples Gupta and Kirmani (1987) may be referred. It is to be observed that for n = 1, that is the univariate set up, p - H G e=~p-FR; p-K ~=~ p-ST and p - M M R L

¢:~ p - M R L .

7. Equilibrium distribution in the context of repairable system

For a repairable system, let {S(n) : n = 1 , 2 , . . } be the time of the nth failure of the system such that 0 = S(0) < S(1) < ... < S(n). DefineX(n) = S(n)-S(n - 1), n = 1 , 2 , . . , as the time between ( n - 1)th and nth failure. Thus S(n) will be a renewal process, corresponding to which let N(t) being a counting renewal process.

Define Xs(n_l),..,s(1)(n ) as the r.v. X(n) given S(n-1)=s(n-1),.., S(1) = s(1), where s ( 1 ) , .. ,s(n - 1) are real numbers, n = 2, 3 , . . , and for n = 1 this is trivially taken as X(1), indicating lifetime of the new system. Ebrahimi (1989) defined the counting process N(t) consisting of inter arrival times X(1), X ( 2 ) , . . . , is improving provided Xs(j-l),..,s(1) (J') >ST Xs(i-l),..,s(1)

(i), for

j>i>

every positive integer

1; 0 < s ( 1 ) < s ( 2 ) < . . . < s ( j - 1 )

.

Replacing ST by LR, FR, M R L and E, Deshpande and Singh (1995) and, by VRL and CX, Bagai and Jain (1994) extended the concepts of system improve-

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ment (deterioration). As p-FR and p-ST are more generalized version of usual order relations, replacing ST by p-FR and p-ST the definitions and results of Ebrahimi (1989), Deshpande and Singh (1995) and Bagai and Jain (1994) may be generalized and that has been attempted by Nanda (1997). It is obvious that putting p = 0, 1,2 and 3 the results corresponding to LR, FR, MRL and VRL follow. The concept and notion of minimal repair is of wide interest among reliability practitioners. For a repairable system, by minimal repair it is meant that the failed system is restored to a state of functioning which is as good as the condition just prior to failure. Thus, by definition, a minimal repair enables the system to work keeping the failure rate unchanged. Thus if the system fails at time 't' and undergoes minimal repair, the reliability function Rt(x) of the minimally repaired system is given by

Rt(x,X) = R(x + t,X(1))/R(t,X(1)) ,

(7.1)

where R ( , X ( 1 ) ) is the reliability function of X(1). Denote by N(t), the counting process giving the number of minimal repairs upto 't'. Let, for notational convenience, X(1) be Y~ and X~(,)......(1)(n + 1) be 11ù+1,for n = 1,2, 3 , . . . Yj's are assumed to be absolutely continuous non-negative random variables having reliability fnnction R(x, Yj). Considering the higher-order equilibrium distributions corresponding to R(x, Yj) as RCo) (x, Yj), for non-negative integers p, Nanda (1997) established the following relationship connecting R(p)(x,Y~+l) and

Rc~)(x, r~). RESULT 7.1.

R(p)(x, Y,+I) = [70(Y1)?l(Y1)...~p-l(Y1))/ Y0(Yn+I)YI (Yn+l).-- 7p-1 (Yn+l)l × [Rc~) (x + s(n), r,)/Rco ~(s(n), ~r,)] with 7-1(,) = 1. PRoov. The proof is simple and follows by induction on p. The following theorem gives the effect of aging of X(1) on the improvement (deterioration) of a system undergoing minimal repairs at failures and is proved in Nanda (1997). THEOREM 7.1. For any non-negative integerp, the stochastic process {N(t):t > 0} generated by a minimal repair policy is improving (deteriorating) in: (i) p-FR sense iff R(x,X(1)) is p-DFR (IFR), (ii) p-ST sense iff R(x,X(1)) is p-DFR (IFR). PROOF. Consider the proof only in the case of 'improving'. For °deterioration' the proof holds with obvious modifications.

Equilibrium distribution its role in reliability theory

133

(i) The system is improving in p-FR sense means for all n = 1 , 2 , . . , and nonnegative x,

R(p) (x, Yn+l)/R(p)(x, Yù) is non-decreasing in x,

Yn+l >p-FR Yn

R~,)(x + s(n), rl)/RC~~(x + ,(n -

1), Y1)

is non-decreasing in x by Result 7. 1. <=>

RCo)(x + t, Y1)/R(p)(X, Y1) is non-decreasing in x, t > 0.

¢:> Y1 is p-DFR .

(ii) The system is improving in p-ST sense means for all n = 1 , 2 , . . , and nonnegative x, Y~+I >p-ST Y~ g:} R(p)(X, Yn+l)/R(p)(O, Yn+l) ~ (R(p)(X, Yn)/R(p)(O, Yn)

¢:> Rcõ)(x + s(n), Y1)/R(p)(X + s(n - 1), Y1) >_R(p)(S(n), Y1)/R(p)(S(n - 1)Yl), by Result 7.1. R(p)(X + t, Y1)/R(p)(X, Y1) is non-decreasing in x, t > 0. *:> 111 is p-DFR . Notes: (i) The point process {N(t) : t _> 0} generated by a minimal repair policy is improving (deteriorating) in: (a) (b) (c) (d)

LR sense iffR(x,X(1)) is D L R (ILR), FR sense iffR(x,X(1)) is D F R (IFR), M R L sense iffR(x,X(1)) is I M R L (DMRL), VRL sense iffR(x,X(1)) is IVRL (DVRL).

The proof follows from the above theorem part (i) by putting p = 0, 1,2 and 3, respectively. It is to be observed that (a) (c) have been proved in Deshpande and Singh (1995) while (d) has been proved by Bagai and Jain (1994). (ii) The point process {N(t) : t > 0} generated by a minimal repair policy is improving (deteriorating) in: (a) W L R sense iffR(x,X(1)) is D L R (ILR), (b) ST sense iffR(x,X(1)) is D F R (IFR), (c) H A M R sense iff R(x,X(1)) is I M R L (DMRL). The proof follows from the above theorem part (ii) by putting p = 0, 1 and 2, respectively. As before here also it may be noted that (b) has been proved in Deshpande and Singh (1995).

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134

8. Characterization of exponential distribution Characterization of exponential distribution is abundant in the literature. Mukherjee and Chatterjee (1990) proposed some characterization resnlts based on integral equations involving RC°)(x,X). In what follows two of those results are restated in terms equilibrium distributions of order p i.e. R(p)(x,X), associated •with alternative and more elegant proof. THEOREM 8.1.

rR~+~)(x,X) = rR(p)(x,X) ¢* X has exponential density . PROOF. Observing rR(p)(x,X)=I/2R(p+l)(x,X), the above equality yields rR(p+l) (x,X) 2R(p+l)(x,X) = 1. This is the characterizing equation of R(p+l)(x,X) = exp(-x/c0, c~> 0. In view of Result 4.1

B

,oc)

R(p)(t,X)dt = (~(p+l)(X)/(p-- 1)p(p)(X)) exp(-x/~) .

Differentiating both sides with respect to x, yields R(p) ( x , X ) : (#(p+l)(X)//°:(P -}- 1)p(p)(X)) exp(-x/c~) . As RCo)(x,X ) is a proper reliability function, R(p)(O,X) = 1 gives ~ = #(p+l)(X)/ (p + 1)p(p)(X). Thus R(p)(x,X) = exp(-x/(p(p+l)(X)/(p + 1)p(p)(X))). The exponentiality of X follows immediately. THEOREM 8.2.

rR(p+i) (x,X)R(p+I) (x,X) = fR(p) (X, X)R(p) (x,X) OezX has exponential density . PROOF. By the same observation as in Theorem 8.1, the above equality yields rR(p+l) (x,X),~R(p+l)(x,X) = R(p)(x,X)/R(p+l ) (x, X ) = (~~l(x)/(p),~

l/(x));~~~+~/(~,x)

Thus ~~e+~/(~,x) = 1/x~~+~/(~,x)

=

~(~l(x)/~)~~

yielding

R(p+2)(x,X) = exp(-x/(#Co)(X)/(p)#(p_ U(X))) . The exponentiality of X follows immediately.

~)(x) ,

.

Equilibrium distribution - its role in reliability theory

135

Characterization of several non-exponential distribution e.g. Pareto, L o m a x and Finite Range has been attempted by several authors in univariate set up. In the bivariate set up similar characterizations have been attempted by G u p t a and Sankaran (1998). In fact they have attempted to extend the results to multivariate cases also. For details of the p r o o f and related articles reference may be made to G u p t a and Sankaran (1998).

9. Concluding remarks Although the present article tried to cover almost all important aspects of equilibrium distribution(s), it is in no way claimed to be exhaustive. There are several areas which could not be given due elaboration or attention so as to prevent the paper becoming unnecessarily lengthy. The following among others continue to remain open problems and deserve close attention in future. 1. To establish the closure or otherwise under other reliability operations (not considered earlier) of various dominance relations and aging classes based on equilibrium distributions of higher orders. 2. To establish various closure properties or otherwise under all reliability operations of dominance relations and aging classes based on equilibrium distributions in higher dimensions and of higher order. 3. To study whether FR, ST and M R L imply p - F R , p-ST and p - M R L (integer p > 1) respectively and if such relations hold, how to interpret them. 4. To solve appropriate inference problems mainly related to p - F R , pST and d(p) order relations on the one hand and p - I F R , p - I F R A and p - N B U on the other.

Aeknowledgement The authors are grateful to Professor A. P. Basu, Department of Statistics, University of Missouri - Columbia, USA and Professor J. V. Deshpande, Department of Statistics, University of Pune, India for m a n y helpful suggestions and comments towards preparation of this article.

Referenees Arnold, B. C. and H. Zahedi (1988). On multivariate mean remaining life functions. J. Multivariate Anal. 25, 1 9. Bagai, I. and K. Jain (1994). Improvement, deterioration and optimal replacement under age - replacement policy with minimal repair. IEEE Trans. Reliab. 43, 156 162. Block, H. W. (1977). Multivariate reliability classes. In Applications ofStatistics, pp. 79 88 (Ed. P. R. Krishnaiah). North-Holland, Amsterdam. Block, H. W. and T. H. Savits (1980). Laplace transforms for classes of life distributions. Ann. Prob. 8, 465-474. Boland, P. J., E. E1 Neweihi and F. Proschan (1994). Application of hazard rate orderings in reliability and order statistics. J. Appl. Prob. 31, 180-192.

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Buchanan, H. B. and N. D. Singpurwalla (1977). Some stochastic characterization of multivariate survival. In Theory andApplication ofReliability, Vol. 1, pp. 32%348 (Eds. C. P. Tsokos and I. N. Shimi). Academic Press, New York. Chatterjee, A. (1993). Unpublished Ph. D. Thesis, University of Calcutta, Calcutta, India. Cox, D. R. (1962). Renewal Theory. Methuen, London. Deshpande, J. V., S. C. Kochar and H. Singh (1986). Aspects of positive aging. J. Appl. Prob. 23, 748758. Deshpande, J. V., H. Singh, I. Bagai and K. Jain (1990). Some partiai orders describing positive ageing. Commun. Stat. - Stoch. Models 6, 471-481. Deshpande, J. V. and H. Singh (1995). Optimal replacement of improving and deteriorating repairable systems. IEEE Trans. Reliab. 44, 500-504. Ebrahimi, N. (1989). How to define system improvement and deterioration for a repairable system. IEEE Trans. Reliab. 38, 214-217. Fagiuoli, E. and F. Pellery (1993). New partiaI orderings and applications. Naval Res. Logist. 40, 829842. Fagiuoli, E. and F. Pellery (1994). Preservation of certain classes of life distributions under Poisson shock model. J. Appl. Prob. 31, 458~465. Fishburn, P. C. (1980). Stochastic dominance and moments of distributions. Math. Oper. Res. 5, 94100. Fisher, R. A. (1934). The effect of methods of ascertainment upon the estimation of frequencies. Ann. Eugenics 6, 13-25. Galambos, J. and S. Kotz (1978). Characterizations of probability distributions. In Lecture notes in Mathematics, Vol. 675 (Eds. A. Doldand and B. Eckmans). Springer, Berlin. Gupta, R. C. and S. N. U. A. Kirmani (1987). On order relations between reliability measures. Commun. Stat.: Stoch. models 3, 149 156. Gupta, R. P. and P. G. Sankaran (1998). Bivariate equilibrium distribution and its applications to reliability. Commun. S t a t . - Theory Meth. 27, 385-394. Johnson, N. L. and S. Kotz (1975). A vector multivariate hazard rate. J. Multivariate Anal. 5, 53-66. Erratum 498. Karlin, S. (1968). Total Positivity, Vols. I and II. Stanford University Press, CA. Kebir, Y. (1994). Laplace transform characterization of probabilistic orderings. Prob. Eng. Inf. Sei. 8, 69-77. Marshall, A. W. (1975). Some comments on the hazard gradient. Stoch. Process Appl. 3, 293-300. MarshalI, A. W. and I. Olkin (1979). Inequalities: Theory of Majorization and Applications. Academic Press, New York. Mukherjee, S. P. and A. Chatterjee (1988). A new MIFRA class of life distributions. Calcutta Stat. Assoc. Bull. 37, 67 80. Mukherjee, S. P. and A. Chatterjee (1990). Some characterization of exponential law through integral equations. In Proceedings of the Second Biennial Conference of the A llahabad Mathematical Society, Symposium on Statistics, 1990, pp. 61 67. Mukherjee, S. P. and A. Chatterjee (1991 a). On some closure properties of dominance relations based on reliability measures. Calcutta Stat. Assoc. Bull. 40 (H. K. Nandi Memorial Volume), 325 342. Mukherjee, S. P. and A. Chatterjee (1991b). On some properties of multivariate aging classes. Prob. Eng. Inf Sci. 5, 523 534. Mukherjee, S. P. and A. Chatterjee (1992a). Stochastic dominance of higher orders and its applications. Commun. Stat.: Theory Meth. 21, 1977 1986. Mukherjee, S. P. and A. Chatterjee (1992b). Closure under convoiution of dominance relations. Calcutta Stat. Assoc. Bull. 42, 251-254. Mukherjee, S. P. and A. Chatterjee (1993). Multivariate stochastic dominance with some implications. In Advances in Reliability, pp. 281-290 (Ed. A. P. Basu). Elsevier, Amsterdam. Nanda, A. K. (1995). Stochastic order in terms of Laplace transform. Caleutta Stat. Assoc. Bull. 45, 195-201.

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its role in reliability theory

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Nanda, A. K. (1997). On improvement and deterioration of a repairable system. IAPQR Trans. 22, 107-113. Nanda, A. K., K. Jain and H. Singh (1996a). On closure of some partial orderings under mixtures. J. Appl. Prob. 33, 698 706. Nanda, A. K., K. Jain and H. Singh (1996b). Properties of moments for s-order equilibrium distributions. J. Appl. Prob. 33, 1108 1111. O'Brien, G. L. (1984). Stochastic dominance and moment inequalities. Math. Oper. Res. 9, 475 477. Rao, C. R. (1965). On discrete distributions arising out of methods of ascertainment. In Classical and Contagious Diserete Distributions, pp. 320-332 (Ed. G. P. Patil). Pergamon Press and Statistical Publishing Society, Calcutta. Ross, S. M. (1983). Stochastic Process. Wiley, New York. Shaked, M. and J. G. Shantikumar (1994). Stochastie Orders and Their Applieations. Academic Press, New York. Singh, H. (1989). On partial orderings of life distributions. Naval Res. Logist. 36, 103 110. Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. (Ed. D. J. Daley). Wiley, New York. Van Zwet, W. R. (1964). Convex Transformations of Random variables. Mathematisch Centrum Amsterdam, Amsterdam. Vinogradov, O. P. (1973). The definition of distribution function with increasing hazard rate in terms of the Laplace transforms. Theory Prob. Appl. 18, 811-814. Whitt, W. (1985). The renewal process stationary excess operator. J. Appl. Prob. 22, 156-157. Zahedi, H. (1985). Some new classes of multivariate survival functions. J. Stat. Plann. Infer. 11, 171 188. Zygmund, A. (1959). Trignometric Series, Vol II. Cambridge University Press, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

Reliability and Hazard Based on Finite Mixture Models

Essam K. AL-Hussaini and Khalaf S. Sultan

1. Introduction

In recent years reliability has been formulated as the science of predicting, estimating or optimizing the probability of survival, the mean life, or more generally the life distribution of components or systems. Interest in reliability has been manifested by mathematicians, engineers, medical scientists, economists and those concerned with the environmental and life sciences. Such developments and interest have been the interplay between reliability and statistics. Thus, the theory of statistical inference plays an important role in life testing and reliability problems. This has led to many published papers and books, such as, Barlow and Proschan (1965, 1975, 1981), Nelson (1982) and Lawless (1982). Recently, the book edited by Balakrishnan (1995) surveyed different problems in reliability and life testing, for example: (i) Parametric models and inference, see Doganaksoy (1995), Nelson and Doganaksoy (1995), Balakrishnan and Chan (1995a, b), Smith (1995), Kulasekera and Nelson (1995), Johnson et al. (1995), Pefia (1995) and Bandyopadhyay and Basu (1995), (ii) Survival analysis and multivariate models, see Arnold (1995), Anderson (1995) and Ebrahimi (1995). Also, the book by Elsayed (1996) emphasized more recent developments in engineering reliability. Statistical models play an important role in the field of reliability, for example justification for using the exponential assumption in reliability applications can be found in the early works of Davis (1952), and Epstein and Sobel (1953). Further justification, in the form of theoretical arguments supporting the use of the exponential distribution as the failure law of complex equipments, can be found in books by Barlow and Proschan (1965, 1975, 1981). Recent studies in reliability estimation and applications based on exponential distribution have been given in Engelhardt (1995). Some properties of distributions with periodic failure rates are discussed by Prakasa Rao (1997). The class of Inverse Gaussian distributions is currently challenging and is being used quite commonly as a lifetime model in reliability studies. Because of this, several authors have paid special attention to inferential issues concerning these distributions. References may be found in books by Chhikara and Folks (1989) and Seshadri (1993, 1999). In this connec139

I40

E. K. AL-Hussaini and K. S. Sultan

tion, the book edited by Balakrishnan and Chen (1997) presents extensive discussions and tables for order statistics from these distributions. AL-Hussaini (1999a) obtained Bayesian prediction bounds for observables from a general class of life distributions which includes, among others, the Weibull, compound Weibull (or three-parameter Burr type XII), Pareto, beta, Gompertz and compound Gompertz distributions. He suggested a general class of priors and based his prediction on the one- and two-sample cases. Bayesian prediction bounds of the median of future observations from this general class of distributions have been investigated by AL-Hussaini (200la). The study of homogeneous populations with 'single component' distributions was the main concern of statisticians along history, although Newcomb (1886) and Pearson (1894) were two pioneers who approached heterogeneous populations with 'finite mixture distributions'. Newcomb (1886) applied the normal mixture as models for outliers, whereas Pearson (1894) used the method of moments to estimate the parameters of a finite mixture of two univariate normal components with unequal variances. With the advent of computing facilities, the study of heterogeneous populations, which is the case with many real world populations, attracted the interest of several researchers from about the middle of the 20th century. Titterington et al. (1985) classified the applications of finite mixture models to direct and indirect applications. Direct applications cover fisheries research (lengths of different types of fish, etc.), economics (wage and housing regressions, number of purchases, etc.), medicine (clinical test scores, age of onset of arthiritis, teeth pressures, cholesterol triglyceride data, chromosome association, plasma glucose level, blood pressure, clinical measurements, etc.), geology (palaentology: foraminifer diameters, Egyptian mandible sizes, sedimentology: reflectivity (caol types), limestone cross-bedding, sand grain sizes, metal concentrations, volcanic ash deposits, etc.), electrophorisis (absorption spectrum, protein concentrations, gas chromatograms, etc.), botany (pollent grains, flowering times, plant heights, etc.), agriculture (crop concentrations, barley yields, panicle lengths, etc.), zoology (bird speeds, egg counts, rat species skull sizes, dental distances, etc.), engineering (valve lifetimes, transmitter receiver lifetimes, bus failure times, telephone call lengths, laser lifetimes, reliability, etc.). Indirect applications cover the contaminated normal models, outliers and robustness, normal mixtures as checks on robustness (for example, see Andrews et al., 1972; David, 1981; Huber, 1981; Tiku et al., 1986; Arnold and Balakrishnan, 1989; Gastwirth and Cohen, 1970; Balakrishnan and Geyer, 1995; and Barnett and Lewis, 1993), Gaussian sums, cluster analysis (see, for example, McLachlan and Basford, 1988), modeling prior densities, the use in the empirical Bayes methods, kernel density estimation, approximations of mixture rnodels by non-mixture models, random variate generation, among others. Four books have been written on mixtures: Everitt and Hand (1981), Titterington et al. (1985), McLachlan and Basford (1988) and Lindsay (1995). The first book concentrated on the estimation of the parameters of finite mixtures of normal, exponential and other continuous components and finite mixtures of discrete components, using the graphical, moment and maximum likelihood

Reliability and hazard based on finite mixture models

141

methods. The second listed and discussed real world applications of finite mixtures, the methods of estimation of their parameters and sequential procedures. The third also discussed mixtures with normal components and estimation of parameters using the EM algorithm; see McLachlan and Krishnan (1997). Applications of mixture models to the two-way data sets were discussed, estimation of the mixing proportions and the assessment of mixture likelihood approach to clustering of the three-way data. The fourth book discussed, in addition to other topics, the non-parametric and semiparametric maximum likelihood estimation under mixture models. Estimation of the parameters based on finite mixture models were studied, mainly when the components belong to the same family. For example, the parameters of a mixture of normals were estimated by Pearson (1894), Day (1969) and Quandt and Ramsey (1978), among others. The parameters of a mixture of Inverse Gaussian components were estimated by Ahmad (1982) and Amoh (1983). Kao (1959) used the graphical method in estimating the parameters of a finite mixture of two Weibull components. AL-Hussaini and Ahmad (1984) obtained the information matrix for a mixture of two Inverse Gaussian distributions. ALHussaini and Fakhry (1995) characterized a finite mixture of a general class of k components that are used in life testing. AL-Hussaini and Osman (1997) obtained the median of a finite mixture of k components. AL-Hussaini et al. (1997) used parametric and non-parametric methods to estimate the reliability function P(Y < X) when each of X and Y is a finite mixture of lognormal components. AL-Hussaini et al. (1999) studied the finite mixture of two-component Gompertz lifetime model. AL-Hussaini (1999b) used the Bayesian method to predict observables under a mixture of two-exponential components model. Predictive densities of order statistics based on finite mixture models with general class components have been obtained by AL-Hussaini (2001b). Certain practical situations require finite mixtures with components that do not necessarily belong to the same family. For example, Davis (1952) investigated a normal-exponential mixed model, Mudhlokar et al. (1995) applied the exponential-Weibull model to the bus-motor-failure data presented in Davis (1952), Ashton (1971) applied the 'modified semi-Poisson' model to distributions for gaps in road traffic, Ashour (1985, 1987) studied the Weibull-exponential mixed model and ALHussaini and Abd-EL-Hakim (1989, 1990, 1992) the Inverse Gaussian Weibull model. Teicher (1960, 1961, 1963, 1967) introduced the concept of identifiability and developed a theory to identify mixtures. Other researchers who studied identifiability of mixtures, countable or finite mixtures are Medgyessy (1961), BarndorffNielson (1965), Patil and Bildikar (1966), Yokowitz and Spragins (1968), Tallis (1969), Rennie (1972), Chandra (1977), AL-Hussaini and Ahmad (1981), Ahmad and AL-Hussaini (1982), Tallis and Chesson (1982) and Ahmad (1988). Other references may be found in Titterington et al. (1985), McLachlan and Basford (1988) and Maritz and Lwin (1989).

E. K. AL-Hussaini and K. S. Sultan

142

2. Concepts, notation and definitions In this section, we introduce, some basic concepts, notation and definitions. These include reliability function, hazard rate function, mean residual lifetime and mixture of distributions. The instantaneous rate of failure or death at time t, given that an individual survives up till t is known in demography and actuarial science as the force of mortality. In the theory of reliability, it is called hazard rate, failure rate or agespecific failure rate. In extreme-value theory, it is known as the intensity rate and its reciprocal is termed Mills Ratio in economics, see, for example, Mann et al. (1974), Kalbfleisch and Prentice (1980), Elandt-Johnson and Johnson (1980), Nelson (1982) and Lawless (1982). We shall use the term hazarfl rate funetion (HRF), denoted by r(t) throughout this chapter. If T is a a non-negative random variable which has a (continuous) cumulative distribution function (cd0 F(t) and a corresponding probability density function (pdf) f(t), then the H R F is given, for t > 0, by

r(t) = lim ( P[t <- T <- & t + &'T >-

=f(t)R((O '

(2.l)

where

R(t) = P[T > t] = 1 - F(t) ,

(2.2)

is known as the reliability (survivor) function (RF). Knowing any of the density, reliability or HRFs leads to the knowledge of the other two by using the relationships

R(t) = e x p [ - fotr(s)ds]

,

(2.3)

1 d

r(t) -

R(t) dt [R(t)] ,

f(t) = -R'(t)

.

(2.4) (2.5)

Examination of (2.3) indicates that r(t) is a non-negative function with fö r(s)ds < ec, for some u > 0 and f ö r(s)ds = oc. The H R F describes the way in which the instantaneous probability of 'death' for an individual changes with time. The function takes several shapes such as (monotone) increasing (IHR), decreasing (DHR), bathtub (BTHR) or other shapes. Models with I H R are used the most as they represent aging (wear out) in time. Information about the nature of the H R F is helpful in selecting the model. For detailed discussion of the HRF, see, for example, Lawless (1982). The expected value of the random variable T is known, in engineering, as the mean time to failure (MTTF), see, for example, Billinton and Allan (1983). A cdfF(t) is said to have an increasing hazard rate average (IHRA), i f - (1/t) log[R(t)] is increasing for t > 0. Reference may also be made to Shaked and Shanthikumar (1994) and

143

Reliability and hazard based on finite mixture models

Chapter 33 in Johnson et al. (1995) for elaborate discussions on these reliability functions and their inter-relationships. A closely related concept is the mean residual lifetime (MRLT) which is defined, at time t, to be

m(t) = E [ T - t l T > t ] ,

t>O

F

1 (T - t ) f ( T ) d T R(t) oo ft+T 7 f0 exp [-- Jt r(x)dx]dT.

(2.6)

It can be shown that the M R L T is related to the RF by the following relation:

~S

ra(t) =

(2.7)

R(x)dx .

The RF is related to the M R L T by the relation

m(0)

F

p dx ]

R(t) = ~ t ( t ) - e x p [ - j 0 m~ßJ '

(2.8)

Furthermore, it can be shown that lim m(t) = }im

t~OO

(

- d td logf(t)]

)

(2.9)

The M R L T at time t, is a measure of the expected remaining life for an individual who survived up till time t. The above concepts extend naturally to more than one variable. For example, the multivariate survival function is defined by R(t_) = P[T1 >_ t l , . . . , Tn >_ tn],

ti > O, i = 1 , 2 , . . ,

n ,

where t = (tl, t2,..,tn). Johnson and Kotz (1975) (see also Kotz et al., 2000) defined a hazard rate vector r(t) by (2.10)

r(t) = (rl (t), .., r~ (t)),

where ô rj(t) = - ~ j ( l o g [ R j ( t ) ] ) ,

j= 1,2,..,n

.

Suppose that F(tlO ) represents a cdf of a random variable (vector) T given that O = 0 and that G(O) represents the cdf of the random variable (vector) O. The function H(t), defined by H(t) =-

Y

F(tlO)dG(O ) ,

oo

(2.11)

144

E. K. AL-Hussaini and K. S. Sultan

(which is the marginal cdf of T) was called by Fisher (1936) compound distribution of F and G. Teicher (1960) called H a mixtnre of F and G. F(tlO ) is known as the kernel and G the mixing distribution. If the entire mass of the corresponding measure of G is confined to a countable number of points 0 ~ , 0 2 , . . and the masses at Oj Ü = 1 , 2 , . . ) are G(@, then (2.11) takes the form (30

H(t) = Z F(t]Oj) G(0y) .

(2.12)

j=l

In this case H(t) is a countable mixture cdf. The noncentral X2 distribution is an example of a countable mixture of Poisson and central Z2 distributions. If the entire mass of the corresponding measure of G is confined to only a finite number of points 0 1 , 0 2 , . . , 0k, then (2.11) becomes a finite mixtnre of k components, whose cdf is given by k

H(t) = Z F(tlOy)G(Oy) .

(2.13)

j=l

To simplify notation, write pj

=_ G(Oj) and/~)(t) = F(t IOj), so that (2.12) becomes

(2<3

H(t) = Z p y F y ( t ) ,

(2.14)

j=l

wherepj _> 0, j = 1 , 2 , . . and ~pj j=l

=

o(oj) = 1 . j=l

Also, (2.13) becomes k

H(t) = ~ p j ~ ( t )

,

(2.1»)

j~l

where pj _> 0, j =

1 , 2 , . . , k and

k

~pj j=l

k

: ~

G(oj>:

1.

j=I

In (2.14) and (2.15), pj is known as the j t h mixing proportion and Fj(t) the jth eomponent in the mixture. In (2.15), k represents the number of components. It may be noticed that the choice Fj.(t) _= F(tlOj) restricts the cdfF(tlOj) for all values of j, to belong to the same family of distributions. However, forms (2.14) and (2.15) are written in the most general forms in which each of the cdf's Fy(t) could belong to a distinct family. The only requirement here is that, for any j, Fy(t) is a cdf.

Reliability and hazard based on finite mixture models

145

Whether T is continuous or discrete, we shall write f(t]O) and h(t) to represent the conditional and marginal density (or mass) functions, respectively. If, in (2.11), G(O) is absolutely continuous, a pdf O(O) exists such that g(O) = G~(O) and if h(t) and f(t]O) are the p d f s corresponding to the cdfs H(t) and F(tlO), then from (2.11), we have

h(t) =

/?

f(t]O)g(O)dO .

(2.16)

O0

Similarly, corresponding expressions to (2.14) and (2.15) are given by OO

h(t) = Z p j J ) ( t )

(2.17)

j=l

and k

h(t) = ~pjfs.(t)

,

(2.18)

j--1

where J)(t) is the j t h component density (or mass) function corresponding to the cdf Fj(t). Barlow and Proschan (1981) defined the hazard transform of the mixture (2.11) as t/(u) = - log

F

e -"° dG(0) ,

OQ

where the vector u has elements uo, 0 < uo < oc, - o c < 0 < ec. They also showed that the hazard transform of a mixture is concave and that the mixture H(t), given by (2.11) is decreasing hazard rate D H R if each F(t I0) is D H R . It is decreasing hazard rate average D H R A if each F(t] O) is D H R A . They remarked that mixtures of I H R (IHRA) distributions are not necessarily I H R (IHRA). Ler

~n,,n = {F(tlO): t C IR", 0 C IR~} , be a family of cdfs and = { H : H(t) = f

JN 7

F(tlO)dG(O), G E N}

be a class of mixture distributions H generated by ~-~,m, where n and m a r e integers _> 1, IR~' is a Borel subset of Euclidean m-space IRm, F(t]O) is measurable in IR" × IR~' and .~ is the class of n-dimensional distributions G whose induced measures ga assign measure one to IR~. The class H of mixtures of ~-ù,m is called identifinble, according to Teicher (1961), if the mapping of N onto H is one-toone. Simply, G c N is said to be identifiable in the mixture H C ~ if a unique solution G of the integral in (2.11) can be found. Any H C ~ is termed a

146

E. K. AL-Hussaini and K. S. Sultan

countable (finite) mixture if the entire mass of the corresponding #c is confined to a countable (finite) set of points in IP,T. Identifiability of finite mixtures requires that k

k*

Ep~~I,I = EpT;(tl, j=i

v,,

j-1

implies that k = k*, pj = p* (j) and Fj = Fi*(/), for j = 1 , 2 , . . , k and some permutation i on 1 , 2 , . . . , k. Obviously, the identifiability question should be settled before one can meaningfully discuss the problem of estimating the mixing cdf G on the basis of observations from the mixture H. So estimation of the mixing distribution, testing hypotheses about G, etc., can be meaningfully discnssed only if the family of mixing distributions is identifiable. Several methods are used in estimating the RF of a mixture. For a given t, the RF is a function of the parameters. So, estimating the R F amounts to estimating a function of the parameters, in some methods. In others, estimating the RF does not involve a direct estimation of the parameters. The graphical method, the methods of moments, moment generating function, ML, Bayesian, minimum distance and the numerical decomposition of mixtures have been used. For details, see, for example, Titterington et al. (1985). The reliability function R(t) corresponding to a finite mixture of k components is given by k

R(t) = E p j R j ( t ) j=l

,

(2.19)

where R j ( t ) , j = 1, 2 , . . , k, is the RF corresponding to the j t h component in the mixture. According to the invariance principle, the MLE of a function of the parameters (RF or H R F for a given t) is the function of MLEs of the parameters. So, to obtain the MLEs of an RF or H R F , we simply replace the parameters by their MLEs. The problem then becomes finding the MLEs of the parameters of the mixture. For the finite mixture (2,18), with k components, the H R F is given by

r(t) - h(t) R(t)

(2.20)

'

where h(t) and R(t) are given, respectively, by (2.18) and (2.19). In this chapter, reliability, hazard and mean residual lifetime functions for some finite mixture models will be considered. It can be shown that in the case of only two components (i.e., k = 2), the H R F and M R L T of a mixture may be written in terms of the H R F s and MRLTs of the two components as follows:

r(t) = A(t)rl (t) + [1 - A(t)]r2(t) ,

(2.21)

m(t) = A(t)ml (t) + [1 - A(t)]m2(t) ,

(2,22)

ReIiability and hazard based on finite mixture models

147

where

A(t)

= p R 1 (t) /~gR1

B(t) = (1

- -

(t)

-~-

(1 - p)R2(t)] = [1 + B(t)] -1 ,

p)R2(t)/[pRl(t)]

(2.23)

.

So that

r'(t) : A(t)r' 1(t) + [1 -

A(t)]/2(t ) -

A(t)[1 - A(t)l[rl (t)

r2(t)] 2

-

(2.24)

m'(t) = A(t)m'~ (t) + I1 - A(t)]m'2(t ) - A(t)[1 - A(t)] [ml (t)

-

m2(t)]

2

.

(2.25) By observing that A(t) and 1 - A ( t ) assume values in the interval [0,1], for all t, it follows from (2.24) that if rj(t) < 0, for all t, j = 1,2, then r'(t) < 0, for all t. Therefore, a mixture with D H R components has DHR. However, if the components have IHRs their mixture need not have IHR. It may also be remarked that in the case of identical components (having identical parameters), the mixture reduces to the single component case. The same interpretation can be drawn for ra(t). Expressions (2.21), (2.22), (2.24) and (2.25) shall be used later. Since this chapter will be devoted to only finite mixture models, we present two groups of such models according to whether the components belong to the same family or to different families of distributions.

3. Models with components belonging to the same family Seven finite mixture models in which the components belong to the same family of distributions shall be introduced in this section.

3.1. Mixtures of normal components If, in (2.18), aä(t) is the N(]1:, a 2) density funetion, given, for j = 1 , . . ,k, by B(x) -

1 exp v~~crj

-2\

«: / - - O0 < X < O0,

--00<]12<00

, lYj )

O ,

(3.1)

then (2.18), represents a mixture of k normal components. Finite mixtures of normal components have been discussed by several authors, such as Behboodian (1970, 1972), Quandt and Ramsey (1978), Everitt and Hand (1981), Titterington et al. (1985) and McLachlan and Basford (1988). Teicher (1963) had shown that a finite mixture of normal components is identifiable. The reliability function corresponding to the jth component, j = 1 , . , k, is given by

148

E. K. AL-Hussaini and K. S. Sultan

R,(,) = 1 - ~('-

\«j/

~J~

(3.2)

and the HRF r(t) is given by (2.20), where for j = 1 , . . , k, J)(t) and Rj(t) are given by (3.1) and (3.2), respectively. Everitt and Hand (1981) surveyed finite mixtures of univariate and multivariate normal components with a good coverage of historic review starting with Pearson (1894). Descriptive properties of mixtures of normal components have been reviewed with emphasis on the conditions under which a mixture of two normal components is unimodal or bimodal. Estimation of the parameters of a mixture of two normal components using the method of moments as suggested by Pearson (1894) has been presented; see also Chapter 13 of Johnson et al. (1994). The five resulting equations can be reduced to a nonic equation. Everitt and Hand (1981) described the methods available for finding the required roots of the ninth-degree polynomial. Day (1969) studied moment estimation for multivariate normal mixtures with two components assuming equality of vatiance-covariance matrices of the two components densities. He showed, by means of simulation, that only in the univariate case the moment estimators behave almost as well as the MLEs. It may be advisable, therefore, not to use the method of moments in the higher dimensions (than one), as the moment estimators, in such cases, perform very poorly. For a discussion on mixtures of multivariate normal and associated estimation problems, see Kotz et al. (2000), Everitt and Hand (1981) also reviewed the MLEs in detail, for ungrouped and grouped data, in the univariate and multivariate cases. The EM algorithm of Dempster et al. (1977) is used, among other algorithms to obtain the MLEs. Hathaway (1985) warns that the method of ML leads to an ill-posed optimization problem in the case of a mixture of normal distributions. A problem associated with MLE arises from the unboundedness of the likelihood function LF (Day, 1969). A global MLE always fails to exist. The unboundedness of LF causes failures of the EM (Redner and Walker, 1984) as well as the quasiNewton Raphson (Fowlks, 1979) algorithms. Spurious maximizers corresponding to parameter points having same component standard deviations very small relative to others, can create, like the unboundedness of the LF, difficulty when using the EM or quasi-Newton-Raphson algorithms. Hathaway (1985) proposed the use of the simple constraints: m i n i j { a i / « j } > c > 0 where ai, i = 1 , . . , k is the standard deviation of the ith component, c is a real number to be determined. He proved that the suggested constraints have a strongly, consistent, global solution. The graphical method as well as other estimation methods as localizing the different components of a normal mixture using Fourier transformation methods, the moment generating function, minimum )~2 and stochastic approximation methods have also been reviewed by Everitt and Hand (1981). They made their own comparisons between some of the estimation methods of the parameters of two (and more) univariate normal components in the mixture. Kocherlakota and Balakrishnan (1986) studied the effects of a mixture of two normal components:

Reliability and hazard based on finite mixture models

149

N(#, te) and N(aa + #, «2) on the tolerance limits which control percentages in both tails. More recently, Roeder and Wasserman (1997) presented some practical methods for coping with the problems associated with the use of mixtures of normals for estimating densities in Bayesian framework. Magder and Zeger (1996) proposed a method of estimating mixing distribution using ML over a class of arbitrary mixtures of normals subject to the constraint that the component variances be greater than or equal to some minimum value h. They showed that the resulting estimate will always be a finite mixture of normals, each having variance h and that the non-parametric M L E can be viewed as a special case with h=0. Suppose that 0 is the vector of parameters involved in a finite mixture of k N(#j,a}) components, (j" = 1 , . . ,k). Then 0 is a (3k-1)-dimensional vector. Suppose that 0 is an estimator of 0 obtained by some reliable method of estimation, this estimator may be used to estimate the RF. Then, for a given t, Rj(t) is given by Rj(t) = 1 - ~[(t - ~j)/«s]

,

(3.3)

where ~b(ff) is the standard normal integral at ~, the RF of the mixture R(t), is then given by k

R(t) = Zt~jRj(t) .

(3.4)

j=l

The H R F of a mixture of k components is given by (2.20), where h(t) is given by (2.18) and R(t) by (2.19). In those expressions, fj(t) and Rj(t) are given, for j = 1 , 2 , . . , k by (3.1) and (3.2), respectively. A mixture of two normal components has a H R F of the form (2.21), where A(t) is as given by (2.22) and the H R F of the j t h component by rj(t) = fj(t)/Rj(t), j = 1,2, where fj.(t) and Rj(t) are given, respectively, by (3.1) and (3.2). It should be remarked that in reliability studies the random variable T represents times and so must be non-negative. It then follows that the normal distribution is not an acceptable description of failure times unless f°o~fj.(t)dt is negligible quantity, say # > 3«. If, however, # < 3fr, a truncated normal distribution could be used (see Davis, 1952). Figures l(a) and (b) show the graphs of pdfs and HRFs of two components of normal and their mixture for some value of the vector of parameters 0 = (p, # , # » «~, «~).

3.2. Mixtures of lognormal components If, in (2.18), j0(t) is the lognormal density function given, for j = 1 , . . ,k, by

fJ(t) = ~ a j t e x p { - l

[(logt- #j)/~j]2}, t > 0 , - o c < #j < oc, crj > 0 ,

(3.5)

150

E. K. AL-Hussaini and K. S. Sultan

(a)

q

2.1

'

h(i)[

......

fl (t) I

"

f2(t)

'

."

"-.

1.6

n 1.1

0.6

0.1 0.0

0.5

1.0

1.5

t (b) 13 ........ .......

rlr(t)(t) I r2(t)

ù"

ù . ..... -'''"""

L

0.0

1.o

0.5

1.5

t Fig. 1. (a) Density functions: normal components and their mixture 0=(0.5, 1.0, 2.0, 0.04, 0.64). (b) HRFs: normal componentsand their mixture 0=(0.5, 1.0, 2.0, 0.04, 0.64).

then, (2.18) represents a mixture of k lognormal components. Many papers describe various applications of finite mixtures of lognormal distributions in different fields. For example, (i) Atmospheric science, see Patterson and Gillete (1977), Whitby (1978), Larsen (1969, 1977), Neiburger and Weickman (1974), Ott (1978), Tsukatani (1981), Gilliom and Helsel (1986); (ii) Geology, see Lepeltier (1969); (iii) Length of telephone calls, see Fowlkes (1979); (iv) Stress-strength reliability, see AL-Hussaini et al. (1997), for some other applications see Crow and Shimizu (1988). Ahmad (1988) had shown that a finite mixture of lognormal components is identifiable.

Reliability and hazard based on finite mixture models

151

The reliability function of a finite mixture of k components is given by (2.19), where, for j = 1 , . . , k

Rj(t) = l _ 4~(!og __t- lzJ) «/

(3.6)

and the H R F r(t) is given by (2.20), where j)(t) and Rj(t) are given for j = 1 , . . ,k, by (3.5) and (3.6), respectively. Aitchison and Brown (1966) and Crow and Shimizu (1988) have provided book-length accounts of the lognormal distribution. Johnson et al. (1994) devoted Chapter 14 to this same distribution. The history of the distribution, estimation of the parameters, and applications are described there. Sultan (1992) studied the estimation of the vector of five parameters of two lognormal components whose density function is of the form (2.18) with k = 2. He obtained the MLEs of the parameters based on Type I and Type II censored samples. He suggested using mixtures of two lognormal components, with parameters (/~j,er/), denoted by A(/~/, «2), as a lifetime model. The MLEs of the vector of parameters and hence, the M L E of the RF of the mixture are obtained as in (2.19), where Rj(t) = 1 - ~[(log t - [tj)/ä2], for a given t and j = 1,2. AL-Hussaini et al. (1997) estimated the stress-strength reliability R ~ P ( Y < X ) in both the parameteric and non-parameteric cases, when the random variables X and ]( are independent and each of which is a mixture of two lognormal components. Suppose that X and Y are two independent continuous random variables whose density functions hx(x) and hy(y) are given, respectively, by the finite mixtures kl

(3.7)

hx(x) = ~ p i f i ( x ) i=1

and hr(y) = ~ q / I j ( Y )

,

(3.8)

j=l

where Pi and q/are mixing proportions, J}(x) and 9/(Y) are density functions on the line. Since X and Y are assumed to be independent, then

R =

hy(y)hx(x)dy dx = oo

co

piqjRi« , j=l

(3.9)

j=l

where for i = 1 , . . , k l , j = 1 , . . , k z ,

Ri/=

/[ OO

gj(y)f~(x)dy dx . CO

(3.10)

152

E. K. AL-Hussaini and K. S. Sultan

If, in particular J~(x) = A(#j, a2) and gj(y) = A(vj, 7~), i = 1 , . . , kl, j = 1..., k2, then kl

k2

R=~

~piq/b(6ij)

,

(3.11)

i=1 j=I

where a« = (#, - v j ) / ~ 2 ~ 2 + >2 . By restricting the number to two components each, 2 2 and ~2= (q~,v1,v2,71,72), then follows that 1 0 0 ( 1 by: 2

RL = ~

(3.12)

of components in the mixed densities (3.7) and (3.8) (i.e. kl = k2 = 2), the MLEs of O = COl,#l, #2, a2, «2) are obtained by the use of the EM algorithm. It e)% approximate confidence bounds of R a r e given

2

Œ»~0«efa«(L)] ,

(3.13)

i=1 j=l 2 RU = Z

2 ~-~~ßiß ~D[(~Ü(U)] »

(3.14)

i=I j=l

where /

at(L) : a« -

a X I/2

z~/2 ~ 2;i;]

(3.15)

and (3.16)

/

^2

\mi

^2

nj] (3.17)

\mi-

1 + nj- U

B« is as given by (3.12) after replacing the parameters by their estimates, m i = M]ô i and nj = NOj, M and N are the (known) sizes of the samples drawn from the mixed densities hx(x) and hy(y), l~i, qj are the MLEs ofpi and qj, and Z~/2 is the upper c~/2 percentage point of the standard normal distribution.

153

Reliability and hazard based on finite mixture models

In the non-parametric case, AL-Hussaini et dl. (1997) computed the Birnbaum-McCarty (1958) and Govindarajulu (1968, 1976) confidence bounds of R = P ( Y < X ) . Such bounds (denoted by BM and G) are given, respectively, by /):TeBM

and

/)TeG ,

(3.18)

where eßM = f i / x / ~ + N , fi depends on the confidence coefficient 1-c~ and K = M/(M+N), which can be obtained from a table in Owen et dl. (1964), eo _> (4L)-1/2~-1(1- et/2), L = min{M,N}, and ~(.) is the standard normal integral. AL-Hussaini et dl. (1997) compared the point and interval estimates of R in the parametric and non-parametric cases, for different sample sizes and different populations via simulation study. The H R F of a mixture of two lognormal components model can be written as in (2.21), where rj(t) and R j ( t ) , j = 1,2 are, respectively, the corresponding jth hazard rate and reliability functions in the mixture model. By using the expression of r(t) given by (2.21), Sultan (1992) showed that the failure rate of the mixture of two lognormal components increases on the interval (0, tmj) and decreases on (tmj, oc) reaching the value zero as t -+ oc, where lm« is the point at which r j ( t ) , j = 1,2, attains its maximum. The H F R curve also takes the upside down bathtub shape when p is close to the boundaries (0 or 1). The M R L T corresponding to the jth lognormal component is given by mj(t) =

exp(~/+ a2/2)[1 - dB(Sj(t))] 1 - (B(cSj(t)) - t ,

(3.19)

where Sj(t) = log t - (#j + a 2) , aj

and

6«(t) _ log t - #j crj

(3.20)

Figures 2(a) and (b) show the graphs of pdfs and HRFs of two components of lognormal and their mixture for some value of the vector of parameters 0 = Co, # , #» a~, d ) .

3.3. M i x t u r e s o f lnverse Gaussian components

If, in (2.18), J)(t) is the Inverse Gaussian (IG) density function, given, for j = 1 , . . , k , by B(t)=

~75

2#~t

j,

t>0,

#j,•>0

,

(3.21)

then, (2.18) represents a mixture of k IG components. For details, see for example, Wald (1947), Tweedie (1957a, b), Sankaran (1968), Wasan (1969), Ahmad (1982), AL-Hussaini and Ahmad (1984), Amoh (1983), Chhikara and Folks (1989), Johnson et dl. (1994) and Seshadri (1993, 1999). AL-Hussaini and Ahmad

154 E. K. AL-Hussaini and K. S. Sultan

(a) P I

2.6 2.1

•

.........

f~ ....... r::~,l

~- 1.6 O.

1.1 0.6 0.1 i

2

3

i

4

5

6

t

(b)

-r t

35

r(t) ~

,'

rl (t) [ r2(t) f

30

,,.....

25 u_

r~ 20 I 15 10 S

.

i J

1

2

3

4

5

6

t Fig. 2. (a) Density functions: lognormal components and their mixture 0 = (0.8, 0.9, 2.5, 0.16, 0.64). (b) HRFs: lognormal components and their rnixture 0 = (0 8, 0.9, 2.5, 0.16, 0.64).

(1981) proved that a finite mixture of IG components (univariate and multivariate, as proposed by AL-Hussaini and Abd-E1-Hakim, 1981) is identifiable. The reliability function for this mode! is given by (2.19), where f o r j = 1 , . . , k,

ej(t) = 1 -- eV~

[~TJ

,-

/,~- 1 t

+ 1

,

(3.22)

155

Reliability and hazard based on finite mixture models

and the HRF r(t) is given by (2.20), where for j = 1 , . . , k , J)(t) and Rj(t) are given by (3.22) and (3.23), respectively. Ahmad (1982) estimated the live parameters of a mixture of two IG components using moments, maximum likelihood and moment generating function methods of estimation. A mixture of two IG components is given by (2.18), where k = 2 and for j = 1,2, fa.(t) is given by (3.22). The five-dimensional vector of parameters, 0, is denoted by 0 = (]7,/21,/22, )~l, 22)

•

Ahmad (1982) used the method of moments to estimate the vector of parameters 0 by equating the first five sample moments, where rar=

x,

r=l,...,5

,

based on a random sample of size n drawn from the two-component IG mixture, to the corresponding first five population moments, given by Vr = p V l r q- (1

--],9)V2r,

r= 1,.,5

,

where for j = 1,2, r-= 1 , . . , 5 , r-1

(F -- 1 -1- i)!

i=0

0!(2x;//22)

He then solved the system of five equations in the five unknowns by using Newton-Raphson iteration scheme. Ahmad (1982) also obtained the maximum likelihood estimates of the five parameters 0 and the estimates based on the moment generating function (MGF) method which was suggested by Quandt and Ramsey (1978). In the MGF method, a random sample Tl, T 2 , . . , Tù is drawn from the mixed population and the vector of parameters 0 is estimated by minimizing 5

&(t; 8) = Z{/?n(te) -

M(te; 8)]2

,

(3.23)

g--1

where Y,(ae) = aF, 1 ,i=l exp(texi) and M(te; 8) =pMl(te) + (1 -p)M2(te), the moment generating funetion of the jth IG component given by

Mj(t)=exp{-~jjI1- (1-2/22t'11 -~-j Bj 1/2}

Mj(t)

is

(3.24)

Based on the three methods, the mean-square errors of the estimates were computed and compared by Ahmad (1982) via a simulation study using data generated from seven different populations. Results indicated, among other remarks, that the ML method was superior to the other two methods of estimation.

156

E. K. AL-Hussaini and K. S. Sultan

Amoh (1983) also studied the M L estimation of the vector of parameters 0. The performance of the estimates based on small sample was examined via a simulation study. The estimates were compared with those based on completely classified samples. Amoh (1983) concluded that the MLEs perform well if the components of the population are well separated, otherwise, the performance is not so good. AL-Hussaini and Ahmad (1984) obtained the information matrix about the five-dimensional vector of parameters 0 of a mixture of two IG components. Laguerre-Gauss quadrature, power series expansion and Simpson's method were used to compare the information about 0. The asymptotic variance-covariance matrix was then obtained. The H R F of a mixture of two IG components has the following properties (see Ahmad 1982): 1. For fixed /~j and large )v, (J = 1,2), a mixture of two IG components is approximately that of two normal components, each with parameters ( # j , / 2 3 / 2 j ) , j = 1,2. Therefore, for fixed #j and large 2j, ( j = 1,2), r(t) ~ rN(t), where rN(t) is the H R F of a mixture of two normal components. 2. For fixed 2j and large/2j, Ü = 1,2), a mixture of two IG components tends to be approximately a mixture of two components, each of which is the distribution of a reciprocal gamma random variable (RG) with parameters (1/2,2/)v),j= 1,2. Therefore, for fixed 2j and large #j, ( j = l , 2 ) , r(t) --+ rRG(t), where rnG(t) is the H R F of a mixture of two distributions each of which is that of a reciprocal of gamma random variable with parameters (1/2,2/27) , j = 1,2. 3. The H R F of a mixture of two IG components tends to zero as t ---, 0. It increases up to a point t* = min(tml, tm2), where tmj is the modal value of a component IG density given by

( 9g) Ij2 3g 2)~y On (t~, co), r(t) is monotonically increasing if2j,j = 1,2 are large; otherwise, it attains one maximum, and in some cases two, reaching a constant value as t -+ oo. This asymptotic value of r(t) is given by lim

r(t) = ~ 21/2/22i'

,~oo

t )~2/2/22'

21//2~<~ 22//22, ;1//22 > ;2//2 2

It can be shown that a point at which equation 3 _ ,~; c9=2/22+2-7 2T "

r(t)

attains its maximum satisfies the

(3.25)

The critical value t and the corresponding maximum r(t) can be approximately determined either graphically or numerically by solving the above equation.

Reliability and hazard based onfinite mixture models

157

Ahmad (1982) obtained the M R L T ra(t) for a mixture of two IG distributions

as 2 m(t) = ~~_,pj{ (lijt)~)[u] 4- (lij ÷ t)exp(2#/lij)q)[vl}

j=l × [j~l [ . =pj{ ~ ü]] -- exp(2,~j//lj)~ [1;]}]-1 ,

(3.26)

where

u = (2j/t)l/2(t/#j -- 1)

and

v = -(#/t)'/2(t/lij

4- 1) .

By using the above form of ra(t), Ahmad (1982) showed that lim ra(t) = p & + q#2 = !~ ,

t-+0

lim ra(t) = ~ 2/~~/21, t-~oo

[ 2/~2/Ä2,

//1/;12~ ")~2/2~» //2/)~2 --> ;2/~'2 "

(3.27) (3.28)

From (3.28) and (3.29), the M R L T will eventually exceed the mean lifetime ,u of the mixed model whenever min(2/~2/21,2/~2/22) >/~ . Figures 3(a) and (b) show the graphs of pdf's and HRFs of two components of Inverse Gaussian and their mixture for some value of the vector of parameters

0 = (p» ~/1,/'/2, il'l, Ä2). 3.4. Mixtures of exponential components If, in (2.18), J)(t) is the exponential density, given for j = 1 , . . ,k, by fj(t) = ~ e x p { - t / c g } ,

t > 0, c9 > 0 ,

(3.29)

then, (2.18), represents a mixture of k exponential components. The exponential distribution has been the most commonly used distribution in reliability studies. Titterington et al. (1985) reported the use of mixtures of exponential components in medicine (renal xenon levels, conception times), geology (particle sizes), failure times (valve lifetimes, transmitter receiver lifetimes and reliability) and radioactive tracers. Mendenhall and Hader (1958) estimated the vector of parameters 0 -- (p, «1, c~2) of a mixture of two exponential components based on Type I censored data. Balakrishnan and Ambagaspitiya (1989) used a two-component mixture exponential as an alternative in evaluating some goodness-of-fit tests for exponentiality. Papadopoulos and Padgett (1986) estimated the parameters and reliability function of a mixture of two exponential compo-

158

E. K. AL-Hussaini and K. S. Sultan

(a) h(t) fl(t) f2(t)

0.8

0.6

0.4 •.

t '~

0.2

0.0 0

2

(b)

4

I

6

r

ù .-'"

1.2

r(t)

• -"

.........

,"

rl(t) r2(t)

0.8 -'1-

0.4

0.0

";

t /

.... ' ......................................

«

I

1

2

4

6

Fig. 3. (a) Density functions: Inverse Gaussian components and their mixture 0 = (0.5, 1.5, 2.5, 1.0, 17.0)• (b) HRFs: Inverse Gaussian components and their mixture 0 = (0.5, 1.5, 2.5, 1.0, 17.0).

nents based on right censored samples and using the Bayes method. AL-Hussaini (1999b) obtained the Bayesian prediction bounds for future observables based on Type I censored sample drawn from a finite mixture of two exponential components. Other studies were made by Rider (1961), Proschan (1963), Kleyle and Dahiya (1975), McClean (1986) and Willamain et al. (1992), Engelhardt (1995), among others. Chapter 19 of Johnson et al. (1994) and the book by Balakrishnan and Basu (1995) are devoted to exponential distributions, and they do include discussions on mixtures. Teicher (1963) showed that a finite mixture of k exponentiäl components is identifiable.

Reliability and hazard based onfinite mixture models

159

The reliability and hazard rate functions corresponding to a mixture of k exponential components are given by (2.19) and (2.20), respectively, where for j=l,...,k

Rj(t) = exp{-t/c~j}

(3.30)

and 1

r/(t) = - -

c9

.

(3.31)

For a finte mixture of two exponential components (k -- 2): (i) if cq = c~2 = c~, the mixture reduces to a single exponential component with a constant H R F c« (ii) if cq ¢ (E2, it follows from (3.32) that r}(t) = 0, for all values of t (j = 1,2). Therefore, from (2.24), r'(t) < 0, for all values of t, so that r(t) is decreasing on (0,oo). Case (il) is the (mixture) case which says that the mixture has a D H R eren though each component in the mixture has a constant hazard rate. Explanations for this result, considered as a paradox, were made by Barlow (1985), Rodrigues and Wechsler (1993) and more recently by Mi (1998).

3.5. Mixtures of Rayleigh components Il, in (2.18), ~(t) is the Rayleigh density, for j = 1 , . B(t) = 75exp -

,

t > o, ~j > 0 ,

,k, given by (3.32)

then, (2.18) represents a mixture of k Rayleigh components. The reliability function of a finite mixture of Rayleigh components is given by (2.19), where for

j= l,...,k

[ «)~]

Rj(t) = exp -

~

,

(3.33)

and the H R F 4 0 is given by (2.20), where ])(t) and Rj(t) are given for j = 1 , . . ,k, by (3.33) and (3.34), respectively. A mixture of two Rayleigh components is given by (2.18), where k = 2 and for j = 1,2, ]~(t) is given by (3.33). The three-dimensional vector of parameters 0 is denoted by 0 = (p, Ctl, az). Vodä (1976) discussed the moments method of estimation of the vector 0. This work is an extension of that by Krysicki (1963) who had earlier discussed the problem of estimation for a two-component mixture of Rayleigh distributions (see Chapter 18 of Johnson et al., 1994). It can be shown that the maximum

160

E. K. AL-Hussaini and K. S. Sultan

likelihood estimates of p, cq and c~2of the vector 0, may be obtained by solving the following system of three equations: /7

~~~(t,)

= 0,

- Y2(t,)]/h(t,)

i=1

~[bj(t,)fj(ti)]/h(ti) ] = O, j =

1,2

»

i=I

where h (t~) is given for k = 2 by (2.18) and fj(t~) by (3.33), j = i, 2 after indexing t by i and for j = 1,2, bj(ti) = 2 ( t ~ - c~~)/~3. The resulting estimates can then be used to obtain the MLEs of the RF and H R F for the mixed model. The H R F for a mixture of two Rayleigh components may be examined by using the fact that the H R F rj(t),j = 1,2 increases linearly. Then H R F for the mixed model increases (decreases) if (2.23) takes the form A (t)rll (t) + [1 - A (t)]r~ (t) - A (t)[1 - A(t)] [ra (t) - r2 (t)] 2 > (<)0 , where

A(t) = [1 + B(t)] -I,

B(t) = (1 -p)/pexp[-(1/c~ 2 - 1/cq2)t2]

and 2

r}(t) = ~ ,

j = 1,2

Figures 4(a) and (b) show the graphs of pdf's and HRFs of two components of Rayleigh and their mixture for some value of the vector parameters 0 = (p, C~I~~2).

3.6. Mixtures of Weibull components lf, in (2.18), j~(t) is the Weibull density, for j = 1 , . . ,k, given by

B(0

=cjj~.? ~«. J.

expl,-

\~j/

j,

t>0,

(~j>0,«j>0)

,

(3.34)

then, (2.18) represents a mixture of k Weibull components. The reliability function for this model is given by (2.19), where for j = 1 , . . , k, Rj(t) = exp{-(t/cg) «j}

(3.35)

and the H R F r(t) is given by (2.20), where, for j = 1 , . . , k, J)(t) and Rj(t) are given by (3.35) and (3.36), respectively. The H R F of component j is given by

r a t ) = cjtJ«-1,/e/«j •

(3.36)

It may be observed that the Weibull distribution includes both the exponential and Rayleigh distributions as special cases (when cj = 1 and 2, respectively).

Reliability and hazard based on finite mixture models

(a)

P

161

I

t

ù """,,

ù(t)[

n

0,0

0.1

0.2

0.3

( b ) 60

s

.'"

r(t) ........ .......

rl(t) r2(t)

.-" ... -"

40 « «

«

"1"

«

20

f 0.0

.......

". . . . . . . . . . .i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i 0.1

0.2

' . . . . -0.3

Fig. 4. (a) Density functions: Rayleigh components and their mixture 0 = (0,1, 0.25, 0.1). (b) HRFs: Rayleigh components and their mixture 0 = (0.1, 0.25, 0.1).

Natural applications for the Weibull distribution have been found in reliability (see, Malik, 1975; Frank, 1988). Makino (1984) approximated the normal by a Weibull distribution and found that cj = 3.43927 provides the two distributions to be close. Due to the flexibility and its possible use in many fields, the distribution has received much attention in the past 30 years. Johnson et al. (1994) devoted Chapter 21 to the "Weibull Distributions", history, inference, characterizations, applications, related distributions and a list of references. Ahmad (1988) showed that a finite mixture of k-Weibull (and hence k-exponential and k-Rayleigh components) is identifiable. Kao (1959) was the first to

162

E. K. AL-Hussaini and K. S. Sultan

propose the use of a mixture of two Weibull components in fitting the life distribution of electron tubes. One of the components in the mixture represented sudden (or catastrophic) failure and the other represented wearout (or delayed failure). Since wearout failure usually occurs after sudden failure, a location parameter, representing the delay time, was introduced in the second component of the mixture. The time ~ at which the proportion of the sudden failure is equal to that of the wearout failure is obtained by equating the reliabilities, at z, in both cases. That is, by equating RI(r) and R2@), where Rj(z),j = 1,2 is given by (3.36), to get = exp

[52 l°g °~2-- c1 log °q] . C2 C1

(3.37)

Kao (1959) used the graphical method in estimating all of the parameters involved in the mixture. Jiang and Murthy (1995) used this method to decide on the appropriateness of the mixed Weibull model to fit a given failure data set. Rider (1961) used the method of moments to estimate the vector of parameters 0 = (p, cq, c~2,cl, c2). Falls (1970) suggested the use of a mixture of two Weibull components as a model for atmospheric data. He estimated the parameters involved in the mixture by using the method of moments, except for p, which was estimated by using the graphical method; for details, see Falls (1970). Ashour and Jones (1976) used the ML method to estimate the vector of parameters 0, assuming that ci = c2, based on type 1 censored data. Olssen (1979) pointed out that the solution of the system of five ML equations is intractable. Cheng and Fu (1982) used the weighted least-squares method to estimate the vector 0 when data are grouped and censored. Ashour (1987) and Sinha (1987) used the Bayes method to estimate the vector of parameters 0, based on type 1 censored data. Ashour used a natural conjugate prior, as given in Raiffa and Schlaifer (1961). Sinha used a five-dimensional prior, in which p ~Unif(0, 1), ci ~Unif(0, 5) and a joint 'vague prior' for the two-scale parameters. He also obtained the Bayes estimate of the reliability function. Kimber (1996) suggested a test for heterogeneity that is based on the Weibull distribution. Figures 5(a) and (b) show the graphs of pdt's and HRFs of two components of Weibull and their mixture for some value of the vector of parameters

0 = (/!3,C1, C2, 0~1,0~2).

3.7. Mixtures of Gompertz components Il, in (2.18), J)(t) is the Gompertz density, for j = 1•..., k, given by

fj(t)=bjexp{ajt-~(eaJt-1)},

t>0,

aj, bj>O,

(3.38)

then, (2.18) represents a mixture of k Gompertz components. AL-Hussaini et al. (1999) showed that a finite mixture o f k Gompertz components is identifiable. For

Reliability and hazard based on finite mixture models

(a)

I

10

h(t)

........

I

fl(t) I

163

I

,,'"

..

"',,

-õ

0.0

0.1

0.2

0.3

(b)

i s

120 ........ .......

tt

r(t) rl(t) r2(t)

t t s s/ t /

80

J t t I J "/r

40

0.0

t

t

0.1

0.2

0.3

Fig. 5. (a) Density functions: Weibu11 components and their mixture 0 (0.2, 3.0, 5.0, 0.25, 0.2). (b) HRFs: Weibull components and their mixture 0=(0.2, 3.0, 5.0, 0.25, 0.2). TM

details on mixtures of Gompertz components, see, for example, Gordon (1990), Adham (1996) and AL-Hussaini et al. (1999). The reliability function of this model is given by (2.19), where for j = 1 , . . , k

Rj(t) = e x p { - ~ ( e ajt- 1)} and the HRF r(t) is given by (2.20), where for j = 1 , . . , k , fj(t) and given by (3.38) and (3.39), respectively.

(3.39) Rj(t) are

E. K. AL-Hussainiand IC. S. Sultan

164

The Gompertz distribution has been used as a growth model, among other uses. AL-Hussaini et al. (1999) surveyed some of the uses of the Gompertz distribution. A finite mixture of Gompertz components model may represent different proportions that die due to different causes. In the above reference, estimates of the parameters, RF and H R F are obtained under a mixture of two Gompertz components model based on Type I and Type II censored samples. The likelihood and Bayes methods have been used in the estimation. We shall write X ~ Gomp(a, b) to denote that X follows a Gompertz distribution with parameters a, b. Let X ~ Gomp(a, b) with pdf given by (3.38) and T = bX. Then T ~ Gomp(2) with pdf

f(t)=exp{2t-~(e~t-l)},

t>0,

2>0,

(3.40)

where 2 = a / b . It may be observed that a Gomp(2) is actually a Gomp(a -- 2, b = 1). The likelihood function based on Type I censored sample is given by n

L()o; t) = II[h(ti)] ô~[R(to)l~i ,

(3.41)

i=1

where h(t) and R(t) are given, respectively, by (2.18) and (2.19), 2 = (21,)L2), 6i is an indicator function, given by 6i=

{1, 0,

Tl < to, T/>t0

(3.42)

and to is a predetermined time. By taking the logarithm of L in (3.41), differentiating with respect to the respective parameters, setting to zero and solving, we can obtain the MLEs )-1, )~2 of 21,2» The M L E of R(t), for a given t, is then given by

kML =-pRi(t) + (1 - p)k2(t) ,

(3.43)

where, for j = l, 2,

Rj(t) = e x p - ~ ( e x p ~ i t - 1 )

.

(3.44)

Similarly, the likelihood function based on Type II censored sample is given by

L*(Z; t) - ( ~ !

h(t«/)

[e(t(r/)] n-r ,

(3.4»)

where h(-) and R(.) are as given by (2.18) and (2.19) and t(i) are the ordered times for/=l,...,r. For a given t, the M L E R~L(t ) is then given by (3.43), after replacing ~by *, where R~(t) is given by (3.44) after replacing ^ by , 2"1 and 2*2 are the MLEs based on Type I1 censoring.

165

Reliability and hazard based on finite mixture models

It is well known that the posterior density function is proportional to the product of the prior density and likelihood function. So, if ~z*(_2lt),fr(_2) and LF denote the posterior, prior and likelihood functions, respectively, then (3.46)

=*(_21_t) c< rc(_2)LF(_2,_t) .

The Bayes estimate of a function 0(.2) of the vector of parameters 4 is given by

B - E[0(4)IZ = _tl = £ 0(.2)~*(_2l_t)dL = [ 4(4>(_2)LF(_2;_t)d_2///~(_2)LF(_2;_t)d_2 . ùle

(3.47)

/Je

AL-Hussaini et al. (1999) used the following density function as a prior: :z(.2) = 3:1721exPl

41-17142__-1.]72 ' 4 j > 1, (Tj > 0 ) , j = l , 2

.

(3.48)

For a given t, if we set q5(4) = R(_t), given by (3.32), we can then obtain the Bayes estimate RB(t) ofR(t) based on Type I censoring if LF(_2, t) in (3.46) is replaced by L(4, t_) given by (3.41) and the Bayes estimate R~(t) of R(t) based on Type II censoring if LF(.2, t) is replaced by L*(4, t) given by (3.45). The ratio of the integrals in (3.47) may be approximated by using a form due to Lindley (1980) which reduces, in the case of two parameters, to the form

B = 0(2_) + S/2 + Pl&2 + P2S21 -~- [h;ov12 q- h~1c12 -- h~2c21 q- h;3v21]/2 , (21,42), S = ~~:1 •j2« Ovaij. Vor i,j = 1,2, Oij = ~2~/(~4i~,~j), 0, 1,2, 3, t/+ v = 3, aij = (i,j)th element in the matrix Z, where Z = -IJ(.2)1-1, J(.2) = [e2e/(e4,a#)], e = log LF(.2;_0 Vor / ¢ j, Sij : 4ilYii -- ~Sjaji , l)ij = (4iŒii ~- 4j(7ij)Œii, cij = 30iaii~Tij q- ~)j(aii(7jj ~- 2a2), where qSf = ~qS/ô4i and p~ = ~p/~4i, p = log n(.2), n(_2) is as given by (3.48). In all cases, the functions used in the approximation form are to be evaluated at the MLE of 4. Figures 6(a) and (b) show the graphs of pdf's and H R F s of two Gopertz components and their mixture for some value of the vector of parameters where _2

hùv aù+~e/(e470~), t/, v #

0 = (p, a l , a 2 , b l , b 2 ) .

4. Models with components belonging to different families

Different types of failure could lead to finite mixture models with components belonging to different families of distributions. In this section, four such models will be presented.

166

E. K. AL-Hussaini and K. S. Sultan

(a) h(t)

........

fl(t)

.......

.

."" ss«

,, -" ....

-, "',

ItpS

•• % %

0.2

0.1

0.3

0.4

(b) 70

l

r(t) tl(t) r2(t)

........ .......

l #

l

7

l

# tl l

50

ll

Il_

Il s

-1"

i s #1

30

llJ

10

0.1

0.2

0.3

0.4

Fig. 6. (a) Density functions: Gompertz components and their mixture 0=(0.2, 10.0, 15.0, 0.1, 0.2). (b) HRFs: Gompertz components and their mixture 0=(0.2, 10.0, 15.0, 0.1, 0.2). 4.]. Normal-exponential m i x e d model

Suppose fl(t)

1 = ~exp{-~

1

(~)

2

"~,

J -oo
cr>O

(4.1)

and B(t) = - l e x p ( - - t ' ~ , O~

C

0~)

t,c~ > 0 ,

(4.2)

Reliability and hazard based on finite mixture models

167

are substituted in (2.18) with k = 2, then the resulting pdf is that of a normalexponential mixed distribution. The remark, regarding the domain of the normal distribution, given at the end of Section 3.1 holds here as well. Davis (1952) considered the normal-exponential model to study the bus motor failure by considering data from 191 buses operated by a large city bus company. Failure was either abrupt, in which some part broke and the motor would not run; or, by definition, when the maximum power produced, as measured by a dynamometer, fell below a fixed percentage of the normal rated value. Failures of motor accessories, which could be easily replaced, were not included in the considered data. By comparing the data by Z2 index with the normal and exponential curves, he found in some cases that, the distributions of the considered data appear to be approximated by the normal curves, while in some others appear to be approximated by the exponential curves. He also found a case in which the failure of the bus motors cannot be described by either normal or exponential curves, but looks like a combination of both (mixture of two components). As a meaning of each component and the mixed model for the considered data, he indicated that a normal distribution of mileage to some cases can be expected, for the moving parts of the motor slowly abrade each other until, when a sufficient amount of metal is worn away, they either break or no longer perform their function with satisfactory efficiency. This expectation is confirmed by some failures being caused singly or in combination by worn cylinders, pistons, piston rings, valves, camshafts, connecting rod or crankshaft beatings, etc. There were, however, a considerable number of failures in which moving parts fractured at low mileages, indicating manufacturing errors or inadequate or improper maintenance or repair. These latter conditions were exponential-type occurrences which should follow a different distribution. But these data on the cause of failure were so fragmentary and incomplete that a segregation of the two types of failure could not be accomplished. The indication of failure resulting from two distinct causes however was too strong to be disregarded and the mixed model might be considered to describe such cases. The RF and H R F of this model are given by (2.19) and (2.20), where for j = 1,2 fl (t), f2(t), R1 (t) and R2(t) are given, respectively, by (4.1), (4.2), (3.2) and (3.32), after dropping the index j. The four-dimensional vector of parameters 0 in this case is given by 0 = (p, #, cr2, c~). The maximum likelihood estimates of the parameters can be shown to satisfy the following system of four equations: n

~7~ (~~) -/~(t~)]/h(t,) = o, i=1

ti -- I~

-T-

i~lp

- 1 A(t;)]/h(t,) = 0,

(ti /h(ti) = 0 ,

E. K. AL-Hussaini and K. S. Sultan

168

~(t@2~)f2(ti)]/h(ti)

= O,

i=1

where h(ti) is given for k = 2 by (2.18), fl(ti) by (4.1) and f2(ti) by (4.2), after indexing t by i. The solution of this system of equations may be obtained by using some numerical scheine such as Newton-Raphson or EM algorithms. The resulting estimates can then be used to obtain the MLEs of the RF and H R F for the mixed model. The H R F for the normal-exponential mixed model may be examined by using (2.21) and (2.24). Figures 7(a) and (b) show the graphs of pdfs and HRFs of the normal-exponential mixed model and the components for some value of the vector of parameters 0 = (p,/~, a 2, c~).

4.2. Inverse Gaussian-Weibull mixed model (IG-W) The IG distribution was first suggested to be used as a lifetime model by Chhikara and Folks (1977). They proposed its application when there is a high occurrence of early failures that dominate the life distribution. Accordingly, the IG distribution should be more appropriate in describing sudden failure than the Weibull distribution, since sudden failures usually start as the system is exposed to operation. The Weibull distribution has been extensively used in the wearout type failure. Motivated by such considerations, an IG W mixture model was proposed by AL-Hussaini and Abd-E1-Hakim (1989), in which the IG and W components represent the sudden and wearout failures, respectively. Kao (1959) suggested for such failures a mixture of two Weibull components; see Section 3.6. The I G - W mixture model is capable of covering six different combinations of failure rates, as one of the components (IG) has an upside down, bathtub or increasing failure rate and the other component (W) has a decreasing constant or increasing failure rate. A study was made for the mixed H R F based on these six combinations by AL-Hussaini and Abd-E1-Hakim (1989). Suppose that

f(t; O) = plfl (t; 01 + p2f2(t; 02) ,

(4.3)

where fl (t; 01) is the density function of IG(/~, 2) component, given by (3.22) and f2(t;O2) is that of W(~,c), given by (3.35), 01 = ( ~ , 2 ) , 02=(c~,c), 0 = (p,/~, 2, ~, c),/)2 = 1 - p and P1 = P. The RF, corresponding to the mixed density, is given by

R(t; O) = plR1 (t; 01) + p2R2(t; 02)

,

(4.4)

where R1 (t; 01) and R2(t; 02) are given by (3.23) and (3.36), respectively. Motivated by the fact that iterative ML estimation in mixture models usually requires a large sample size or well-separated components in order to obtain

169

Reliability and hazard based on finite mixture models

(a)

i

1.0 .

• ....

i '

I

.

•

0.8

........ .......

.

h(t)

fl(t) I f2(t)

-lo 0.6 0_ 0.4 0.2 0.0

ù . . ''',

t

i

1

2

3

(b) 11

r(t) rl(t) I r2(t)

........ .......

9

..." .'" ...'"

7 ù . .....''''''''

5 3 1 0

1

2

Fig. 7. (a) Density functions: normal-exponential components and their mixture 0=(0.5, 1.0, 0.16, 1.2). (b) HRFs: normai-exponential components and their mixture 0 = (0.5, 1.0, 0.16, 1.2).

reliable estimates, Hosmer (1973) suggested the following two schemes (models) of sampling: Model 1 (M1). in which a random sample Tl1,.., T/,~, i -- 1,2, is drawn from the mixed population. The log-likelihood function for this M1 is given by 2

L1 = ~

ni

Z

j = l j--I

m

log f,(tij; O) + ~

log f(te; O) .

(4.5)

«--1

Model 2 (M2). in which a random sample T,.1. . . , T~ni, i -- 1,2, is drawn from the subpopulation with ith component, where nl -- n2 -- sn. The remaining ran-

170

E. K. AL-Hussaini and K. S. Sultan

dom sample m = (1 - s)n is drawn from the mixed population. The log-likelihood function for M2 is given by

L2 = L1 -I-

log[L (sn'~p"'p n2] t f t l ) 1 2J' p2=I-Pl,

(4.6)

where L1 is given by (4.5). Regular sampling (when the total sample is drawn from a mixed population), is termed Model 0 and denoted by M0. In this case, the log-likelihood function is given by n

L0 = Z log f ( t j ; O) . j-1

(4.7)

Introduction of Lagrange multiplier to L0, given by (4.7), leads to the weighted maximum likelihood function, given by Pi - 1

L; = Lo - 7

.

(4.8)

AL-Hussaini and Abd-E1-Hakim (1990) obtained the MLEs by maximizing L0, L1 and L2. The weighted maximum likelihood estimates (WMLEs) were obtained by maximizing L~. The efficiency of the schemes of sampling from the I G - W mixture model was studied by AL-Hussaini and Abd-E1-Hakim (1992). AL-Hussaini and Abd-E1-Hakim (1989) had shown that this mixed model covers six combinations of HRFs. They are: (IHR, DHR), (IHR, CHR), (IHR, IHR), (UBTHR, DHR), (UBTHR, CHR) and (UBTHR, IHR). To study the behavior of the H R F r(t), they proposed following. PROPOSITION. For r(t) and r'(t) given by (2.21) and (2.24), we have limr(t)= t--+0

lim r(t) = t+oo

li+ör'(t) =

{ {

oc, (1-p)/c¢, 0, 0, 1 ~eg 2/(2#2),

(4.9)

c 1/c~, « > 1 or ( « = 1 and 2/(2# 2) < 1/c~) ,

- p ( 1 - p)/~2,

c = 1,

oc, 2(1 --p)/o: 2,

1 < c < 2, c = 2,

O,

c>2

lim r'(t) = 0 . t~OO

c < 1, c=1, c>l ,

(4.10)

(4.11)

, (4.12)

Reliability and hazard based on finite mixture models

171

The M R L T corresponding to the mixed model is given by (2.22) (see Abd-E1Hakim, 1985), where ml (t) is given by (3.27) and

m2(t) = c~exp[(t/c)C]F(1 + 1/«; {t/~} c) - t , with £(/~;z) = f ~ x~-12e-Xdx. Figures 8(a) and (b) show the graphs of pdfs and H R F s of the Inverse Gaussian-Weibull mixed model and the components for some value of the vector of parameters 0 = (p,/~, 2, «, c).

(a)

~

~

~

1.5

" ,' .

-,

-

ù'"

,

',

I

- ........

h(t) fl(t) f2(t)

I

".,

1.0 13-

•

0.5

0.0

.

t

i

h

0.1

0.6

1.1

1.6

1.1

1.6

(b) 3

i ----.L.--

i ' ' "

«

r(t) rl(t)

,'

Un," "1-

0.1

0.6 t

Fig. 8. (a) Density functions: I G - W components and their mixture 0 = (0.7, 1.0, 1.0, 1.0, 4.0). (b) HRFs: I G - W components and their mixture 0 = (0.7, 1.0, 1.0, 1.0, 4.0).

172

E. K. AL-Hussaini and K. S. Sultan

4.3. Weibull-exponential mixed model Suppose that, in (2.18), k = 2, fl (t) and f2(t) are given, respectively, by fl(t)=

im)

r(~)~l

expl-

(~)',~ ~

,

t>0,

c~>0, c > 0

(4.13)

and

1

B(t) = ~ e x p [ - t / f l ] , P

t > 0, fi > 0 .

(4.14)

The resulting model, h(t) = Pfl (t) + (1 - p ) f 2 ( t ) , is known as the Weibull-exponential mixed model; see, for example, Ashour (1987). The reliability function of this model is given by R(t) = pR1 (t) + (1 -p)R2 (t), where R 1(t) = e x p [ - ( t / @ C]

(4.15)

R2(t) = expl-t/il] .

(4.16)

and

The H R F corresponding to the mixed mode1 is given by (2.21), where rs(t) = («/c~)(t/~) «-1

and

r2(t) = 1/fi .

Ashour (1985) obtained the MLEs of 0 = (p, a,c, fi) and the approximate asymptotic variance-covariance matrix, based on Type 1 censored data from the mixed Weibull-exponential model. Ashour (1987) considered the same problem by using Bayesian technique and compared the Bayesian estimates, based on a conjugate prior, with the MLEs. It is well known that the Weibull distribution has decreasing, constant or increasing hazard rate according as c < 1, c = 1 or c > 1, respectively. It then follows that the Weibull--exponential mixed model has D F R if c < 1, where c~¢ fi. Figures 9(a) and (b) show the graphs of pdfs and HRFs of the Weibull-exponential mixed model and the components for some value of the vector of parameters 0 = (p, ~, c, fi). 4.4. Modified semi-Poisson model In studying the distribution of time gäp in road traffic, Buckley (1962) suggested the use of a 'displaced' exponential (or two-parameter exponential), with density function

1

I

t-c~

to be a good fit for low-medium flow of traffic.

173

Reliability and hazard based on finite mixture models

(a)

t

{

ù

1.5

I

I

I

"..

........ .......

h(t) I fl(t) f2(t)

1.0 I%

0.5

0.0 0.1

(b)

0.6

1.6

i

i

2.1

2.6

10.00 r(t) rl(t) r2(t)

........ .......

7.75

u_

1.1

5.50

n-

3.25

1.00 .

,."

0.1

i

i

i

i

0.6

1.1

1.6

2.1

2.6

Fig. 9. (a) Density functions: Weibull-exponential components and their mixture 0 = (0.7, 1.0, 4.0, 0.5). (b) HRFs: Weibull-exponential components and their mixture 0 = (0.7, 1.0, 4.0, 0.5),

A mixture of two dispaced exponentials, with density function, given by (2.18) with k = 2, where for j = 1,2, B(t)=~exp

],

t>~j, ej>0,

flj>O

(4.18)

was suggested to fit a situation in which the flow was made up of two types of vehicle, normal (free-flowing) and heavy (restrained). Ashton (1971) modified the 'semi-Poisson' model, proposed by Buckley (1962), by assuming the 'zone of emptiness' to follow the gamma instead of the normal distribution suggested by Buckley. In this model, a distance of length Z behind

E. K. AL-Hussaini and K. S. Sultan

174

each vehicle, which other vehicles never enter, defines the 'zone of emptiness'. A proportion p of vehicles stay at exactly distance Z from the preceding vehicle. The remaining proportion 1 - p follow at a distance Y behind the preceding vehicle. The random variable Ig is assumed to follow the exponential density with parameter 2. Ashton's modified semi-Poisson model is thus a finite mixture of two components with density function f ( t ) = Pf1 (t) + (1 - P)f2 (t), where 1

-1

fl(t) - F(-:)fi ¢t~7

exp(-t/fl),

t > 0, 7 > 0, /~ > 0

(4.19)

and B(t)

--

2 exp(-2t) F(~) I(?;t//~)(l+2fl) '~,

t>0,

2>0,

7>0,

/~>0 , (4.20)

where I(a; b) is the incomplete gamma function defined by

I(a, b) =

wa-le -w dw .

The component fl (t) is the density function of the random variable Z. Ashton (1971) derived f2(t) by first observing that its cdf is given by

F2(0 = P[Y _< tlY > z] - »[z _< t] ' P[Y< >y Z]

(4.21)

where Z has the pdf (4.19) and Y is assumed to have the exponential pdf

9(Y)=2exp(-2Y),

y>0,

2>0

.

By differentiating both sides of (4.21) with respect to t, we obtain

f2(t) = ddt {P[Z < Y < t]}/P[Y > Z]

d {fo,foyfl (z)9(y)dz dy }/f~ f o~fl (z)9(y)dy dz

= dt

.

(4.22)

The form off2(t), given by (4.20), then follows by substituting fl(z) and 9(Y) in this expression. For details, see Ashton (1971). The name modified semi-Poisson model is probably due to the fact that/'2 (t), in (4.20), is related to a partial Poisson sum through the incomplete gamma integral. In order to avoid the possibility of zero gaps, t is replaced by t - c~, in both of (4.19) and (4.20), to insure a minimum gap of c« A special case of f (t) is obtained by putting 7 = 1, which corresponds to an exponential zone of emptiness. In this case, f~(t)=~exp

-

,

t>~,

c~>0, / ~ > 0

(4.23)

Reliability and hazard based on finite mixture models

175

and B(t) =2(1 + 2]~)exp { - 2 ( t -

~)}

x {1-exp[-(~)l},

t>cq c~> 0, f i > 0 .

(4.24)

Ashton's study was actually based on a finite mixture model of the two components with pdf's given by (4.23) and (4.24). Such a model was compared with the displaced exponential model given by (4.17) and mixture of displaced exponential model given by (4.18), which were suggested by Buckley (1962).

(a)

P

0.6

I

I

--

h(t) fl(t)

........ .

.

.

.

.

.

.

f2(t)

0.4

0.2

0.0 2

J

L

3

4

5

t

(b)1 5 ........ .......

r(t) rl(t) r2(t)

1.0

-----

. . . . . . . . . . . . . . . . . . . . . . . . . . .

0.5

jt t

0.0

i i

i

t

2

3

4

t

Fig. 10. (a) Density functions: modified semi-Poisson components and their mixture 0 = (0.5, 1.0, 1.5, 1.0). (b) HRFs: modified semi-Poisson components and their mixture 0 = (0.5, 1.0, 1.5, 1.0).

E. K. AL-Hussaini and K. S. Sultan

176

Table 1 Reliabilities and hazard rates corresponding to the jth component in a mixtue of k components belonging to the same family jth Component

Rj(t)

rj(t)

1 - gO[uj(t)] b

( Ju21//1 \(,!j~l/2ex ~) p{--g j f/~

(1) Normal: N(#j, a2) (2) Lognormal: A(#j, a2) (3) Inverse Gaussian: IG(@, 2j)

2% -exp{Tf}g)Ivj(t)l (4) Weibull: W(cj, ~j) (5) Exponential (~j) (6) Rayleigh («j) (7) Gompertz (aj, bi)

Cb[uj(t)l.

-exp{~} (b[vj(t)])

exp[-(t/~j) Cj]

(cj/c~j)(t/c~j)~~-1

exp[- ~ (e«,t - 1)]

bj exp(ajt)

a 05(.) is the pdf of standard normal distribution ~(.). b L/j(t) ~ (Äj/t)I/2(t/[2j - l) and vj(t) =

(2)/t)l/2(t/@ ÷ 1).

If a mixture of the two densities fl(t) and f2(t) given in (4.23) and (4.24), respectively, could be used in a lifetime situation, then the reliability functions of the components are given by Rl(t)=exp[-

( ~ - ~ ) l,

t>c~

and

It then follows that the H R F s of the components are given, for t > c~, by 1 r 1(t) = ~ = a c o n s t a n t hazard rate,

r2(t)=)~{1--exp [ - - ( ~ ) ] } / { 1 - - ( 1

+2~B2~)exp E - - ( ~ ß - ~ ) ] } "

Figures 10(a) and (b) show the graphs of pdfs and H R F s of the proposed mixed model and the components for some value of the vector of parameters 0 = (p, ~,/~, ~). 5. Summary and remarks

In this chapter, finite mixture models have been presented with discussion on their reliability, hazard rate functions and mean residual lifetimes. The models have been classified into two groups depending on whether the components of the

177

Reliability and hazard based on finite mixture models

Table 2 Reliabilities corresponding to each component in a mixture of two components belonging to two different families Two-component mixture model

Forms of R1 (t) and R2 (t)

(1) I G - W

Rl(t)

L-t~)

1-(j~[@)l/2(~_ l)l - e~~F [2,1/2,tip+

1)1

Ra (t) = expE-(t/d)o] (2) Normal-exponential

R1 (t) = 1 - ¢ ( ~ ) R2 (t) = exp[-t/c~]

(3) Weibull--exponential

R1 (t) = exp[-(t/c~)cl R2 (t) - - expl- t/e]

(4) Modified semi-Poisson

R, (t) = e x p { - ( t ~ ) } , t > R2(t) = {1 + 2 f i - 2 f i e x p [ - ( ~ ) ] } e x p [ - ) o ( t - ~ ) ] , t >c~

mixture belong to the same family or to two different families of distributions. The first group covers seven models of finite mixtures with components belonging to the normal, lognormal, Inverse Gaussian, exponential, Rayleigh, Weibull and Gompertz families. The second group covers four models of two components each comprising the normal-exponential, Inverse Gaussian-Weibull, Weibullexponential and modified semi-Poisson models. Applications of the underlying mixed models as well as their components have been pointed out. Estimation of the RFs and HRFs of such models, by using different methods of estimation, have also been discussed. In addition, the MRLT of some of these models have also been examined. The RFs and HRFs of the components are tabulated in Tables 1 and 2 for these two groups of models.

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N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 ElsevierScienceB.V. All rights reserved.

Mixtures and Monotonicity of Failure Rate Functions

Moshe Shaked and Fabio Spizzichino

In this chapter we have selected to review some aspects of mixed distributions that have been of interest in our own research. First we describe some results that involve the preservation (or the lack of it) of monotonicity of the hazard rate functions of mixed distributions. Next we cover some recent advances in the study of the limiting behavior of the hazard rate function of a mixed distribution. Then some discussion on a multivariate ultimately negative aging property of multivariate exchangeable mixtures is given. Finally we describe some results on the distance of a univariate scale mixture from its parent distribution.

1. Introduction

The analysis of qualitative properties of the failure rate function of an individual, who is a member of a population of (at least apparently) 'similar' individuals, is fairly important in many fields where waiting times, attached to single individuals, are the object of study (by 'similarity' of individuals we may understand that the unidimensional distributions of single lifetimes are all identical). In the field of reliability theory, such properties are crucial when deciding whether a burn-in procedure or some type of preventive replacement politics are to be contemplated. For instance, a burn-in procedure may be recommendable when the failure rate function is decreasing, at least in some right neighborhood of the origin (this is the case, for example, when the hazard rate function has a bathtub shape). For reasons that we shall see in part later in this chapter, such a behavior can manifest itself in the case of mixed distributions. This is the reason why, in the field of reliability theory, the theme of burn-in is strictly tied with that of mixed distributions, and with the related orte of heterogeneous populations (see, for example, Block and Savits, 1997 and references cited therein). In this chapter we review some works which model observed lifetimes as observations from a mixed distribution. This is a common assumption in survival analysis and in reliability theory when it is believed that the sampled population is heterogeneous. Because of the multitude of applications of mixed distributions in 185

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these areas of research, m a n y advances have been made during the last 30 years in studying such distributions and their properties. We obviously cannot cover all the research that has been done with mixed distributions in reliability theory. In this chapter we have selected to review some aspects of mixed distributions that have been of interest in our own research. In Section 2 we describe some results that involve the preservation (or the lack of it) of monotonicity of the hazard rate functions of mixed distributions. In Section 3 we cover some recent advances in the study of the limiting behavior of the hazard rate function of a mixed distribution. Some discussion on a multivariate ultimately negative aging property of multivariate exchangeable mixtures is given in Section 4. Finally, in Section 5 we describe some results on the distance of a univariate scale mixture from its parent distribution.

2. Monotonicity of the hazard rate of a mixture

Imagine yourself in a queue for a server, say a bank teller, or an airline representative at an airport. The customer ahead of you has just stepped forward to the teller, and you are the next in line. That is, once the service given to the person ahead of you is done, then it is your turn. You probably feel optimistic - in just a short while it is going to be your turn, and then your waiting in the line will be over. However, after a short while you notice that the customer ahead of you is still being served... A little more time passes, and this customer is still being served... You start to have that foreboding feeling that the customer ahead of you "is one of those whose service takes for ever" (indeed, in most such situations you later realize that it has taken quite a while until you were finally served). This is the case even when each customer requires an exponential service time (that is, the service time has the lack-of-memory property) with a mean that m a y vary from one customer to another. W h a t has been described above is an illustration of the mathematical fact which states that a mixture of distributions with decreasing failure rate ( D F R ) is D F R (see, for example, Theorem 4.7 in page 103 of Barlow and Proschan, 1975). In particular this is the case even when the mixed distributions are exponential (that is, having constant failure rates). As can be seen from the above description, 'the more you wait, the more likely you are to further wait'. In other words, your waiting time seems to be D F R . The fact that sometimes a mixture of distributions, that do not have decreasing failure rate functions, m a y be D F R has fascinated m a n y researchers (see some references below). In particular, a mixture of distinct exponential distributions (each of which, of course, has a constant failure rate function) must have a strictly decreasing failure rate function. This fact may look at first paradoxical: If each component in a mixture has a constant failure rate function, that is, the lack-ofmemory property, should not the mixed distribution have the same lack-ofm e m o r y property, rather than the strict D F R property? The answer is that this need not be the case. When one observes an item with a distribution that is a

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mixture of exponential distributions, one actually gains information (rather than having a lack of memory) as time passes, as long as the item does not fail. The information that has been gained by time t > 0, if the item is still alive then, indicates that the actual realization of the mixing distribution is relatively more likely to be an exponential distribution with a large mean than with a small mean, as compared to the initial information at time 0 when no survival has been observed yet. In fact, survival at time t' > t indicates even more strongly that the actual realization of the mixing distribution has a large mean. To see it more clearly, imagine yourself again in a queue as described earlier. It is reasonable to assume that different customers have different service distributions. When you observe a customer (for instance, the one ahead of you) that has used the server for some length of time, and that is still being served, you are justified in supposing that this particular customer is likely to be one with a 'larger than average' mean, and therefore you are justified in expecting the rest of the service time to be stochastically larger than the service of an 'average customer'. In fact, the longer the customer ahead of you uses the server, the more convinced you are that he has a large mean, and the more likely it is that your remaining waiting time is going to be large. That is, your waiting time has the D F R property, even if the service time of each customer is exponential (with different means for different customers). The above discussion can be formalized and extended as follows. Let O be the parameter determining the survival function of the observed waiting time T, that is, let us set P{T>t[O=0}-F(tl0),

t>O

.

Also, let M be a mixing distribution (that is, a probability distribution for O), and let S be the set of possible values of O. Then the survival function of T is of the form

ffM(t) ~ PM{T > t} = fsff(tlO)dM(O),

t>0

(2.1)

For simplicity let us assume that Æ(t]0) > 0 for all t > 0, 0 E S. I f w e also assume that Æ(.I 0) is absolutely continuous for all 0 E L, then we obtain that /~M is absolutely continuous as well, and its associated failure rate function is given by

IM(t)

liml

4- At) - FM(t)

1 .fs[F(t + AtlO)-F(tlO)]dM(O) - ~ - At+oAt fsF(tlO)dM(O) =1' 1 1 f F(t + AtlO)-- F(tlO) )~,õ At fsF(t]O-)dM(O) Js ~=="~-~ltjo) F(tkO)dM(O) . l

= im-

Notice at this point that, whenever interchange of the limit operation with integration is allowed, we can write

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ru(t) = ffs r(tlO)dM(OIt),

t>0 ,

(2.2)

where r(.lO) denotes the failure rate function corresponding to Æ(.lO), and where

f(tlO)dM(O) dM(OIt ) = f ~ )

.

(2.3)

For T > 0, the mixing distribution M(OJt) defined by (2.3) can be given two, slightly different, interpretations, according to the interpretation given to M.

Frequentist interpretation. We deal with a multitude of similar individuals with associated waiting times /'1, T2,.., each T~ with its own parameter 0i. If M is interpreted as the distribution of the values of Oi's, then M(-It ) is the distribution of the subpopulation of the 'residual' Oi's, attached to the only individuals for which T« > t.

Subjective interpretation. We consider a single, observable, lifetime T and an associated non-observable parameter O with the assigned set of conditional survival functions/~(t[0) for T given O = 0. I f M is the 'a priori' distribution for O, then M(.[t) is, by Bayes formula, the 'a posteriori' distribution of O, after observing the survival data T > t (see Spizzichino, 1992). In this view we can write

r~4(t) = E[r(tlO)lT > tl,

t _> 0 .

(2.4)

It is clear from (2.2) and (2.3) that qualitative properties of r(tlO ) and M(O) provide relevant information about the monotonicity behavior of rM(t). However the analysis is greatly complicated by the fact that the mixing distribution M(.It) actually varies with t; thus general results are very hard to achieve. It can be of interest, in this marter, to take into account possible properties of stochastic monotonicity of T in O. Notice that those properties can reflect on stochastic monotonicity of O in T and then on monotonicity properties of the distribution M(.]t) as a function of t. Assume, for instance, that S is a subset of the real line and that [T[O = 0] is increasing in 0 with respect to the hazard rate ordering; that is, ff(tlO')/F(t]O ) is increasing in t whenever 0 _< 01 (see, for example, (1.B.2) in Shaked and Shanthikumar, 1994), of, in other words, that F(tl0 ) is a totally positive (TP2) function of t and 0 (see, for example, Karlin, 1968). It is easily seen from (2.3) that, in such a case, dM(O)) taust be TP2 as well; that is, for 0 < t < t I we have [ol:r >

t]

--~lr [OlT > t'l ,

where _
[op:r > t] _<s~ [olr > t'] ,

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where <_st denotes the nsual stochastic order (again, see, for example, Shaked and Shanthikumar, 1994). Another related result can be found in Fahmy et al. (1982). Using the notation of (2.2) we point out the following facts (the first one was already mentioned above as Theorem 4.7 in page 103 of Barlow and Proschan, 1975): • If • If

r(t]O) r(t]O)

is decreasing in t for all 0 C S then rM(t) is decreasing in t as well. is increasing in t for all 0 E S then rM(t) need not be increasing in t.

In the following we review several results and remarks presented in recent literature, which can be viewed as useful, even though partial, contributions to a general study. Some of the contributions to be reviewed highlight the effects that differences in qualitative behavior between r(tlO ) and rM(t) should have in an accurate statistical analysis. In particular, we note the possibility of wrong conclusions where heterogeneity in a population is ignored and a statistical analysis is misleadingly carried out at an aggregate level (the examples presented in Vaupel and Yashin, 1985 are illuminating in this respect). Hougaard (1986) has noted that the initial observed decrease in the mortality rate of patients recovering from a certain heart disease may be explained by the fact that different patients have different (though constant) hazard rates. Cohen (1986) noted that child mortality in Korea had decreasing hazard rate function when heterogeneity was ignored, but it was found to be increasing when the analysis allowed for possible heterogeneity. Brooks et al. (1994) demonstrated decrease in mortality rate with age in an entire population of nematodes even though each of several subpopulations showed continuously increasing age-specific mortality. Vaupel and Yashin (1985) have presented graphs of the hazard rate functions of several mixtures. They have shown graphically (in their Figure 1) that the hazard rate function of a mixture of two distinct exponential distributions is strictly decreasing on [0, ~ ) . Vaupel and Yashin (1985) and Wang et al. (1998) have shown graphically that the hazard rate functions of some mixtures of two distributions may orten strictly decrease over some region of [0, oe), even when all of the distributions that are being mixed have strictly increasing hazard rate functions. Gurland and Sethuraman (1994) considered some further examples of mixtures that have strictly decreasing failure rate functions although each of the distributions that are being mixed has a non-decreasing failure rate function. They showed, for example, that a truncated extreme value distribution (which has a strictly increasing failure rate function), when it is only slightly mixed (5%) with an exponential distribution, gives rise to a mixed distribution that has a failure rate function that is virtually strictly decreasing over the whole region t > 0. Rajarshi and Rajarshi (1988) and Gupta and Akman (1995) have described some mixture models that yield bathtub failure rate functions. The reader may have picked up the impression that every mixture of distributions with increasing failure rate (IFR) has a failure rate function that strictly decreases over some region of [0, oc). This is, of course, not true. For example, Lynch (1999) has noted that a convolution of two I F R distributions is IFR, and such a convolution can also be viewed as a mixture of I F R distributions. There

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are many other examples of mixtures of IFR distributions that are IFR. This motivated Lynch (1999) to look for general conditions under which the IFR property is preserved under mixtures. Lynch (1999) proved two results of the kind described above. Consider a survival function/~M of the form (2.1).

FM(t) = f Æ(tlO)dM(O),

«> 0

where _P(-]0) is a survival function for each 0, and M is some mixing distribution. The first result of Lynch (1999), which follows at once from Prekopa's Theorem, says that if: (i) F(t[0) is logconcave in (t, 0), and if, (ii) the mixing distribution M(O) has a logconcave density ra(O), then ÆM is logconcave, that is, it has the I F R property. The second result (Theorem 2.1 of Lynch, 1999) says that if: (i') il(tl0) is logconcave in (t, 0) and is non-decreasing in 0 for each t, and if, (ii') the mixing distribution M(O) is IFR, then, again, FM has the I F R property. It is remarkable to note that the result that a convolution of IFR distributions is IFR, is a special case of Theorem 2.1 of Lynch (1999). Some further studies of mixtures in the context of monotonicity (or lack of monotonicity) of associated hazard rate functions can be found in Gurland and Sethuraman (1995). They obtained necessary and sufficient conditions for a mixture of two IFR distributions to be DFR. They also studied the special case of (2.1) where

FM(t) =

f #(t)dM(O), ~>_0 ,

(2.5)

that is, when the distributions that are being mixed have proportional hazard rate functions. Again, when F is IFR, they gave necessary and sumcient conditions for FM to be DFR, when M is either a two-point distribution or a continuous distribution. Gupta and Gupta (1996) also studied mixtures of the form (2.5). More explicitly, they studied (and also gave a few references to other stndies of) some particular mixtures, such as gamma mixtures of Weibull distributions, and inverse gamma mixtures of Weibull distributions. It is worthwhile to note that the assumption of proportional hazards (2.5) is very common in survival analysis because of its mathematical tractability. An interesting study of mixtnres of piecewise exponential distributions, with applications to biology and to mining disaster data, can be found in Zelterman et al. (1994). We have mentioned above that a mixture of, say, m subpopulations, each with a distinct constant failure rate, taust have an overall decreasing failure rate. This

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is the case when each of the m subpopulations is modeled independently of each other. That is, when the m subpopulations are assumed not to interact with each other. In other words, this is the case when a 'death' of an individual means a removal of that individual from the respective subpopulation and also from the aggregate population. But in m a n y demographic studies, a removal of an individual from one subpopulation may mean its transfer to another subpopulation. For example, a labor force exit may mean an entry of the individual to another subpopulation. Similarly, changes in marital status mean transitions among subpopulations. Rogers (1992a, b) modeled such transitions and showed that then, even when each subpopulation has a constant failure rate, the aggregate failure rate need not decrease. A more general analysis of aging properties for heterogeneous population, where individuals can change subpopulations along time, is au interesting field and, at the best of our knowledge, one still open for research.

3. Limiting behavior of the hazard rate of a mixture

As at the beginning of Section 2, imagine yourself again at the head of queue, waiting for the current customer to finish his service - at that time your wait in the queue will end. Suppose now, however, that there are only two kinds of customers: Either 'fast' ones that on the average require a short service time, or 'slow' ones that require on the average a lengthy service time. When it is also assumed that customers require exponential service times, then the above statement means that the service time of the 'fast' customers has a high (constant) failure rate, whereas the service time of the 'slow' customers has a low (constant) failure rate. Now, the longer it takes for the customer ahead of you to conclude his service, the more convinced you become that that customer is one of the 'slow' ones (this statement can be translated into a formal mathematical statement and then easily proven). After a while you become quite convinced that the current hazard rate of the service time of the customer ahead of you is actually the hazard rate that is associated with the 'slow' customers. That is, a long observed 'survival' t of the customer ahead of you at the server leads you to believe, with a probability that tends to 1 as t ---+ec (this can be proven mathematically), that the current hazard rate, at time t, is the low one which corresponds to 'slow' customers. What has been described just above is an illustration of the mathematical fact which states that the hazard rate function of a mixture of distributions tends to follow in the long run the lowest failure rate function (if such one exists) among the failure rate functions of the distributions that are being mixed. In particular, if the lowest failure rate function is increasing (decreasing), then the failure rate function of the mixture will tend to have the same property in the limit (that is, for large t). A formal mathematical statement that conveys the above intuitively clear idea may not be easy to state and prove. However, in some special cases (for example,

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when there are only two distributions that are being mixed, or when the distributions that are being mixed are exponential, or have decreasing failure rate functions) this formal mathematics has been worked out in the literature. We informally describe below some of the advances in this area of research. Consider a mixture of the form given in (2.1). Ler S denote the support of M, and for any 0 E S let r(.lO ) denote the hazard rate function associated with F(.IO ) (that is, r(tl0 ) = -(~/~t)log F(tlO), provided it exists). Suppose that the limits a(O) = lim~__+~r(tlO ) exist for all 0 E S, and ler ~ = inf{a(0) : 0 C S} (that is, c~is the limiting hazard rate corresponding to the 'strongest' distribution (in the limit) among the distributions that are being mixed). Let rM denote the hazard rate function associated with FM (that is, rM(t)=-(d/dt)log/~M(t), provided it exists); see also (2.4). In a study of burn-in and mixed populations, Block et al. (1993) showed, under some regularity conditions, that rM(t) --+ c~ as t --+ oc. In other words, in the limit, the failure rate of the mixture follows the smallest limiting failure rate; see also Block and Savits (1997). Mi (1999) further refined this observation by finding conditions under which the above intuitive conclusion also holds when it is formally restated by means of limiting conditional survival probabilities, or by means of limiting mean residual life functions. Rather than looking just at the limiting value of the hazard rate functions, but also looking at the monotonicity properties in the limit, Block and Joe (1997) refined the results Block et al. (1993) in a different direction. Most of their results apply to mixtures of two lifetime distributions. Ler us denote by rl and r2 the hazard rate functions of the two distributions that are being mixed, and as before, let rM denote the hazard rate function of the mixed distribution. Suppose that one component is stronger in the limit than the other one in some sense; without loss of generality, the stronger component may be selected to be the second one. Block and Joe (1997) postulated it formally as the requirement that rl (t) _> r2(t) for any large enough t. Block and Joe (1997) then found conditions under which, for example, (rM(t)/r2(t)) ,[ 1 as t ---+ee. They also extended these results to more general mixtures. Gurland and Sethuraman (1995) have shown that mixing exponential distributions with various other distributions lead to failure rates that are ultimately decreasing (that is, that are decreasing on regions of the form (to, oc) for some to < oc). Block and Joe have noted that these results of Gurland and Sethuraman (1995) are really special cases of their results in Block and Savits (1997). Let us give here a word of warning. It should be pointed out that the study of the limiting behavior of a hazard rate function is of importance only in situations where long survivals are crucial and are emphasized. The reader taust have noted already that if the mixed distribution FM is the distribution of the life of an item, then the probability is often small that the age of the item ever reaches the time region in which the above mathematical results apply. Most items will expire a considerable time before that. Thus, in situations where the interest centers around the 'average' lifetime of the item (such as in the study of light bulbs in a theater hall), one should avoid trying to apply the above mathematical results, unless one has a convincing reason for doing that.

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4. Negative aging properties of multivariate exehangeable mixtures In this section we review some results of Spizzichino and Torrisi (2001) regarding exchangeable mixtures and their negative aging properties. The starting point of Spizzichino and Torrisi (2001) is that if Tl, T2, • •, T~ are non-negative independent and identically distributed random variables, then each is D F R if, and only if, for any choice of ages h , . . . , ti-1, t l + l , . . . , tj-1, tj+l,.., tn (i < j), and any ~ > 0, the following implication holds: ti < tj =~ P { Ti > ti + ~ID,~} ~ tj +

~lDn}

,

(4.1)

where Dn = {Tl > tl,T2 > t2, ..,Tn > t~} . The validity of the implication (4.1) provides a notion of negative aging also for the more general case when the lifetimes Tl, T 2 , . . , T~ are merely assumed to be exchangeable rather than independent and identically distributed. This notion can be viewed as a natural analog of the D F R property for the case of exchangeable lifetimes. For more discussion on this point of view, see Bassan and Spizzichino (1999). Sometimes (4.1) may hold only for large enough ti's. That is, suppose that the exchangeable random lifetimes Tl, T 2 , . . , Tn satisfy the implication ti < t] ~ P{Ti > ti + z[Dn} < P{T] > tj+ z[Dù},

whenever tz > t, l = 1 , 2 , . . , n

(4.2)

for some t _> 0. Then we say that Tl, T 2 , . . , Tn have the multivariate ultimately negative aging property with respect to ~. In particular, when n = 1, then (4.2) reduces to the assumption that P{T1 > t~ +v]T~ > tl} is non-decreasing in h E It, ec) for all -c > 0; in such a case we say that T1 (of its survival function) is ultimately D F R with respect to t. Spizzichino and Torrisi (2001) studied the following model of multivariate exchangeable mi×tures. Consider a population of individuals 1 , 2 , . . , n . Let (Zi, Ti) be a pair of random variables associated with individual i. We think of as an observable lifetime of individual i, and of Zi (taking on values in ~ for all i) as an unobservable quantity (often called frailty) which determines the distribution of Tl.. Since Zi is random, the marginal distribution of T~ is a mixture. Suppose that the individuals are 'similar' in the sense that given ~ = z, the distribution of Tl. is determined by the value z, independently of i. That is, suppose that there exists a family of univariate survival functions {G(.Iz), z E ~ } such that given (Zl, Z 2 , . . , Z,) = (zl, z2, .., zn), the conditional survival function of Tl. is G(.Izi). We also suppose that given Z = z, the random variables Tl, T 2 , . . , Tn are independent. We assume below that Z1, Z 2 , . . , Z , , are exchangeable and this implies that T~, T 2 , . . , Tn are also exchangeable.

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Let us suppose that G(.[z) is ultimately D F R with respect to sorne t for all z E ~q. Furthermore, let us suppose that

ä(tlz') - -

G(tlz)

in non-increasing in t _> ~ whenever z < z' .

Spizzichino and Torrisi (2001) have shown that then T ] , T 2 , . . , T n have the multivariate ultimately negative aging property with respect to t. Some variations of this result, which give conditions under which each marginal distribution is ultimately DFR, are also described in Spizzichino and Torrisi (2001).

5. S o m e results on univariate scale m i x t u r e s

In this section we specialize (2. l) to mixtures of the following form: FM(t) =

/0 ~

F(Ot)dM(O),

(5.1)

t > 0 ,

where M is a distribution function of a positive random variable O. In terms of random variables we have (=st denotes equality in law) T =st Y/O

.~

where Y and O are independent random variables with distribution functions F and M, respectively, and T is distributed according to F~t. If we denote by p the failure rate function associated with the F in (5.1), then the failure rate functions, conditional on O = 0, are given by r(tlO) = Op(Ot),

t >_ 0

(5.2)

and, by specializing the formula (2.4), we obtain rM(t) = E [ O p ( O t ) l T > t I ,

(5.3)

t >_ 0 .

Both, the proportional hazard model described in (2.5), and the scale model described in (5.1), are obtained by somehow incorporating a random effect to a baseline survival function F. These two models are then parallel, in a sense, but, in general, are different one from another. The two models coincide when the baseline survival function is exponential. For a same initial mixing distribution M, it can be of interest to compare r~ in (5.3) with the expression of rM associated with the model given by (2.5). In this respect, using the symbols r ~ ) and r ~ ) for the expression in (5.2) and for the corresponding expression associated with model (2.5), respectively, we can write

r(~/ -r(~ ~ -----Cov(«/E(O,

p ( O t ) ) l r > t] + p(t)Æ(»)[OI r > t]

+Æ(s)[oIr > d " e(s)[p(ot)l

r

>

t],

t >_ o ,

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195

where the suffixes (S) and (P) remind us that the conditional distribution of O given T > t is computed according to the model in (5.2) or to the one in (2.5), respectively. In particular,

EIPt[OIT > t]

E(s)[OIT > t]

L O[Æ(t)]° dM(O) = L[p(t)]OdM(O ) ,

t »_ 0 ,

fL O[Æ(Ot)] dM(O) -- LE~(0t)] d M ( 0 ) '

t > 0

Note that, trivially, Cov (s) [(69, p(Ot))lT > tl) is positive or negative according to whether p is an increasing or a decreasing function, independently of the value of t. General results concerning the comparison among r(~ ), r~ ) and p are hard to achieve. Several aspects of the scale change models are examined in Anderson and Louis (1995), and, for some particular cases, the differences among the scale change model, the proportional hazard model and the baseline model without heterogeneity are illustrated graphically. It is a common practice to approximate the reliability of an item with distribution function FM, given in (5.1), at time t (that is, FM(t)) by ~e(t), if orte has a reason to believe that EO or EO -1 is nearly equal to 1. Thus it is of interest to obtain bounds on suptlF(t ) --FM(t)I = suptlÆ(t) --ÆM(t)I. Few authors have obtained various general results which give useful bounds in some interesting special cases. Here we will not describe these general results, but we will describe the resulting bounds in some special cases. Suppose that FM is a scale mixture of Weibull distribution functions with shape parameter c~ > 0; that is, suppose that in (5.1) we have

F(t)

= 1 - exp{-~t~},

t > 0

for some fixed 2 > 0. Shaked (1981) has shown that in this case sup IF(t) - FM(t)I < E[O ~ - 11 , t

provided the expectation exists. Note that equality holds if, and only if, the mixture FM is pure; that is, if, and only if, O is degenerate at 1. Take ~ =- 1 to obtain that if FM is a scale mixture of exponential distribution functions then sup IN(t) - FM(t)[ <_ EIO - II ,

(5.4)

t

provided the expectation exists. Equality holds if, and only il, the mixture FM is pure. Brown (1983) obtained another bound for a scale mixture of exponential distribution functions. He showed that if E Y = 1, E O -1 = 1, and E O 2 < cc then

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sup IF(t) - FM(t)I < E ( O 2 _ 1) .

(5.5)

t

It is not difficult to see that for some distribution functions M of O the bound (5.5) is tighter than (5.4), and for some other M's it is the other way around. Inequality (5.4), however, does not require the above moment conditions. The papers by Brown (1983, 1985) contain further inequalities of interest. Next suppose that FM is a scale mixture of gamma distribution functions with shape parameter ~ > 0; that is, suppose that F in (5.1) has the density function f ( t ) = [F(«)]-a2«t~-1 exp{-2t},

t _> 0

for some fixed 2 > 0. Shaked (1981) has shown that in this case sup IF(t) - F M ( t ) [ < E I o ~ - 11 , t

provided the expectation exists. Equality holds if, and only if, the mixture FM is pure. As a final example suppose that/~~u is a scale mixture of linear-hazard-rate distribution functions with parameters A and B; that is, suppose that in (5.1) we have F(t)= l-exp{-At-½Bt2},

t>_O .

Shaked (1981) has shown that in this case sup IF(t) - FM(t)I <_ EIO z - 11 , t

provided the expeetation exists. Equality holds if, and only if, the rnixture FM is pure. The papers by Heyde and Leslie (1976), Hall (1979), and Shaked (1981) conrain some results of the type described above, but for general scale mixtures. Ler F, M and FM be related as in (5.1), and let O have the distribution function M. Heyde and Leslie (1976, p. 328) showed that sup l E ( t ) - FM(t)l < G ( E I O - 1 -

1[) 1/2 ,

t

where C1 is a constant that depends only on F. Hall (1979, Theorem 2) showed that if for some ~ > 0, E O -1 = 1 then sup l E ( t ) - F~(t)l _< C2E(O - ~ -

1) 2 ,

t

where C2 is a constant which is determined by F. Hall (1979, Theorem 3) also showed, without assuming E O 1 = 1, that sup ]F(t)--FM(t)] <_ C3E]O - ~ -

1] ,

t

where C3 is another constant which depends only on F.

Mixtures and monotonicity of failure rate functions

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References Anderson, J. E. and T. A. Louis (1995). Survival analysis using a scale change random effects model. J. Amer. Stat. Assoc. 90, 669-679. Barlow, R. E. and F. Proschan (1975). Statistical Theory of Reliability and Life Testing, Proóability Testing. Holt, Rinehart and Winston, New York. Bassan, B. and F. Spizzichino (1999). Stochastic comparison for residual lifetimes and Bayesian notions of multivariate aging. Adv. Appl. Probab. 31, 1078-1094. Block, H. and H. Joe (1997). Tail behavior of the failure rate functions of mixtures. Lifetime Data Anal. 3, 289-298. Block, H. W., J. Mi and T. H. Savits (1993). Burn-in and mixed populations. J. Appl. Prob. 30, 692-702. Block, H. W. and T. H. Savits (1997). Burn-in. Stat. Sci. 12, 1-19. Brooks, A., G. J. Lithgow and T. E. Johnson (1994). Mortality rates in genetically heterogeneous population of Caenorhabditis elegance. Science 263, 668-671. Brown, M. (1983). Approximating IMRL distributions by exponential distributions, with applications to first passage times. Ann. Probab. 11,419-427. Brown, M. (1985). A measure of variability based on the harmonic mean, and its use in approximations. Arm. Probab. 13, 1239-1243. Cohen, J. E. (1986). An uncertainty principle in demography and the unisex issue. Amer. Stat. 40, 32-39. Fahmy, S., A. de B. Pereira, F. Proschan and M. Shaked (1982). The influence of the sample on the posterior distribution. Commun. Stat. - Theory Meth. 11, 1757 1768. Gupta, P. L and R. C. Gupta (1996). Ageing characteristics of the Weibull mixtures. Probab. Eng. Inf. Sci. 10, 591 000. Gupta, R. C. and H. O. Akman (1995). On the reliability studies of a weighted inverse Gaussian model. 3". Stat. Plan. Inference 48, 69-83. Gurland, J. and J. Sethuraman (1994). Reversal of increasing failure rates when pooling failure data. Technometrics 36, 416-418. Gurland, J. and J. Sethuraman (1995). How pooling failure data may reverse increasing failure rates. J. Amer Stat. Assoc. 90, 1416 1423. Hall, P. (1979). On measnres of the distance of a mixture from its parent distribution. Stochastic Process. Appl. 8, 352365. Heyde, C. C. and J. R. Leslie (1976). On moment measnres of departure from normal and exponential laws. Stochastic Process. Appl. 4, 317 328. Hougaard, P. (1986). Survival models for heterogeneous populations derived from stable distributions. Biometrika 73, 387-396. Karlin, S. (1968). Total Positivity. Stanford University Press, Palo Alto, CA. Lynch, J. D. (1999). On conditions for mixtures of increasing failure rate distributions to have an increasing failure rate. Probab. Eng. Inf Sci. 13, 33 36. Mi, J. (1999). Age-smooth properties of mixture models. Stat. Prob. Lett. 43, 225-236. Rajarshi, S. and M. B. Rajarshi (1988). Bathtub distributions: a review. Commun. Stat. - Theory Meth. 17, 2597-2621. Rogers, A. (1992a). Heterogeneity and selection in multistate population analysis. Demography 29, 31-38. Rogers, A. (1992b). Heterogeneity, spatial population dynamics, and the migration rate. Environ. Plan. A 24, 775-791. Shaked, M. (1981). Bounds on the distance o f a mixture from its parent distribution. J. Appl. Prob. 18, 853-863. Shaked, M. and J. G. Shanthikumar (1994). Stochastic Ordet~' and Their Applications. Academic Press, Boston. Spizzichino, F. (1992). Reliability decision problems under conditions of ageing. In Bayesian Statistics, Vol. 4, pp. 803-811 (Eds. J. Bernardo, J. Berger, A. P. Dawid and A. F. M. Smith). Clarendon Press, Oxford.

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Spizzichino, F. and G. L. Torrisi (2001). Multivariate negative aging in a exchangeable model of Heterogeneity. Star. Probab. Lett. (to appear). Vaupel, J. W. and A. I. Yashin (1985). Heterogeneity's ruses: some surprising effects of selection on population dynamics. Amer. Star. 39, 176-185. Wang, J.-L., H.-G. Müller and W. B. Capra (1998). Analysis of oldest-old mortality: lifetables revisited. Arm. Stat. 26, 126-163. Zelterman, D., P. M. Grambsch, C. T. Le, J. Z. Ma and J. Curtsinger (1994). Piecewise exponential survival curves with smooth transitions. Math. Biosci. 120, 233-250.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 ElsevierScienceB.V. All rights reserved.

"7 /

Hazard Measure and Mean Residual Life Orderings: A Unified Approach

Majid Asadi and D. N. Shanbhag

The hazard rate ordering is applied frequently in reliability to compare two probability distributions on R+(= [0, oc)) such that they are both absolutely continuous (w.r.t. Lebesgue measure) or both purely discrete (concentrated on the set of non-negative integers) via their hazard rates. Kotz and Shanbhag (1980) extended the concept of hazard rate, introducing a new concept of hazard measure, applicable to any arbitrary distribution on the real line; in particular, this concept avoids the restriction that the distribution be absolutely continuous or purely discrete. These latter authors have also extended the concept of mean residual life function, and have given certain representations for distributions in terms of the characteristics referred to. In this paper, we introduce the concepts of hazard measure ordering and mean residual life ordering to compare two arbitrary probability distributions and study their basic properties.

1. Introduction

Partial orderings relative to probability distributions are frequently studied in reliability and allied topics. There are several ways in which one can assert that a random variable X (or equivalently, the corresponding distribution function F) is greater than another random variable (or its distribution). The simplest way to compare two distribution functions is via their means (if they exist) or their variances (when these exist and the means are equal). However, such comparisons usually are not very reliable, because they are based on only one or two characteristics that are not very informative. Among more commonly studied partial ordering of distributions in reliability are those based on the corresponding survival functions, hazard rates (hr's) or mean residual lives (mrl's). There is substantial literature on such orderings as well as on their preservations under operations such as convolving and mixing. Most of the literature assumes that the distributions are absolutely continuous (with respect to Lebesgue measure) or purely discrete. See, for example, Shaked 199

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M. Asadi and D. N. Shanbhag

and Shanthikumar (1994) for the literature on partial orderings of the type referred to. Meilijson (1972), Swartz (1973) and Kotz and Shanbhag (1980) gave representations for distributions that are assumed to be neither absolutely continuous nor purely discrete, in terms of the respective mean residual life function. Analogous representations for distributions in terms of the respective hazard function or hazard measure, where there is no assumption that the distribution be either absolutely continuous or purely discrete, were established implicity or explicitly by Cox (1972), Jacod (1975), and Kotz and Shanbhag (1980). In particular Kotz and Shanbhag (1980) have unified several of the results including representations, in characterization and reliability theories, and have extended these to distributions that are not necessarily concentrated on R+. (Incidentally, there do exist in the literature references such as Keilson and Sumita (1982) giving important reliability results for distributions that are not necessarily concentrated on R+, but still requiring that the distributions are absolutely continuous or purely discrete.) In spite of the existence of the general representation theorems for distributions for sometime, only Kupka and Loo (1989) have produced major findings in reliability involving these in the spirit of our results. Taking a clue from what is achieved by Kupka and Loo, we now introduce here some general partial orderings without assuming the distribution to be either absolutely continuous or purely discrete or as that concentrated on R+, and generalize many of the existing results in the literature to those based on these new concepts. Among the results that we establish, we have a characterization of the hazard measure ordering based on the ratio of survival functions and various other results revealing what exactly this ordering means in relation to other orderings such as the stochastic ordering and the mean residual life ordering. We also establish, under appropriate assumptions, certain results concerning the hazard measure orderings or related properties, including, in particular, some closure properties of the hazard measure ordering, in relation to sums of independent random variables, order statistics and record values. The organization of the paper is such that in Section 2 we give some basic definitions and auxiliary results, and in Sections 3 and 4 we present the aforementioned findings of the paper in the respective order.

2. Some basic definiüons and auxiliary results We need the following definitions and auxiliary results in the present investigation. DEVINmON 2.1. Let X be a real-valued random variable with a real-valued Bord measurable function m on R satisfying

ra(x) = e ( x - x l x _> ~)

E(X+) <

~ . Define

(1)

for all x such that P(X > x) > O. This function is called the mean residual life function (mrl function for short).

Hazard measure and mean residuallife orderings: A unified approach

201

DEFINITION 2.2. Let F be a distribution function on R. Consider the measure VF on (the Borel « - f i e l d oÜ R such that VF(B) =

B (1 - F1( x - ) ) dF(x)

(2)

for every Borel set B. This measure is called the hazard measure (hin) relative to F. THEOREM 2.3. Let b( < cx~) denote the right extremity of the distribution function F of a random variable X with E(X +) < oc and m be its mrl function. Further, let A = {y : limxTyra(x) exists and equals 0}. Then b = oo if A is empty and b = inf{y : y C A} ifA is non-empty. Moreover, for every - o c < y < x < b, 1-F(x-) 1 - F(y-)

ra(y) -

r a ( x ) exp

{

~x dz} -

(3)

m~

and, for every - o c < x < b, 1 - F ( x - ) is given by the limit of the right-hand side of (3) as y --+ -oc.

COROLLARY 2.4. Let X be a non-negative random variable with distribution function F and E(X) < oo and ler b be the right extremity of F. Then, for every x c [0, b), (4)

1-F(x-)=m(~ßx))exp{- foXm~} where m is' as defined in Theorem 2.3.

THEOREM 2.5. Let vF be as defined above and V~Fbe continuous (non-atomic) part of VF and let H«(x)= VCF(--oc,X] in R. Denote by b the right extremity of F. Then b -- sup{x : VF(X,X + (~) > 0 for some c5 > 0}, and the survival function Æ(x) = 1 - F ( x - ) is given by {'(x) = I x g x ( 1 - - VF{Xr})] exp{--H«(x)} ,

x
(5)

where Dx is the set of all points y E (-oc, x) such that VF{y} > 0 ((5) also holds with 'x < b' replaced by 'x c R' provided we define exp{-oc} to be equal to zero).

COROLLARY 2.6. I f --oC < C~< OC and the restriction of F to (-o% c~) is continuous (i.e. if vF is continuous or non-atomic on (--oc, @), then Æ(x) = exp{-H(x)}

for all x E ( - e t , c~) ,

where H(x) = VF((--oGx]) and we define exp{-oc} = 0.

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M. Asadi and D. N. Shanbhag

We have taken the definitions and results appearing above from Kotz and Shanbhag (1980). Specialized versions or variants of these have appeared in Cox (1961, 1972), Jacod (1975) and other places. The following are some of the standard definitions appearing in the literature, but with modifications to take into account recent improvements to these suggested by Kotz and Shanbhag (1980) and Keilson and Sumita (1982). Most of the literature still restricts itself, while addressing the issues linked with the definitions involved, to the distributions concentrated on R+ with infinite right extremities, and the cited references are among those that have made an effort to study larger classes of distributions with members that are not necessarily concentrated on R+. DEFI~TION 2.7. The random variable X is said to be smaller than the random variable Y in the usual stochastic order (st), denoted by X _<stY (or by Y _>stX) if B(x)-
for a l l x E R ,

(6)

where Æ(x) and G(x) are survival functions of X and Y, respectively. DEFINITION 2.8. Suppose X and Y have absolutely continuous distributions w.r.t. Lebesgue measure or have purely discrete distributions, with hazard rates 2F and 2c, respectively. Then the random variable X is said to be smaller than the random variable Y in the hazard rate order (hr), written as X -
hrX), if 2F(X) > 2c(X)

for almost all [PF + Pc]x Œ R ,

where PF and Pc are the measures determined on R by F and G, respectively. The usual stochastic ordering is being used in many areas of statistics and applied probability. For some standard references on this, we refer the reader to Marshall and Olkin (1979), Ross (1983) and Shaked and Shanthikumar (1994). DEFINITION 2.9. Let X and Y be two random variables with X + and Iz+ integrable and the mean residual (mrl) functions mF and inc, respectively. Then X is said to be smaller than Y in the mrl order, denoted by X -<mrl Y (or by 1z ~mrl X), if mF(x) <_ ma(x)

for all x < min{bF, bc} ,

(7)

where bF and bc are the right extremities of the distributions of X and Y, respectively. It can be easily shown that X -<mrl Y if and only if, in obvious notation, fx~ ff(t)dt f ~ ä(t)dt is non-increasing on {x : G(x) > 0}, and it implies that bF _< bG. Also, note that if X -

Hazard measure and mean residual life orderings." A unified approach

203

distributions whenever either of them is so, in view of what is observed in Shanbhag (1979) and Kotz and Shanbhag (1980), where bF and ba are as above (i.e. as in Definition 2.9).

3. Hazard mcasure ordering and its relationships with other partial orderings Let X and Y be two real random variables with distribution functions F and G, respectively, and let VF and vc be the hazard measures and mF and mG, the mean residual lives relative to F and G, respectively. In the following, we extend the concept of hr ordering. DEFINITION 3.1. The random variable Y is said to be less than the random variable X in the h a z a r d measure order (hin), denoted by Y ~hm X (or by X ->hin Y), if vG = vF + # with ~ as a non-negative measure on R, where vF and vc are the hazard measures of F and G, respectively. In particular, if F and G are absolutely continuous (w.r.t. Lebesgue measure), the hazard measure ordering reduces to the hazard rate ordering and one can denote the ordering with 'hr' in place of 'hin'. It is well known that in the restrictive case referred to here, distribution functions F and G possess the 'hr' ordering if and only if E'(x)/G(x) is an increasing function on {x E R : G(x) > 0} and VF({X C R : G(x) = 0}) = 0. We shall now show that the result in question holds without the assumption that F and G be absolutely continuous. THEOREM 3.2. Let X and Y be random variables distributed with df's F and G, respectively. Then Y ~hm X if and only if F'(x)/G(x) is" an increasing function on {x E R : G(x) > 0} and VF({x E R: G(x) = 0}) = 0. Moreover, if VF({X e R : G(x) • 0}) ~---0 and { X n : rl = 1 , 2 , . . } and {Yn: n = 1 , 2 , . . } are

sequences of randorn variables converging in distribution to X and Y, respectively, such that Yn ~hm Xn, then Y ~hm X. PRoov. We have from the representation (5) =exp

Z

(log(1 - v«{xr}) - l o g ( 1 - va{xt}))

l ~H~F~ ~x~ - 4 a l ( x ) ) ~,

x < min{bF, bG} ,

(8)

J where b F and bc are the right extremities of F and G, respectively, D (F) is the set of discontinuity points of vF lying in ( - o o , x ) and H(~F)(x) = VF,«((--OO,X]) with VF,« as the non-atomic part of VF, and D~(c) and H~(G)(x) are defined similarly for G. If y _< x < min{bF, bG} and Y <-hinX, then it follows from (8) that:

204

M. Asadi and D. N. Shanbhag

/>(~) >/>(y) .

(9)

G(x) - G(y) to see this note that (in obvious notation)

/>(x)

/>(y)

f

O(x) = G - ~ e x p ~

OO

x,.6(D(F)uD!G))\(D~F)uD~,G)) k=l +

v~,«((y,x]) - ~~'«((Y'~J)/ '

assertion Y- bc ve((bG, oo)) > vc((b~, oo)), and bF < ba ~ for an a smaller than bF and sufficiently close to bF, VF((a,bF]) > vG((a, bF]).) Because of the left continuity of/> and G we have then the "only if" part of the first assertion in view of (9). To prove the "if" part of the first assertion, note that under the given condition, The

be = ba and the exponent of (8) is an increasing leit continuous function on (--0o, bF) with limit as x ~ - o c to be zero. Consequently it follows that for each x E (--oC, bF), VF{X} <_Vc{X}, and we have VG,«(.n (--oO, bF)) =VF,«(. N (--oo, bF)) + #(') with # as a non-negative measure. As VF({Z ¢ R : G(z) -- 0}) = 0, we have then that Y ~hm X, {Note that ifbF is a discontinuity point o f F i.e bF < oc with VF{bF} = 1, it is also a discontinuity point of G giving vc{b~} = 1.} Hence we have the first assertion. This, in turn, establishes the validity of the second assertion, since each survival fnnction is left continuous and the assumption concerning {Xn} and {Yù} implies that fi(x)/G(x) is increasing on {x: G(x) > 0 and x is a continuity point of /> and G}. Consequently we have the theorem. [] The following theorem and example show that the hazard measure ordering is stronger than the usual stochastic ordering. THEOREM 3.3. If X1 and X2 are two random variables such that 322 --1 and/?2 denote the survival functions and vl and v2 denote the hazard measures of X1 and X2, respectively. Then by the representation (5) we have the right extremities of/71, the distribution function of X1, and F2, the distribution function of X2 to be equal with

~.(x)~lx.~(1--VF,{Xr})lexp{--H,.«(x)}

x E ( - o o , b), i = 1 , 2 ,

(10)

where b is the common right extremity of F1 and F2, Dix is the set of all points y E ( - o c , x ) such that v»~,{y} > 0 and//~«(x) = v~(-oc,x] with v~ as the continuous part of vF., VF. being the hazard measure relative to F,. Now, since re2(D) > VFz(D) for every Borel set D, we can easily see that F2 (x) < fil (x) for every x E R. []

Hazard measure and mean residual liJè orderings: A unißed approach

205

EXAMPLE3.4. Let X2 be a non-degenerate random variable with survival function f2 and right extremity b < o« Define X1 = X2 - 1; the survival function of X1,/~1, is given by F l ( X ) = f f 2 ( x + l),

xER

.

Clearly then X1 _<stX2 but the right extremities of X1 and 3(2 are different and hence they do not obey the corresponding hazard measure ordering. There exist also other examples in the literature (see for example, Alzaid, 1988; Shaked and Shanthikumar, 1991). The mrl ordering is a well-known concept in the literature, where, as implied earlier, it is assumed usually that random variables are non-negative. It is known that if we restrict ourselves to absolutely continuous random variables or purely discrete random variables, the mrl ordering is weaker than the hr ordering (see Shaked and Shanthikumar, 1994). Indeed this last result with 'hr' replaeed by 'hm' holds for more general random variables i.e. for those that are not necessarily absolutely continuous and non-negative. THEOREM 3.5. I f Y ~hm X , and X + and Y+ are integrable, then Y ~mrl X. PROOF. Suppose Y -
x,x+tE

(--oO, bF) and t > O ,

(11)

giving fob~-X O(x + t)dt < för-x fi(x + t)dt ~(x) P(x) ' (12) implies Y ~mrl X.

x c ( - ~ , ò~) .

(12) []

Singh and Vijayasree (1991), using a counter example, showed that the mrl ordering is not closed under the formation of k-out-of-n systems. A comparison of random sums based on the mrl ordering is studied by Pellerey (1993), while Shaked and Shanthikumar (1994) showed that under some conditions the mrl ordering is preserved under the operation of taking convolution. Shaked and Shanthikumar (1991) also proved that under the condition that the ratio of the mrl's of X and Y is increasing, the hr ordering and the mrl ordering are equivalent. The following theorem now shows that the last result of Shaked and Shanthikumar (1991) stated above remains valid even when the assumption that random variables are non-negative and absolutely continuous is dropped.

206

M. Asadi and D. N. Shanbhag

THEOREM 3.6. Let Xi, i = 1,2 be two random variables with X~+, i = 1,2, integrable and mean residual life functions mi, i = 1,2, respectively. Let bFh, i = 1,2, be the

right extremities of the distribution funetions of Xi, i = 1,2, respectively. Suppose that ml(x)/m2(x) increases for x<min{bF~,bF2}. Then )(1 ~mrlÄr2 implies Xl ~hmX2 • PROOF. The assertion of the theorem can be proved as follows. As observed earlier, X1 --<mrlX2 implies that bF~ is less than or equal to b&. The increasing nature of ml (x)/m2(x) for x < min{b<, bF2} implies that bF, >_bF2 and hence we have that bF, = bF2. On the other hand, we have ff2(X) __ tal(X) fx/P2(t)dt B(x) m2(x) f/~~(t)dt'

x < min{bFh, be2} .

Under the assumptions of the theorem, in view of what we have observed immediately after Definition 2.9 and the fact that bF~ = bF2, we have the right-hand side of the last equality, and hence its left-hand side, to be increasing on {x E R:/~1 ( x ) ) 0}, and in obvious notation, VF2({x E R : P1 (x) = 0}) = 0. By Theorem 3.2, we have then X1 ~hm X2, and the theorem is proved. []

4. Some closure properties of the hazard measure ordering In this section, we extend some of the known closure properties of the hr orclering involving either sums of independent random variables, or order statistics relative to a random sample, to the corresponding properties of the hm ordering. However, before doing so, we give the following two important results of auxiliary natnre which tell us that under appropriate assumptions, the hm ordering implies the st ordering of certain conditional distributions. THEOREM 4.1. Let Y ~hm AT and Z be a continuous random variable independent

of X and Y such that P{X > Z} > 0 (and hence also such that P{Y >_Z} > 0). Then (XlX ~ Z) ~st ( Y I r

_> z)

(13)

PROOF. Let us denote by F, G and H, respectively, the df's of X, Y and Z. We can then see that (13) is equivalent to

B,oo) H(y)dF(y) fR H(y)dG(y) - /

Jfx,oo)

H(y)dG(y) / H(y)dF(y) >_0 for all x ¢ R . dR

We can see that (13) is equivalent to the condition that

(14)

Hazard measureand mean residuallife orderings:A unifiedapproach

207

B,~) H(y)dF(y) £~,x) H(y)dG(y) - J~,~) H(y)dG(y) x~

H(y)dF(y)>O

a(- OO~X)

for a l l x C R ,

(15)

which, in turn, is then seen to be equivalent, in view of Fubini's theorem, to ùfRmin{P(x),P(z) } dH(z) >.f-oo,x](G(z) - G(x))dH(z)

-

]£ min{ G(x), G(z)} dH(z) __][-oo~](Æ(z)- F(x))dH(z) >_0 for all x c R. (16)

As the inequality in (16) is met trivially when G(x) = 0, it is clear that to have (13), it is sufficient if we show that for each x with G(x) > 0 and hence F(x) > 0,

~(x)O(x){~min{1Æ(z)

-fRmin{1

(G_-(z)

1) dH(z)

\~- 1)dH(z) } > 0 G-(z)-'~dH(z) f(_~,x](~(z)

' C(x)J

(17)

-

In view of Theorem 3.2, we have that (17) holds for each x with F(x), G(x) > O. Hence, we have the theorem. [] COROLLARY4.2. Let Y ~hm X and Z be a eontinuous random variable independent

of X and Y. Then, for all z with P{X + Z >_z} > 0 (and henee P{ Y + Z >_z} > 0), we have

(xlx+z >z) >_st(:qy + z _>z)

(18)

PROOF. The result follows on applying the theorem with z - Z in place of Z with z arbitrary. [] COROLLARY 4.3. If the assumptions of Corollary 4.2 are met with distribution of Z as absolutely continuous having an interval support and an increasing hazard function, then

Y + Z <_hmX+Z •

(19)

PROOF. Let F, G and H be the dfs of X, Y and Z, respectively. The result follows from Corollary 4.2 as (19) is equivalent to that (for some version of h): L h(z - x)dY(x)

L h(z - x)dG(x)

Biär(z_ x)dF(x) < ~ «

~ ,

z < bH +be ,

(20)

208

M. Asaß and D. N. Shanbhag

where bF and bw are the right extremities of F and H, respectively, h and H are, respectively, the density and the survival function relative to Z. (Note that /~(z - .) is the d f of z - Z and we have, in obvious notation, bF = bG, and that (20) is equivalent to stating that

\H(z-X)

-

-

\ H ( z - r )

-

[] REMARK 4.4. Theorem 3.2 implies that given Y _ and a function of the form (1 - c~) + c~a~r(x),x E R, with c~ E [0, 1] and a~r as a survival function on R.) Also, if Z is independent of X and Y, we can claim the existence of a sequence { Z ~ : n = 1 , 2 , . . } of random variables with distributions that are absolutely eontinuous with respect to Lebesgue measure (with Z~ ~ Z i f Z is already assumed to be absolutely continuous), so that for each n, Z~ is independent of Xù and Y~ and the sequence {Z~ : n = 1 , 2 , . . } converges in distribution to Z. In view of these observations, it is clear that Theorem 4.1, Corollaries 4.2 and 4.3 follow also from the corresponding results when X, Y and Z have distributions that are absolutely continuous with respect to Lebesgue measure. While we are addressing the problem of obtaining a closure property of the hm ordering involving conditional distributions, it is worth pointing out that the following holds: REMARK 4.5. Let (X, Z1) and (Y, Z2) be independent random vectors such that XIZ1 <_hin Y[Z2

almost surely .

Then we have X _x}P{Y>y}<_P{X>y}P{Y>x}

ifx_
(which is implied by P{X > x]Z,}*'{Y > ylZ2} _< P{X > y]Z1}P{Y > x]Z2}

a.s., x < y , )

and that X and Y have the same right extremity. (This result essentially extends Theorem 1.B.8 of Shaked and Shanthikumar, 1994.) Appealing to Corollary 4.3, we can easily get the following results as further corollaries of the theorem. THEOREM 4.6. Let X and Y be two independent random variables with distributions that are absolutely continuous (with respect to Lebesgue measure) with supports as intervals and increasing hazard rates on the respeetive supports. Then X + Y has an

Hazard measure and mean residual life orderings: A unified approach

209

absolutely continuous distribution with interval support and increasing hazard rate on the support. PROOF. A random variable Z with absolutely continuous distribution having infinite right extremity and continuous density has an increasing hazard rate if and only if Z -- 0; the "only if" part of the assertion holds even when the assumption that the density is continuous is not met. We can construct a sequence {Xn :n = 1 , 2 , . . } of random variables converging in distribution to X, independent of a random variable distributed as Y, such that X,',s have increasing hazard rate absolutely continuous distributions with infinite right extremities. (For example, i f F is the survival function of X and, for each n _> 1, x,, is a point such that Ü(xn) = 1/(n + 1) and the hazard rate atxn isyn, then one can take Xn such that P{X~ > x} = f •(x) I. F(xn)e -y'(x-x')

if x _< x~, i f x _> x,, .)

We have then X,, _ 0 and n -- 1,2,.... Suppose Y* is the random variable distributed as Y and independent of {Am}. As Y* has an absolutely continuous distribution with interval support and increasing hazard rate on the support, Corollary 4.3 implies that Xù + Y* ~hm Xn -c Y* 4- t for each t > 0 and n = 1 , 2 , . . . As {X,* + Y*} converges in distribution to X + Y and X 4- Y has an absolutely continuous distribution, the stability theorem for hazard measures given, for example, in Kotz and Shanbhag (1980) implies then that for every Borel subset B of ( - o c , b), where b is the right extremity of the distribution of X 4 - Y, vH(B)>_vH(B-t)

for a l l t > 0

,

(21)

where H is the distribution function of X 4- Y (with vH obviously as the corresponding hazard measure). It is obvious that the distribution of X 4- Y has its support to be an interval and density (i.e. some version of it) to be continuous. In view of this, (21) implies that the theorem holds. [] COROLLARY 4.7. Let (Xi, Y/), i = 1 , 2 , . . , m , be &dependent random vectors such that Yi _~hm)(/, i : 1 , 2 , . . m. [je Xi and Yi, i : 1 , 2 , . . , m, have absolutely continuous distributions with interval supports and increasing hazard rates on respective supports, then m

m

i=l

i-1

(Also Theorem 4.6 implies that the distributions of ~im~xi and ~im=l Yi have interval supports with increasing hazard rates on respective supports.) PROOF. We shall obtain the result by induction. Assume that it is valid when m = k, where k is a fixed positive integer. Then, if we define (Xk+x, Y~+I) to be a

M. Asadi and D. N. Shanbhag

210

random vector independent of (X,., Y~), i = 1 , 2 , . . , k and distributed as 0(1, Y1), we have by Corollary 4.3 and Theorem 4.6, Yii

q- Yk+l __~hm

Xi

47 Yk+l

i

+Xk+,

i=1

-
(22)

k Z i has an increasing hazard rate (on noting that Theorem 4.6 implies that ~i=1 absolutely continuous distribution). Hence we have that the result holds for m = k + 1. As the result trivially holds for m = l, it follows then inductively that the result holds for all m. [] REMARK 4.8. In 1.B. 1 on p. 12 of Shaked and Shanthikumar (1994), a misleading statement has appeared. Without clarifying what really is meant by the hazard rate corresponding to a distribution that is not absolutely continuous with respect to Lebesgue measure, the authors claim that their definition of hazard rate ordering, possibly with a modification, holds even when the distributions are not assumed to be absolutely continuous. Our findings in this paper provide one with a clear picture of the situation in this. REMARK 4.9. Involving the construction in the proof of Theorem 4.6, it can be seen that there is no loss of generality in taking {Xù}, {Y~} and {Zn} in Remark 4.4 to be such that the distributions of their members have infinite right extremities. REMARK 4.10. For an absolutely continuous distribution function F with density function f , the reverse hazard rate is defined by f ( x ) / F ( x ) on {x :F(x) > 0}. Taking a hint from this, if G is a d f on R, we can define the reverse hazard measure relative to G as the measure v~ on R such that for every Borel set B

where Pc is the measure determined by G, on R. Note that for every Borel set B, ~b(B) = ~ ~ ( - B )

,

where H is the d f given by H(x) = 1 - G((-x)-),

x ER

and vLr is the hazard measure relative to H. Implications of this to our study are self-evident. The following three results (i.e. two theorems and a corollary) follow via shorter proofs in view of Theorem 3.2. However, as their direct proofs have some interesting features, we have decided to produce them here.

Hazard measure and mean residual life orderings." A unißed approaeh

211

THEOREM 4.11. Let (Xi, Yi), i = 1 , 2 , . . , m be independent two-component random vectors such that Xi _
corresponding survival function, and, for each c~ > O, ler X~ and Y~ be random variables with survival functions ES(x), x E R and 1 - F « ( x - ) , x E R, respectively. Then, with obvious terminology, we have {X~ : c~ > 0} decreasing in hazard measure and {Y~ : c~ > 0} increasing in hazard measure. PROOV. If 0 < e < ec, then, for any Borel subset B o f the set of points at which F is continuous, we have the value o f the hazard measure relative to the survival function P~ to be f»(c~/F(x))dF(x), and, for any x C R or in particular, as a discontinuity point of F, we have the value (in obvious notation) o f the hazard measure o f {x} to be 1 - (1 - VF{x}) ~. Hence the first part o f the assertion easily follows. To prove the second part o f the assertion, note that if c~ and B a r e as above, then the value for B o f the h a z a r d measure relative to the survival function 1 - F ~ ( x - ) , x E R, equals fe(c~F ~-1 (x))/(1 - F~(x))dF(x), which is decreasing in c~ if ~ is allowed to vary because, for each y c (0, 1), ( z / y f f -1 dz

1 - y=

Moreover, in this latter case, for any {x} with x as a discontinuity point of F, we have the value o f the hazard measure as

F~(x) - F«(x - ) t - F~(x - )

( =

1 - F«(x)

,~

.1 + F«(x ) _ F:(x_) )

-1

,

which is decreasing in c~when c~is allowed vary because if 0 < y < z < 1, we have

,z~ f,{//

~~ _ ,~

--

(~)

~, d u }~ d v

.

212

M. Asadi and D. N. Shanbhag

In view of what we have observed, we have then the second part of the assertion and hence the theorem. [] COROLLARY 4.13. Let {X~ : n = 1, 2 , . . } be a sequence o f independent identically distributed random variables. Then, f o r each integer m > 2, min{Xi,... ,X,~,} ~hm min{X~,.. ,Xm-l} and

max{X1,.. ,Xm 1} --~hmmax{X1,.. ,Xm} • PROOF. If we denote the distribution function and survival function of X1 by F and F, respectively, then for each positive integer n, we have the survival functions of min{X1,.. ,X~} and max{X1,.. ,X,~} to be, respectively, P'(x), x E R, and 1 - F n ( x - ) , x E R. On appealing to the theorem, we have the corollary. [] REMARK 4.14. Theorem 1.B. 15 in Shaked and Shanthikumar (1994) is not valid in its existing form. (We do not claim here that typos or minor blemishes have nothing to do with this.) The following example illustrates as to why this so. EXAMPLE 4.15. Let Xj and X2 be independent random variables with absolutely continuous (with respect to Lebesgue measure) distribution functions F1 and F2, respectively, with supports [0, a] and la, ec), where a E (0, ec). Note that we have here min{X1,X2} = X1 almost surely and max{X1 ,X2} = 2(2 almost surely, and it is not true that X1 ~hr 22, which contradicts Theorem 1.B. 15 of the cited reference (claiming that, in the present case, min{X1,X2} _

k = 1,2, . . , m

.

(Note that if {(Zln),.., Z(mn)) : n = 1 , 2 , . . } is a sequence of m-component random vectors with i.i.d components, converging in distribution to an m-component

Hazard measure and mean residual life orderings."A un(ßedapproach

213

random vector ( Z 1 , . . , Zm) with i.i.d components, then, in obvious notation, for e a c h k = l , 2 , . " , m , { Z (k) (n) : n = 1 , 2 , . . } converges in distribution to Z(k); hence, the general result holds from its corollary relative to the case when X/s and Y,-'s have absolutely continuous distributions.) REMARK 4.18. The definition of Y ~hm X in this paper is tailored so as to subsume the definition in Shaked and Shanthikumar (1994) of Y <_hrX as a special case. However, the definition given in Shaked and Shanthikumar (1994) is not universally followed; indeed, there are places in which the ordering "Y _
with w(x) integrable. Then, provided w(x) > 0 for almost all [F1 x E R,

Xw ~hm X" if w is increasin9 a.e. [F1

(23)

Xw <_hinX

(24)

and if w is de«reasin9 a.e. [F] ,

where Xw is a weighted randorn variable relative to X with weight function w. PROOF. It is clear that the right extremities of F and Fw, with Fw as the d f of Xw,

are equal. Eq. (23) follows because if w is increasing a.e. [F], then, in obvious notation, for every Borel set B,

fB

1 Œw(X) = ~ w(x) dF(x) 1 - Fw(x-) f[~,~) w(y) dF(y) <-

B

1

dF(x) .

1 - F(x-)

(25)

As (25) with "_>" in place of " < " holds if we have w decreasing a.e. [F], it is then obvious that (24) also holds. [] COnOLLARY 4.20. I f F & a continuous distribution function, then the corresponding sequenees {Ri : i --= 0, 1 , 2 , . . } and {Si : i = O, 1 , 2 , . . ) of ascending record values

and descending record values, respectively, satisfy

Ri+l _>hinRi for all i >_ 0 and Si+l ~hmSi

for all i > O .

214

M. Asadi and D. N. Shanbhag

The corollary follows easily on nothing that, for each i _> 0, Rz+~ can be viewed as a weighted random variable relative to Ri with weight function that is equal to - l n ( 1 - F ( x ) ) for each x for which P{Ri > x} > 0, and Sz+l can be viewed as a weighted random variable relative to Sz with weight function that is equal to - lnF(x) for each x for which P{Si < x} > O. REMARK 4.21. For F whose right extremity is not one of its discontinuity points, even when we do not assume it to be continuous, we have

Ri

-->hin R0

for all i _> 0 .

Similarly, for F whose left and right extremities are not its discontinuity points, even without the continuity, we have

Si <-hmSo for all i > 0 . These results can be shown to be valid, essentially applying in Theorem 4.19. REMARK 4.22. I f X and Y are independent continuous random variables with the same right extremity, then, it is easily seen that, in the hazard measure order, min{X, Y} is less than each of X, Y, and max{X, Y}. However, the result does not hold if the continuity assumption of X and Y is dropped. It is also worth pointing out here that the second result in Remark 4.21 does not hold if the condition that the right extremity of F is not a discontinuity point of F is not met. References Alzaid, A. A. (1988). Mean residual life ordering. Stat. Papers 29, 3543. Cox, D. R. (t961). Renewal Theory. Methuen, London. Cox, D. R. (1972). Regression models and life-tables (with discussion). J. R. Stat. Soe. B 34, 187-208. Jacod, J. (1975). Multivafiate Point Processes: Predictable Projection, Radon-Nikodym derivafives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31, 235-253. Keilson, J. and U. Sumita (1982). Uniform stochastic ordering and related inequalities. Canad. J. Stat. 10, 181 198. Kotz, S. and D. N. Shanbhag (1980). Some new approaches to probability distributions. Adv. Appl. Prob. 12, 903421. Kupka, J. and S. Loo (1989). The hazard and vitality measures of ageing. J. Appl. Prob. 26, 532-542. Marshall, A. W. and I. Olkin (1979). Inequalities: Theory of Majorization and its Applieations. Academic Press, New York. Meilijson, I. (1972). Limiting properties of the mean residual life function. Ann. Math. Stat. 43, 354-357. Ross, S. M. (1983). Stoehastic Processes. Wiley, New York. Pellerey, F. (1993). Partial ordering under cumnlative damage shock-models. Adv. Appl. Prob. 25, 939-946. Shaked, M. and J. G. Shanthiknmar (1991). Dynamic mnltivariate mean residnal functions. J. Appl. Prob. 28, 613-629. Shaked, M. and J. G. Shanthikumar (1994). Stochastie Orders and their Applications. Academic Press, New York. Shanbhag, D. N. (1979). Some refinements in distribution theory. Sankhyä, Ser. A 41, 252-262. Singh, H. and G. Vijayasree (199 t). Preservation of partial orderings nnder the formation of k-out-ofn:G systems of i.i.d, components. IEEE Trans. Reliab. 40(3), 273-276. Swartz, G. B. (1973). The mean residual life function. IEEE Trans. Reliab. 1/-22, 108-109.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

K_Y

Some Comparison Results of the Reliability Functions of Some Coherent Systems

die Mi

The expression of reliability of k-out-of-n system as a function of the reliabilities of the components forming the system is complicated. In the literature a result has been established which can compare the reliability functions of k-out-of-n system. This article relaxes the condition assumed in the literature, and further extends the result to coherent systems. Applying this result to different models, we obtain sufficient conditions that can be easily verified, under which such reliabilities can be compared. This result can be used to indicate how the reliability of some coherent system can be improved. Due to the relationship between the reliability of a k-out-of-n system and the distribution function of a Poisson binomial random variable the result of this article can also be applied to establish stochastic ordering of Poisson binomial random variables.

1. Introduction

A system with n components is called a k-out-of-n system if it operates if and only if at least k(1 < k < n) of the n components function. These n components do not have to be independent, but in this paper we assume they are. Let 0 < Pi < 1 be the reliability of component i(1 < i < n) and p ~ ( P l , . . . , P ~ ) E (0,1) ~, B,, - {p : p c (0, 1)n}. Then the reliability of the k-out-of-n system consisting of the n components is given as

hkln(p) =

~

II[p~i(1--pi)l-~i I .

(1)

~:el+'"+en~k i=l

The k-out-of-n system is widely applied in industry. It is of practical importance to compare reliability hklù(a) with hk]n(b) for given a,b c ~n. Hoeffding (1956) showed that n

hkl~(P) --> hkl~(P) if p E ~~ a n d Z p i=l 215

i~ k

J. Mi

216

and n

hein(P) -< hkl~(0)

if p C ~tl and ~ P i

_< k - 1 ,

i=1

where p - ( ~ i ~ = l p i / n , . . . , ~i~:lpi/n). Glesser (1975) extended Hoeffding's result and showed that hkr~(P) is Schur-convex in the region { p : ~ i ~ lpi > k + 1} and Schur-concave in the region {p:~j-~4~=lpi <_ k - 2}. Boland and Proschan (1983) extended the regions of Schur-convexity and Schur-concavity of the function h~l~(p). This showed that hkl~(P) is Schur-convex in the region {p:(k-1)/(n-1)<_pi_
{p:0
2. Main result The derivation of out result requires use of the theory of majorization. For convenience we state the relevant definition below. For more details see Marshall and Olkin (1979). 1. For x, y E IRtl let x[1 ] > • -. ~ x[n] denote the components of x in decreasing order, and x(1) <_ ... <_ x(~) denote the components o f x in increasing order. We say that x is majorized by y (y majorizes x) denoted as x -< y if'.

DEFINITION

(i) (ii)

k

~~i_lx[i] <_ ~i=lY[i], k = 1 , . . , n tl ~iL1 x[i] = ~i=1Y[i]"

1; and

DEFINITION 2. For x, y E IRtl we say x is weakly submajorized by y, denoted as k x -<w Y i f ~z~i=~ x[i] ~

k Y[i], k = ~i=1

1 , . . . , n; we say x is weakly supermajorized by y, k k = 1 , . . , n. denoted as x _<w y, if ~ki_ 1 x(i) >_ ~i=1Y(i), It is easy to see that x -4 y implies both x -<w Y and x _<w y.

Some comparison results of the reliabiHtyfunctions of some coherent systems

217

The concept of node criticality introduced by Boland et al. (1989) is central to the following developments. The following notation will be used. Suppose x E {0, 1 }~, i.e. each one of the n components o f x is either 0 or 1. We use (li, 0j, x (ij)) E {0, 1} ~ to denote a vector w h o s e / t h c o m p o n e n t is 1, j t h c o m p o n e n t is 0, and the remaining components are the same as in x, vector (0i, l j, x (i~)) etc. will be defined similarly. DEFINITION 3 (Boland et al., 1989). In a coherent system said to be more eritical than node j for 4, denoted by i > j, ~b(0i, l j , x (ij)) for all x (ij) and strict inequality holds for at ~)(li,Oj, x (ij)) = ~(Oi, lj,x(iJ)),VX ~ {0,1} n, then no«es i and permutation equivalent, denoted by i ~=j.

(S, 4), hode i is if 4(li, Oj,x (U)) >_ least one x (ij). I f j are said to be

TnEOREM 1. Consider a coherent system (S, O) of m nodes 1 , 2 , . . , m. Suppose that l C 2 ~=... ~:n. Let a, b Œ~n (n < m) and q C ~m_n, where ~m-n = {(Œ1, .., C~m-n) : 0 < C~k< 1,k = 1 , 2 , . . ,m - n}. If l n a -< lnb, then h(a,q) _< h(b,q), and the equality holds if and only if a = b. Here h(a, q) is the reliability of (S, 4) when the components assigned to nodes { 1 , 2 , . . , n} and {n + 1 , . . , m} have reliability a and q, respectively. PnOOF. Since 1 c 2 ~ . • • & n from Boland et al. (1989) we know that the reliability function h(p, q) is permutation invariant in p ~ Nn. Thus, we can assume ak = a(k) and bk = b(k), 1 < k < n without loss of generality. Further, since l n a - < lnb, so in a can be derived from ln b by successive applications of a finite n u m b e r of T-transforms (see, e.g., Marshall and Olkin, 1979). We can assume that there are indexes 1 < i < j < n such that

lnai = 2 lnbi + (1 - 2) lnbj ,

(2)

lnaj = 2 lnbj + (1 - ,~) lnbi ,

(3)

for a constant 2 ~ (0, 1), and lnak = lnb~,Vk ¢ i,j. That is, l n a is derived from lnb by one T-transform. F r o m (2) and (3) we obtain ;~ (1-4) ai : b i bj , (4) a j : @ b } 1-2)

(5)

Hence aiaj = bibj. Moreover, it can be shown that ai 4- aj < bi + bi. In fact, the inequality

implies

i.e., ai -~- aj ~ bi ~- bj .

218

J. Mi

This further gives (6)

ai + aj - aia j < bi + bj - bibj .

Since aiaj = bibj yields either ai <_ bi or aj < bj, so it follows that h(a, q) < h(b, q) by Theorem 1 of Mi (2000). The next result relaxes the condition in a -< in b to in a -% in b. [] TnEOREM 2. Suppose that the coherent system (S, ~2) satisfies the same conditions as in Theorem 1. Then f o r any a, b E ~~ satisfying In a - % ln b, it holds that

h(a, q) < h(b, q). PROOF. By the definition of weak submajorization <w, the following inequalities hold: k

k

Zlna[i]_
k= 1,2,..,n

.

i=1

Without loss of generality we can assume ai : a[i] and bi : b[i], 1 < i < n. In the case of ~7=1 ln ai = ~i~_i ln bi, we actually have ln a-< ln b, so by Theorem 1 we have h ( a , q ) < h(b,q). Thus, we will consider the case of n

n

~ i : 1 in ai < ~Y-~4:1In bi in the following. B

If

n

~i~_1 lnai < ~i=1 lnbi, then I]/"--1a; < l~i=l bi. Denote Hi~=l ai > 1, and ler cù = b ù / e and ci : bi, V i # n. We have bn cn = - - < bn = b[n] <_ b[n-1] = b~-i = c~-1

and so en < Cn-1 ~ Cn-2 ~ "'" ~ C2 ~ C1 •

For any 1 < k < n - 1, it is true that k

Ein

k

a[i] _< Z

i=1

k

In b[i] : E

i=1

in c[i]

i=1

and n

In a[i ]

ln(~) n. b.

=

i=1

:

n--1 Elnbi~-hl i-1 n-1

= ~

in c[i] + In c[n]

i=1 n

= ~ i=1

bn o~

in c[il •

n

c~_= I~i=1 b i /

Some comparison results of the reliability functions of some coherent systems

219

Therefore, we have shown lna-< lne, where e = (cl,... ,c~). By Theorem 1 it follows that h(a, q) < h(e, q). Now let us compare h(e, q) with h(b, q). Note that the only difference between b and e is that cn = bù/c~ < bù. As a coherent system, the reliability function h(p,q) of (S, Ó) is increasing in p ¢ ~ù. This means h(e, q) <_ h(b, q) and consequently we have h(a, q) <_ h(b, q). [] The following result generalizes Theorem 2.2 of Pledger and Proschan (1971). COROLLARY 1. For any a,b E Nn, /f lna -% lnb, then hkl~(a) < h~l~(b), where hkLù(P) is the reliability function of a k-out-of-n system. The connection between the reliability function hklù(P) and the distribution function of Poisson binomial random variable enables us to apply Corollary 1 to Poisson binomial random variables. For any given p E N~ suppose that independent Bernoulli random variables X/take 1 with probability p~, 1 < i < n. Denote Sn(p) - ~i~1X/. Then S~(p) is a Poisson binomial random variable. COROLLARY 2. For any a,b • ~ù, tf lna -<w lnb, then S~(a) is stochastically st

smaller than Sù(b), i.e., Sù(a) < Sn(b).

PnOOF. It suffices to note that for any 0 < k < n, hkl~(a) =P(Sù(a) _> k) and hktn(b) = P(Sn(b) _> k). [] EXAMPLE 1. Theorem 1 (i), the main result of Ma (1997) is a special case of our Corollary 1. Its 'if' part is straightforward. To see the 'only il7 part of Theorem 1 (i) of Ma (1997), let p E ~~ and p* = (p,... ,p) where p a n d p satisfy ~[in_lPi > ph. This time we have ~7=~ lnp[i] > n lnp, or ~ i ~ l lnp?]/n >_p. Hence it follows that: ~~=1 lnp[~] > k -

Ein=l

lnp[~] >_ P n

V1 < k < n

and so k

k

Elnp[i]>-kp=Zlnp~l i=1

V1 < k < n .

i=1 st

That is, lnp* ~w lnp and thus Sn(p*) < Sn(p) by our Corollary 1. Theorem 1 (il) of n Ma (1997) can be shown by noticing that I~i=l qi ~_ qn where qi z 1 - p i and q - 1 - p , implies (lnq,... ,lnq) -~w (lnql,... ,lnqn). For the convenience of application we can convert our theorems to be stated in terms of lifetime of coherent system. Let 321,.. ,X~ be independent lifetimes of n KI~ • . . , K~, respectively. Denote components following distributions K = (Kl,... ,K~) and 1et z(K) be the lifetime of the aforementioned coherent system (S, ~). COROLLARY 3. Let (S, ~b) be a coherent system with m nodes and 1 o= 2 o= ... o= n. Suppose that one set of components has independent lifetimes T l , . . . , Tn with

220

J. Mi

distribution functions F1, .. ,Fn and W1, W%..., Wm-n with distribution function H~, ..,Hm_n, and another set has lifetimes S1, ..,Sn which are independent of W1, W2, .., Wù~-n and follow the distributions G1, .., Gn. I f

( l n ~ ] ( t ) , . . , l n F n ( t ) ) -% ( l n G l ( t ) , . . , l n G n ( t ) )

V0 < t < x ,

(7)

wh«r« Fi(t) ~ l - F i ( t ) and Gi(t ) ~- I - Gi(t)» then P ( z ( F , H ) > t) < P(z(G, H) > t), V 0 < t < x, and "c(F,H) is the Iif«time of (S, (o) when lif«tim«s of components assigned to node i(1 O, then st

-c(F, H) _< r(G, H) . REMARK 1. Suppose that (S, Ó) is a coherent system and has a modular decomposition {(A~,z1), (A2, z2)} where A 1 has n nodes and A2 has m - n nodes. Denote the nodes in A1 by 1 , 2 , . . , n and assume that they are permutation equivalent. Then the aforementioned results can be applied to this system. For the definitions of module, modular decomposition see Barlow and Proschan (1981). REMARK 2. For any a = ( a l , . . •, an), b = ( b i , . . . , bh) with ai > O, bi > 0 1 < i < n the majorization a-< b does not imply ln a - % ln b. (Actually, ln a-<w ln b implies a-<wb) However, a -1 -~Wb-a does imply l n a - % l n b , where a - 1 ( l / a 1 , . . , l/an) and b - l _= ( I / b i , . . . , 1/bn). Also, the weak submajorization n H in a -<w In b must imply ~i=1 a~ _< ~i=1 b~, V c~> 0. These results are shown in Marhsall and Olkin (1979).

3. Applications In this section the results developed previously will be used for different applications. For the simplicity of notation, we will state the results for k-out-of-n system. Slight modification can easily give the similar results for coherent system (S, ~b) with rn nodes of which n are permutation equivalent 1 ~ 2 ~ ..- =~ n. In these examples the conclusions will be expressed in terms of lifetime of the system. We will use the same notation such as F, G etc. defined in Section 2. Also, whenever the lifetimes 7 1 , . . , Tn are involved they are assumed to be independent. THEOREM 3. Let F/(t) have failure rate function and cumulative failure rate function ri(t) and Ri(t), 1 < i < n, respectively. Denote the counterparts of Gi(x) as 2/(t) and Ai(t). (i) IF (Rl(t),..,R,,(t))-<W(Al(t),..,A~(t))

V0
(8)

Some comparison results of the reliability functions of some coherent systems

221

then P(z(F) > t ) t ) , (ii) Suppose rl(t)

V0
;

5 " " <_ r~(t) and 21(t) _< ... < 2n(t),V0 < t < n, then

(rl(t),..,rn(t))-<w(fl,(t),..,flù(t)),

V0
(9)

implies P@(F) > t) < P(-c(G) > t),V0 < t < x; st (iii) If the weak majorization in (8) or (9) holds for all t >_ 0, then ,(F) < z(G). PROOF. For any given t we have lnfii(t) = - R i ( t ) and lnGi(t) = -Ai(t) by the

definition of cumulative failure rate function. Hence, from (R~(t),.. ,Rn(t)) -<~ (&(t), .. ,An(t)) we have (-lnFl(t), . . , - l n F n ( t ) ) _<w ( - l n G l ( t ) , . . , - In Gn (t)), which then yields (in fil ( t ) , .., lnle,,(t)) -% (in G1 ( t ) , . . , In Gn (t)). Therefore, by Corollary 3 P@(F) > t) < P(r(G) > t),V0 < t < x. Now consider (il). From the assumption in this oase and the definition of weak supermajorization we have ~ / k l ri(t) > ~ik=l 2i(t),V 1 < k < n, and V0 < t < x. These inequalities immediately imply ~~=lRi(t)=~,~=lfóri(s)ds>~~_l Jó}~i(s)ds = Ek=l Ai(t), V l < Ic < n, and V0 < t < x. Therefore, from (i) we obtain (ii). [] EXAMPLE 2. Suppose F/ is exponential with failure rate ri and Gi is exponential with failure rate fli(1 < i < n). From Theorem 3 we see that if @ 1 , . . , rn) _<w ( 2 1 , . . ,fl~), then ~(F) < r(G). For instance, (3.5,4,7) -< (1,5,8) so the lifetimes of series, parallel and 2-out-of-3 systems consisting of components with exponential distributions exp(3.5),exp(4),exp(7) is stochastically smaller than their counterparts from exp(1),exp(5),exp(8) components even though r2 = 4 < 5 = 22 and r3 = 7 < 8 = 2» The following result generalizes Corollary 2.7 of Pledger and Proschan (1971). COROLLARY (Proportional failure rate). Let r(t) be a failure rate function and L = (21,..,)~~) and I1= (th,...,rl~) be two vectors with positive eomponents.

Suppose that Fi, Gi have failure rate functions 2ir(t) and tlir(t), respectively. I f st z «~ n, then ~(F) «_ ~(c).

PROOV. It is true since (flq,...,~n) _.~w (/'/1,''',/~n)

implies ()qr(t),..,2~r(t)) _~w O h r ( t ) , . . ,~,r(t)). follows.

Hence,

the

desired result []

EXAMPLE 3. Let F~-(t) = F/(t; cti, )~i), Gi(t) =- Gi(t; fii, ~i), 1 «1 . . . . . c~n -z ~ > fi ~ fil . . . . . fin > 0. If

J, Mi

222

( 2 1 , . . ,2n) _<w (r/l,..., t/,), then P(v(F) > t) < P(r(G) > t),Vt > max{1/2i, 1 _< imax{1/2i, l l and thus (2it) ~ > (2it) ~, 1 /9. Then, same as in the proof of Lemma 1, we conclude that ((21t)~,.., (2~t) ~) _<w ((21t)/~,.., (2j)/~),Vt _> max{1/2i, 1 t) _ t),Vt >_ max{1/2~, 1 < i < n}. It is easy to see that in this example if we further assume the shape parameters st c~ =/~ c (0, 1], then the above result can be strengthened as z(F)_< z(G). In st particular, if c~ =-/~ C (0, 1] and 2~ . . . . . 2~ =- f/= ~~~_~ tli/n, then z(F) < ~(G) since ( 0 , . . , 0n) -< ( q l , . . . , %) always holds. In reliability theory, the mean residual life (MRL) function is another important quantity since there is a one-to-one correspondence between the set of lifetime distributions and the set o f M R L functions, In the following, for convenience we assume all the involved M R L functions are continuous without repeat. Out next result will use M R L function to compare the lifetimes of k-out-of-n systems. THEOREM 4. Let the associated M R L functions of Fi(t) and Gi(t) be i~i(t) and vi(t), respectively. Suppose that for a given x > 0 the following conditions are satisfied: 1° For each pair of indexes i < j the inequality #i(t) _< #j(t), V0 < t < x holds. 2 ° For each pair of indexes i < j the inequalities vi(t)< vj(t),V0 < t < x and vi(O)/vi(t) <_ vj(O)/vj(t), V0 < t < x hold simultaneously 3 ° {,ù,(t) t,~,(o),

4° ( ~

)Z1 t ) ' '

uù(t))

~,(0"~ _<~ (~,(t)

- • •, ~ù(o)j

VO < t < x.

t,v, ( o ) , • • • , ~,,(o)2

1)-<w(~@(t)

" " ' ~/3[)

ù"

~~6 )

V0
Then P(z(F) > t) < P(z(G) > t), V0 < t < x. In particular, if the above condist

tions are true for all t > 0, then r(F) < z(G). PROOF. For any pair i < j from condition 2 ° we have -

ds<_-

ds

V0
Also the condition 2 ° implies In

<_ln

Thus, the sequences

V 0 < t < x

.

Some comparison results of the reliability functions of some coherent systems

223

and

are similarly ordered for any t E [0,x]. The definition of similarly ordered sequences can be found in Marhsall and Olkin (1979). The condition 3° implies

or

(lo) From 1° and 2° we have _ _1> . . . > _ _ ]Jl(S)

_

1 _

_ _1> . . . > _ _

~n(S

UI(S ) --

-

1 __

V0<s<x

Vn(S)

and thus by 4 ° further have for any 0 < t < x t

1

t

/0 ) l ~ d s > -

/0

t__1 #,-1 (s) ds + t

1

fo v ~ ) ds'

1

t

1 as >_ t

/o

t__1 ds + v, 1(s)

1

t

//1

ds,

v,~

1

t

v - ~ ds

V0 < t < x .

1

fo #~)ds+'"+fo #~)ds>-f v~ds+'"+ fo v~ ds" The inequalities imply t (-f0

1 ~ds,'",-f0

-<w -

t

1 ~

ds,...,-

d,)

The weak submajorizations (10) and (11) yield

(ll)

224

J. Mi

(ln#~(0)_

#l(t)

* 1

(lnV,(O)

V'w \

,ln#~(0)

t

1

f0 ~-7~ ds''' " ~--~ -- f0 ~t-7~ ds) '1

vl (t ) -- fO v-7-~ ds, " . , ln ~

'1

- foo v - ~ ds ) •

(12)

since the sequences at the right-hand sides of (10) and (11) are similarly ordered. Note that by the inversion formula we have Fi(t ) = #i(0) e-

fót~i~7 1 ds

and thus In Fi(t) = In #i(0) Bi(t ) - f0 t ~ 1

ds .

(13)

ds ,

(14)

Similarly, it is true that in äi(t) = in

-

Comparing (13) and (14) with (12), we conclude that (ln F l ( t ) , . . , l n

F,,(t)) -% (in ä l ( t ) , . . , l n

Gn(t))

V0 < t < x .

Therefore, the desired result follows from Corollary 2 of T h e o r e m 2.

[]

COROLLARY (Proportional MRL). Ler #(t) be a mean residual life function.

Suppose that the M R L function associated with F~(t) and Gi(t) are given as ~---,~ifl(t) and vi(t) = 7ifl(l), respe«tiv«ly. If k -1 _<w q-l, i.e.,

Bi(l)

st

the~ ~(v) < ~(c). PROOF. First of all, note that we can assume 21 _< 2 2 . . . < 2n and 71 -< 72 -< "'" -< 7, without loss of generality. Since #i(O)/#i(t) = vi(O)/vi(t) = #(O)/#(t), 1 < i < n, so conditions 1° and 2 ° of Theorem 4 are satisfied. Condition 3 ° is equivalent to ( 1 , . . , 1) _<w ( 1 , . . , 1) which is certainly true. Finally, condition 4 ° is equivalent to

st

which is what we assumed. Therefore, by T h e o r e m 4 we see that r(F)_<

~(c).

[]

Some comparison results of the reliability functions of some coherent systems

225

T h e f o l l o w i n g sufficient c o n d i t i o n s are h e l p f u l for c h e c k i n g 3 ° a n d 4 ° r e q u i r e d b y T h e o r e m 4. LEMMA 1, L e t #i(t), vi(t), 1 ,(0~v'(t)vwnv< ~~< xV 1 #i(t)/vi(t) increases in t E [0,x], 1 vi, 1 < i < n, t h e n u _~w v. F r o m this fact in case (a) we h a v e 3 °, a n d i n case (b) we h a v e 4 °. N o w i n case (c) the a s s u m e d c o n d i t i o n s i m p l y those r e q u i r e d in (a) a n d (b) a n d t h u s b o t h 3 ° a n d 4 ° are true. []

References Barlow, R. E. and F. Proschan (1981). Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD. Boland, P. J. and F. Proschan (1983). The reliability ofk out ofn systems. Ann. Probab. 11, 760 764. Boland, P. J., F. Proschan and Y. L. Tong (1989). Optimal arrangement of components via pairwise rearrangement. Naval Res. Logist. 36, 807-815. Glesser, L. (1975). On the distribution of the number of successes in independent trials. Arm. Probab. 3, 182-188. Hoeffding, W. (1956). On the distribution of the number of successes in independent trials. Arm. Math. Stat. 27, 713-721. Ma, C. (1997). A note on stochastic ordering of order statistics. J. Appl. Prob. 34, 785-789. Marshall, A. W. and I. Olkin (1979). Inequalities: Theory of Majorization and its Applications. Academic Press, New York. Mi, J. (2000). A Unified Way of Comparing the Reliability of Coherent Systems (submitted). Pledger, G. and F. Proschan (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In Optimizing Methods in Statistics (Ed. J. S. Rustagi). Academic Press, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

12t _J

On the Reliability of Hierarchical Structures

Lev B. Klebanov and Gabor J. Szekely

In this paper a one-parameter family of limiting survival functions of hierarchical structures is associated to (almost) every reliability polynomial. These survival functions are generalizations of the exponential survival function (corresponding to the reliability polynomial of series structures). The problem of reconstruction of reliability polynomials from these limiting survival functions is also settled.

1. Introduction

Let N = N1 be an arbitrary network, and call it the first generation. Let us obtain the second generation N2 from N1 by replacing each component of N1 by N. Similarly, if we replace each component of N2 by N then we get the third generation N3, and so on. Suppose the components of N function independently and each component has the same probability p of proper functioning. Then the chance that N functions properly is a polynomial of p, called the reliability polynomial which will denote by h(p) (we follow the notation of Barlow and Proschan, 1975). There is a lot of information known about h(p) for monotone structures (we shall consider here only monotone structures in terms of Barlow and Proshan, 1975). It is easy to see that the nth function iteration of h is the reliability polynomial of An. Let us denote this nth iteration of h by h°n(p). The theory of iterative functions (see Kuczma et al., 1990) is a good tool to stndy these hierarchical networks. The limiting function F of the survival functions Fn of _~~ with linear normalization (as n tends to infinity) can be reduced to the solution of the equation

il(t) = h(F(zt)) for some z E (0, 1). A similar equation plays an important role in the theory of branching processes but there the role of h and of fi is played by the generating function or characteristic function of the corresponding distribution F (see Harris, 1963 or Athreya and Ney, 1972). This makes the solution of the above equation 227

L. B. Klebanov and G. J. Szekely

228

completely different. Equations of this form were introduced by Poincaré (1980) (see also Valiron, 1954). Poincaré's paper remains a very valuable source. Two of the properties of h(p) for monotone structures we are going to use are that h(p) is an increasing function o f p in the interval [0, 1] and it is S-shaped (for details see Barlow and Proschan, 1975). Thus h(p) can have at most one fixed point in the open interval (0, 1). It is natural to suppose that 0 and 1 are always fixed points: h(0) = 0 and h(1) = 1. Now consider a reliability polynomial h(p) and suppose that it has three fixed points in the closed interval [0, 1]: p = 0, p = 1 and p =Po. Let to and tl (h _< to) be arbitrary non-negative numbers and introduce the following survival function: /70(t) =

{

0, po, 1,

t > to, to--_ > t_> h, t<

(1.1)

t1 .

This function satisfies the following equation:

P0(t) = h(F0(t))

(1.2)

for all t. This means that

Fo(t) = h°n(Fo(t) ) for all t and all n, hence F0 is a 'fixed point' function therefore if the component survival function is/~0 then system survival function is the same for every generation Am, thus Po is a trivial limit survival function. The same holds if po does not exist. In this case all 'fixed point' survival functions are degenerate at some (arbitrary non-negative) point to. In order to get non-trivial limit survival functions we need a suitable linear normalization. In the following section we discuss such a normalization in detail.

2. The main equation and its analytic solutions For simplicity we introduce the notation f instead of F, and with this notation we are going to investigate the functional equation

f ( t ) = h (f(zt) )

(2.1)

for suitable values of the parameter r ~ (0, 1). At frst, consider Eq. (2.1) under the additional restriction that f has an analytic continuation from an interval (0, õ] (6 > 0) to a disc Itl < õ of the complex plain. In this case, using the fact that r ~ (0, 1), it is obvious t h a t f has an analytic continuation to the whole complex plane as an entire function. In the following theorem we suppose that the derivative h~(1) > 1. In the typical S-shaped form of h we have a (unique) solutionp0 o f p = h(p) in the open interval (0, 1). In this case instead of h~(1) > 1 we need h'(p0) > 1. For more details see Section 7.

On the reliability of hierarchical structures

229

THEOREM 2.1. Let h(p) be a polynomial in p such that."

(i) (ii)

h(1) -- 1 and h(p) > 0 in the interval (0, 1]," h'(p) > O for all p C (0, l) and h'(1) > 1.

Put r = 1/h'(1). Then for any a > 0 Eq. (2.1) has a solution f ( t ) analytic in the

disc ltl < c~, (~ > 0) and satisfying the conditions: f ( 0 ) = 1,

f'(0) = -a

.

(2.2)

This solution is' positive, monotone decreasing for t > O, unique in the class of functions satisfying (2.2), and analytic in the disc [t] < 6. REMARK. U n d e r the conditions of this theorem there is no solution po o f p = h(p) in the open interval (0, 1), otherwise (2.1) would imply that h(p) is identically equal to p, and thus h~(1) > 1 cannot hold. Using this observation it is easy to show that {01

F(t) =

- f(t)

for t < 0, for t _ > 0

(2.3)

is a cumulative distribution function. The p r o o f is the following. According to T h e o r e m 2.1 f ( 0 ) = 1, and f is positive, m o n o t o n e decreasing, therefore limt~o~f(t) = v exists. We only need to prove that v = 0. Since f satisfies Eq. (2.1) we can take the limit in b o t h sides of (2.1) as t ---+ oc. Thus we get v = h(v), that is v = 0 or v = 1. But the case v = 1 can be excluded since f ( 0 ) = 1 and f is strictly decreasing. EXAMPLE. If N is a series structure of k > 1 c o m p o n e n t s then h ( p ) = p h , h'(1) = k > 1 , r = l / k , and f ( t ) = e -at is the solution of (2.1) satisfying (2.2). PROOF. The p r o o f is divided into four parts. (i) Set f ( t ) = ~~=oant n, where a0 = 1 and al = - a . Then the coefficients an, (n _> 2) can be determined uniquely f r o m Eq. (2.1). T o see this, differentiate Eq. (2.1) n times with respect to t. F o r n = 1 we get

f ' ( t ) = h'(f(vt))f'(rt)'c

(2.4)

for n _> 2 we obtain n!

( f ' ( ' c t ) ~ mt (f(~)('ct))m"zn 1! J " " \ n! J

(2.5)

f(n)(t)=z---'ml!m~~..mn(h(s)oc(zt))\~"

where the sum ~ is taken over all non-negative integer m l , . . . , m n under condition ~ j = l jmj = n, and s = ml + ... + mù. Plugging t = 0 into (2.4) and (2.5) we see that the value of al can be arbitrary, and using (2.5) the values of an = f(n)(O)/n] for n >_ 2 are determined uniquely by a0 and al. (ii) The series ~n~0 aùt" converges in the disc Itl < 6, (6 > 0).

230

L . B. K l e b a n o v a n d G. J. S z e k e l y

Indeed, putting t = 0 into (2.5) and using h~(1) = 1/r we obtain f"(O) z" n! -- l - - - ~ n - l E m l ! . . .

an--

lm~

1! h/,)(1)a~,,

m°_, ""an-1

(2.6)

here m l + • .. + mù-i = s and the sum ~ is taken over all non-negative mj under the condition ~j~__-]j m j = n. We are going to show that [azJ _< AL l, l = 0, 1 , . . for some constants A > 0, eA < 1/~ and L > 0. Suppose that this inequality is proved for all l _< n - 1 where n is sufficiently large, and let us prove it for l = n. F r o m (2.6) we see that for n > 2 C 1 ih(~)(1)lASL" la"l -< 1 --7~-a Zml[...mn_l! --i-

C L~ 1 C -~ ~ml!...mn_l!

z" <- 1 - 7 " - I L " M Z m l ! . . . m ù

As]h<s)(l)]

1

1!

As ,

(2.7)

where M = max0<s<m ]h(s) (1), and m is the degree of the polynomial h. Let us estimate

Z

1

mm[...mn

1!A

s

'

where the sum ~ is taken over all non-negative integers mj, j = 1 , . . , n satisfying the conditions v~,n2__,j=11J 'mj = n, and s = m l + ... + mù-i We have oc Am1

Zml

1

1

.mn_l[

oo A m 2

A~ < EmB~.v EmB2.v --

. , .

mi=l

• m2=1

oo ..

•

" Z

1

Amù_ l

~

<e(n-1)A _

mn_i=l

--

.

F r o m here and (2.7) if n is sufficiently large we have C Me(ù_I)AL ~ <_ AL ~ . fa"[ < 1 - v n-1

Therefore the series ~ù~0 a S

converges in a disc

Itl <

&

(õ > 0).

(iii) The solution constructed in (i) and (ii) is m o n o t o n e decreasing. F r o m (2.1) we see that f ' ( 0 ) < 0. Therefore f is m o n o t o n e decreasing in interval [0, e). N o w (2.4) and the positivity of h' in (0,1) imply that f ' ( t / z ) = h ' ( f ( t ) ) f ' ( t ) is negative in [0, e/r). Induction shows that f ' ( t ) is negative for all non-negative values of t. (iv) f ( t ) > 0 for all t > O. I f f ( t 0 ) = 0 and f ( t ) > 0 for 0 < t < to then 0 = f ( t o ) = h(f(zto)) > 0 .

This contradiction proves (iv). The statement of T h e o r e m now follows from (i), (il), (iii), and (iv).

On the reliability of hierarchical structures

231

REMARK. It is obvious that if f is any solution of Eq. (2.1) satisfying the conditions of Theorem 2.1, then f has an analytic continuation from any interval of the form (0, 6) to the whole complex plane as an entire function.

3. The general solution of the main equation In this section we determine the general solution f of Eq. (2.1) without assuming that f is analytic. DEF~NrrIoN 3.1. Let fo be the solution of (2.1), satisfying the conditions

f(O)=O,

f'(O)=-I

and all other conditions of Theorem 2.1. In this case we say that fo is the standard solution of (2.1). TnEOREM 3.1. Suppose that the polynomial h satisfies all conditions of Theorem 2.1. Let fo be the standard solution of (2.1), and f be any other solution of (2.1)for t > O. Then there exists a In z-periodic function ~ such that

f ( t ) =fo(tq/(lnt)),

t> 0

(3.1)

and vice versa, for any lnz-periodic O the function (3.1) is a solution of (2.1)for t>0. PROOF. Since f0 is strictly monotone (decreasing), we can introduce the following function: ~(t) = ~fo'

ü'(t))

•

This means that f(t) =f0(t~(t))

.

From the equätions

fo(t) = hOCo(zt)) and

f ( t ) = h(f(rt)) we obtain

hOeo(zt(p(zt) ) ) = h(fo(zt~o(t) ) ) , which is equivalent to (2.1). But both h and f0 are monotone, therefore the previous relation is equivalent to

~o(~t) = ~o(t) .

L. B. Klebanovand G. J. Szekely

232

Put t = e ° and (p(e°) = 0(0). So, we have 0(0) = ~/(0 - in r) that is ~ is a in z-periodic function. The proof is complete.

[]

It can be interesting to find conditions that guarantee that the function ~ in (3.1) is constant. Our Theorem 2.1 shows that it is sufficient to suppose that f has analytic continuation from [0, 6). But this condition is too strong. PROPOSITION 3.1. Let Y be the class of functions given on R+, differentiable in an interval [0, 6) such that the derivative f ' satisfies the Hölder condition (3.2) below (with fixed parameters) in the underlying interval. Suppose further that the poIynomial h satisfies all conditions of Theorem 2.1 and maxzc[0,1] Ih'(z) l = h'(1). Then Eq. (2.1) has only one solution satisfying (2.1) within the class ~ . PROOF. Introduce the class Y, of all monotone decreasing functions from ~satisfying (3. l). All functions f in ~,~, satisfy the Hölder condition If(tl) -- f(t2)] _< KIq - t2lr ,

(3.2)

where K > 0 and r > 0 are universal constants for the class Y » Fix an arbitrary ~ (0, r) and introduce the distance:

«(fl,f2) = ~00°° ] f l ( t ~ ~ [ 2(t) l d t ,

(3.3)

in Æ,. The distance d is finite in o~, and with this distance Yl is a complete metric space. Consider now the operator T defined on ~ , by the relation

Tf(t) = h(f(rt)) .

(3.4)

We have

d(~,

/0 ~ Jh«l(~,/~~~~(~,//Id,

~/

_< h,(1)zl+~ f0 °~ Ifl(0t2+~ -f2(t)l dt =

~~d(A,f2)

.

Here we used the definition of ~ = hr(1). The inequality

d(Tfl, Tl2) <_ "cOrd(f1,f2) means that the operator T is a contraction, and therefore it has only one fixed point in the space ~,~, This makes the proof complete. []

On the reliability of hierarchical structures

233

4. Other normalizations In the previous sections we considered Eq. (2.1) where z = 1/h'(1). In this section we are going to consider other values of z. To avoid any misunderstanding let us rewrite Eq. (2.1) in the form

f(t) = h(f (~ct)) ,

(4.1)

where we do not suppose that ~c--1/h~(1). As before, we suppose 0 < ~ = h1(1) < 1. It is easy to represent the solutions of (4.1) with the help of the solutions of (2.1). If ~0 is a solution of (2.1), that is

~o(t) = h((p(zt) )

(4.2)

then introduce c~= ln z/lntc and define f ( t ) = ~0(t~). This function f clearly satisfies Eq. (4.1), an conversely, if any function f satisfies Eq. (4.1) then (p(t) = f ( t 1/~) satisfies (2.1). (It is needless to emphasize that only the positive values of t are considered.) So, we proved that under suitable conditions every solution of Eq, (4.1) has the form f(t) = ~o(t~), where (p corresponds to the solution of (4.2), and et - In •/in ~c.

5. Reconstruction of the reliability polynomial from the limit distribution Suppose we know the function f(t) (the solution of (2.1)), but we do not know the reliability polynomial h of the network N. Is it possible to reconstruct the polynomial h knowing only f ? In other words, can f be a solution of two different equations

f(t) = hl(f(rlt) )

(5.1)

f(t) = h2(f(~2t))

(5.2)

and

PROPOSITION 5.1. Suppose that the degrees of hl and h2 are bigger than 1. Then the

equations above imply that there exist positive integers n and m such that (i) h~n = h~m, or

(ii) there exist linear fractions that transJòrm hl and h2 to the power functions pn and p m, or to the pair of Chebyshev polynomials Th, Tm, respectively. PROOF. We are going to prove that the polynomials hl and h2 are commutable in the sense that hl o h2 = h2 o h l . To see this observe that from (5.1) and (5.2) we get

f (t) = hl (h2U(~l~2t) ) )

L.B. Klebanov and G. Z Szekeß

234

and f ( t ) = h 2 ( h 1 (f("c2"Clt)))

,

hence hi(h2(f(~iz2t))) = h 2 ( h l ~ ( ~ l ~ 2 t ) ) )

•

(5.3)

But here f is monotone and therefore from (5.3) we get that h l o h2 (p) = h 2 o h l (p) for p in some interval, and therefore for all complex values ofp. The result now follows from the description of all commutable polynomials in Julia (1922) and Fatou (1921) (see, also, Eremenko and Lyubich, 1989).

6. A generalization of the main equation

In this section we are going to prove a generalization Theorem 2.1 for the case when the polynomial h is replaced by an arbitrary analytic function. In order to emphasize the difference, in this section we change the argument of h from p to z. THEOREM 6.1. Let h(z) be an entire function in z such that."

(i) h(1) = 1 and h(z) > 0 in interval [0, 1]; (ii) h'(z) >O for a l l z E (0,1) andh'(1) > 1. P u t ~ = 1/h'(1). Then for any a > 0 Eq. (2.1) has a solution f ( t ) , analytical in a circle It] < ~, (6 > 0) satisfying to the conditions." f ( 0 ) = 1,

S(0) = -a .

(6.1)

This solution is positive and monotone decreasing for t > 0 and unique within the class of analytic solutions in a circle It[ < ~ and satisfying (6.1). PROOF. The proof is very similar to that of Theorem 2.1. Copy (2.6) from the proof of Theorem 2.1 an

fn(O) _ C ~ 1.mn_l!h(,)(1)aT~ ... an-lm"-~ ~! 1 ---~~ 1 Z-~m~!..

(6.2)

for all n > 2. Since the series

j•h••(1) j!

( _ 1): ,z

converges for all z, we have that there exists an M > 0 and an t/ E (0, 1) such that

Ih«/(1)/j!j ~ M#

On the reliability of hierarchical structures

235

for all j = 1 , 2 , . . We are going to show that there also exists an L > 0 and an A > 0 such that ]aj] <_AU j = I,2, .. Suppose that the inequalities lajl <_ALJj= 1 , 2 , . . , n 1 hold where n is sufficiently large, and prove the corresponding inequality for an. Since ~ E (0, 1) we have

C _ 1

~~Ln _~ù-1

Ih(~>(1)[A~L n

1

1 ASlh(S)(1) ] ~-~~ml!...mn-l!

zn

1

«- 1 - ~-;~r-~LnM~ml!. .mù ~f(AC The rest of the p r o o f of the t h e o r e m is the same as that of T h e o r e m 2.1.

7. A general limit result T a k e an arbitrary continuous cumulative distribution function F and consider h°n(F(t)). We are interested in the limiting behavior of this sequence as n tends to infinity. O u r T h e o r e m 2.1 answered this question in case there was no fixed point in the open interval (0, 1). Here we assume that there exits a fixed point Po in (0, 1). (As we have already mentioned before, the typical S-shape form of h guarantees that there cannot be m o r e than one fixed points of h in (0, 1).) Because of the S-shape f o r m of h we can suppose that the graph of h lies below the line y = p in the interval (0,p0) and h is above this line in the interval (Po, 1). Suppose n o w that F is not only continuous but also 0 < F(t) < 1 holds for all real t. In this case there exists a point to such that Æ(t0) = Po, for any t > to we have F(t) < Æ(t0), and it is easy to see that h°n(Æ(t)) --+ to as n ~ oe. Therefore the limit distribution does not exist (the limit function is a constant, not a survival function). Introduce a new function

Fl(t) = F(to + t) . Obviously, ffl (0) : Po. T o study the asymptotic behavior of h °n (fr1 (t)) we need the usual linear normalization. Consider the asymptotic behavior of h°n(#l(Vt)) for fixed z Œ (0, 1). We can study this equation for the cases t < 0 and t > 0 separately, and thus we arrive at the situation we already considered in Section 2.

Referenees Athreya, K. B. and P. E. Ney (1972). Branching Processes. Springer, New York. Barlow, R. E. and F. Proschan (1975). Statistieal Theory of Reliability and Life Testing Probability Models. Holt, Rinehart and Winston. Eremenko, A. E. and M. Yu. Lyubich (1989). Dynamics of analytical transformations. Algebra Anal. 1(3), 1-70, (in Russian).

236

L. B. Klebanov and G. J. Szekely

Fatou, P. (1921). Sur les fonctions qui admettent plusiers théorémes de multiplication. C.R. Acad. Sci. 173, 571-573. Harris, T. E. (t963). The Theory ofBranching Processes. Springer, Berlin-Göttingen-Heidelberg. Julia, G. (1922). Mémoire sur la permntabilité des fractions rationnelles. An. de l' École Norm. Supér. 39, 131~15. Kuczma, M., Choczewski, B. and R. Ger (1990). Berative Function Equations. Cambridge University Press, Cambridge. Poincaré, H. (1890). Sur une classe nouvelle de transcendantes uniformes. J, Math. Pures Appl. 6, 313-365. Valiron, G. (1954). Fonctions Analytiques. Presses Universitaires De France, Paris.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

1

lk ~j

Consecutive k-out-of-n Systems

N. A. Mokhlis

Notation

x/

state of component i (i.e. X~ = 0 or 1 according as component i fails or works) reliability of component i (Pi = Pr{X~ = 1}) failure probability of component i (i.e.

Pi

qi

qi = 1 - pi) 7

R~F(k; n; p l , . . . , p~)

R~o(k;n;pl,..

. ,pù)

R~F(k; n ; p )

R~o(k; n;p)

P~4

R,~F( (r,s); (n,m); Pl,1, . . ,Ph,m)

implies L (linear) or C (circular) reliability of a 7-consecutive k-out-of-n: F system with component reliabilities pi, i= 1,..,n reliability of a 7-consecutive k-out-n: G system with component reliabilities pi, i=l,...,n. reliability of a 7-consecutive k-out-of-n: F system with identical component reliabilities p, i.e. Pi = P for i = 1 , . . , n reliability of a 7-consecutive k-out-of-n: G system with identical component reliabilities p reliability of the lattice-system component located in /th row (circle) and jth column (ray) reliability of y-connected-(r, s)-out-of(n,m): F lattice system with component reliabilities pi,j; for i = 1 , . . n and j=

R~G((r, s); (n, m ) ; P l , 1 , . .

,Ph,m)

l,...,m

reliability of 7-connected-(r,s)-out-of(n, m): G lattice system with components

237

238

N. A. Mokhlis

reliabilities ])ic; for i = l , . . . , n and j= 1,..,m reliability of ~-connected-(r, s)-out-ofR~v((r, s); (n, m);p) (n,m): F lattice system with identical component reliabilities p reliability of 7-connected-(r, s)-out-ofR,yG((r, s); (n, m);p) (n,m): G lattice system with identical component reliabilities p RyF((r, «)-or-(s, r); (n, m);Pl,1,.., Ph,m) reliability of 7-connected-(r,s)-or-(s,r)out-of-(n,m): F lattice system with component reliabilities pi,j; for i = 1 , . . , n andj=l,...,m R~G((r, s)-or-(s, r); (n, m);pl,1,.. ,Ph,m) reliability of 7-connected-(r,s)-or-(s,r)out-of-(n,m): G lattice system with component reliabilities pi,j; for i=l,...,nandj=l,...,m R~v((r, s)-or-(«, r); (n, m);p) reliability of 7-connected-(r,s)-or-(s,r)out-of-(n, m): F lattice system with identical component reliabilities p reliability of pconnected-(r,s)-or-(s,r)R~G((r, «)-or-(s, r); (n, m);p) out-of-(n,m): G lattice system with identical component reliabilities p reliability of relayed unipolar consecutive Ru(k;n;pl,... ,pù) k-out-of-n: F system with component reliabilities pi, i = 1 , . . . , n reliability of relayed bipolar consecutive Rb(k;n;pl,... ,Ph) k-out-of-n: F system with component reliabilities Pi, i = 1 , . . , n reliability of relayed unipolar consecutive Ru(k; n;p) k-out-of-n: F system with identical component reliability p reliability of relayed bipolar consecutive Rb(k; n;p) k-out-of-n: F system with identical component reliability p. largest integer less than or equal to x

1. Introduction

The reliability of consecutive k-out-of-n systems has been studied by many researchers due to their practical importance. Consecutive k-out-of-n systems may be classified into two classes; consecutive k-out-of-n: F and consecutive k-out-of-n: G systems. Moreover, two cases could be considered; the one-dimensional case and two-dimensional case.

Consecutive k-out-of-n systems

239

1.1. The one-dimensional case (a) A eonseeutive k-out-of-n: F(G) system consists of an order sequence of n components, such that the system fails (operates) if and only if consecutive k (k _< n) components fail (operate). The system is called linear or circular depending on whether the components are arranged on a straight line or form a circle. The reliability of linear consecutive k-out-of-n: F system was first studied by Kontoleon (1980), but the name consecutive k-out-of-n: F was first considered by Chiang and Niu (1981). Derman et al. (1982) discussed the circular consecutive kout-of-n: F system for the first time and expressed its reliability in terms of the reliability of linear systems. (b) Relayed consecutive k-out-of-n: F systems was first discussed by Chiang and Chiang (1986). However, Hwang (1988) introduced the notion of relayed unipolar and bipolar consecutive k-out-of-n: F systems. For a relayed unipolar consecutive k-out-of-n: F system the components are arranged in a line such that the system fails if the first component (source) fails or at least k consecutive components fail. While for a relayed bipolar consecutive k-out-of-n: F system, the n components are arranged in a line such that the system fails if the first component (source) fails, last component (sink) fails, or at least k consecutive components fail. The one-dimensional consecutive k-out-of-n systems are of great need in out daylife. Chiang and Niu (1981) and Hwang (1988) indicated their relevance to some telecommunication, oil pipeline and microwave systems. Bollinger and Salvia (1982), and Fu (1986a) remarked that such systems frequently arise in the design of electronic integrated circuits. Kuo et al. (1990) gave an example of railway station. Example 1.1.1 (Telecommunication system given by Chiang and Niu, 1981). A telecommunication system with n relay stations (either satellites or ground stations) numbered from 1 to n. Suppose that a signal emitted from station 1 can be received by stations 2, 3 , . . , k + 1 together, and a signal emitted from station 2 can be received by stations 3, 4 , . . , k + 2 together, etc. Thus when a number of consecutive stations less than k are failed, the telecommunication system still is able to transmit a signal from station 1 to station n. However, if any k consecutive stations fail, the system must fail. Chiang and Niu (1981) introduced a similar example of consecutive 2-out-of n: F system. Example 1.1.2 (Oil pipe system introduced by Chiang and Niu, 1981). A system for transporting oil by pipes from point A to point B has n pump stations. Pump stations are equally spaced between points A and B. Each pump station can transport the oil a distance of k pump stations. If one pump station is down, the flow of the oil could not be interrupted because the next station could carry the load. However, when at least k consecutive pump stations fail, the oil flow stops and the system falls.

240

N. A. Mokhlis

Example 1.1.3 (Railway station introduced by Kuo et al., 1990). A railway station has n lines that receive and send trains. Consider an over size train that requires k consecutive lines in order to enter the station without delay. Then the reliability of the system is the probability that the train enters the station without delay is itself the probability that at least k-consecutive lines which are not in use are available. This is an example of linear consecutive k-out-of-n: G system. Example 1.1.4 (Relayed systems described by Hwang, 1988). In a telecommunication system some object, be it a message, a flow or a signal, is to be relayed from a source to a sink through a sequence of intermediate stations. Case should be taken as to whether the source or the source and the sink are also considered components of the systems, i.e. whether they serve the same functions as the intermediate stations. In telecommunication system source or source and sink and the intermediate stations are all of the same kind of relayed stations. Thus either source or both source and sink are considered components of the systems. In the mobile communication system it is assumed that the source and intermediate stations are all photo-transmitting space-craft bnt not the sink, which could be just an antenna. In these examples where the source (source and sink) is a component of the system, the system fails if the source (source or sink) fails regardless of the value of k. Such a system is an example of relayed consecutive k-out-of-n: F system, unipolar if only the source is included, and bipolar if both source and sink are parts of the system.

1.2. The two-dimensionaI case Salvia and Lasher (1990) introduced the notion of the two-dimensional consecutive k-out-of-n: F systems. Boehme et al. (1992) presented a generalization of such systems as follows. (a) A linear connected X-out-of-(n,m): F(G) lattice system consists of nm components arranged like the elements of (n, m)-matrix, i.e. each of the n rows includes m components and each of the m columns includes n components. The system fails (operates) whenever at least one subset X of connected failed (operating) components occurs which includes failed (operating) components connected in the meaning of connected 2-. (b) A circular connected X-out-of-(n,m): F(G) lattice system consists of n circles, centered at the same point with m rays. The intersections of the circles and the rays represent the elements, i.e. each of the circles includes m components and each of the rays has n components. The system fails (operates) whenever at least orte subset X of connected failed (operating) components occurs which includes failed (operating) components connected in the meaning of connected X. Linear and circular-connected-J(-out-of-(n,m): F(G) lattice systems are generalization of the one-dimensional linear and circular consecutive k-out-of-n: F(G) systems. This is clear by taking X = (1, k), n = 1 and m = n. The subset X could take one of the following forms:

Consecutive k-out-of-n systems

241

(i) (r, s) matrix, r _< n, s < m. (ii) (r, s) matrix or (s, r) matrix, r, s _< n, m, r ¢i s. Connected (r, s)-out-of- (n, m): F(G) and connected (r, s)-or-(s, r)-out-of-(n, m): F(G) lattice systems are found in our daylife in electronic devices, diagnosing diseases, measuring temperature, modeling supervision systems, etc. Example 1.2.1 (Diagnosing disease as presented by Salvia and Lasher, 1990). The presence of a disease is diagnosed by reading an X-ray. The radiologist might not detect the presence of the diseased cells unless they are aggregated into a sufficiently large pattern (say an r x s rectangle). The probability that a small portion (say n x m) of the X-ray is healthy is the same as the reliability of a linear connected (r, s)-out-of-(n, m): F lattice system. Example 1.2.2 (A supervision system was presented by Boehme et al., 1992). A supervision system consisting of 16 TV cameras arranged in four rows and four columns, such that each camera can supervise a disk of radius r. The cameras in each row and column are of the same type and are at a distance r from each other. The supervision system fails if at least two connected cameras in a row or a column fail. This is a linear connected (1, 2) or (2, 1) out of (4, 4): F lattice system. Example 1.2.3 (Measuring temperature system was introduced by Boehme et al., 1992). A cylindrical object covered by a system of feelers for measuring the temperature with five circles each includes four feelers. The measure system fails whenever at least one connected (3, 2) or (2, 3) matrix of failed components occurs. This is a circular connected (3, 2) or (2, 3) out of (5, 4): F lattice system. We shall give a brief survey concerning the exact and approximate formulas as well as bounds for the reliability of one- and two-dimensional consecutive k-outof-n systems. Chao et al. (1995) gave a chronological survey of computing the reliability of consecutive k-out-of-n: F systems.

2. Exact reliability formulas 2.1. Linear consecuti~e k-out-of-n: F s y s t e m s

Assume that the system consists of n linearly arranged independent components. The system falls iff at least k consecutive components fail. Let Pi be the reliability o f / t h component qi - 1 - p » Let Xi denote the state of component i, where X/ = 1 if the component is operating, Xi = 0 if the component is failed. Let X be an n-vector such that the component i of X is 1 or 0 depending on whether component i of the system is operating or not, respectively. Chiang and Niu (1981) presented a recursive formula to compute the exact system reliability by defining the vector X of component states and a random

N . A . MokhlB

242

variable indicating the index of first 0 in the vector of states. They gave recursive equations requiring O(n2k) computing time. The derivation of the recursive formula is based on examining the first sequence of consecutive 0's in the X vector. If the number of consecutive 0's in the first sequence is at least k, then the system is failed. If the number of consecutive 0's in the first sequence is less than k, then the reliability of the system is equal to the reliability of a consecutive k-out-of-n': F system where n' < n. Let Z be a random variable indicating the index of first 0 in X, and M be a random variable indicating the index of first 1 after position Z in X. The reliability of the system is RLF(k; n;p) = Pr {the system is operating} = Z

Pr{the system is operatingIZ = z,M = m}

Z

z

m

Pr{Z = z , M = m} . When Z > n - k + 1, the system will have less than k failed components and Pr {the system is operatingIZ > n - k + 1} = 1 and Pr(Z > n - k + I) = ph-k+1. When m > z + k, the system will have already consecutive k failed components, and the system is failed. Thus n k + l z+k-1

RLv(k;n;p) = ~ z--1

Z

Pr{the system is operatinglZ = z,M = m}

m=z+l

x Pr{Z = z , M = m} +ph k+~ For z + 1 < m < z + k - 1, the first sequence of failed components does not constitute a cut set. Furthermore, since Xm = 1, the event that consecutive k-outof-n: F system is equivalent to the event that a consecutive k-out-of-(n - m): F system is operating. It is clear that Pr{Z = z, M = m} = prqm-r where q = 1 - p. Thus, the recursive formula for RLF(k; n;p) is n - k + l z+k 1

RLv(k;n;p)= ~ z=l

RLF(k;j;p) =

{1,

0,

Z

RLF(k;n--m;p)Prqm-r +p" ~+~ '

(2.1)

m--z+l

O<j<_k,

j < 0 .

Derman et al. (1982) derived an exact formula for RLF(k; n;p) by considering the positions in the system of failed and operating components. The recursive equations require O(n 2) computing time. The expression is given by RLF(k; n;p) = Z N ( j , n - j + 1; k - 1)pn-Jqj ,

(2.2)

j-0

where N(j, r; m) is the number of ways in which j identical balls can be placed in r distinct urns subject to the requirement that at most m balls are placed in any one urn.

Consecutive k-out-of-n systems

243

Suppose t h a t j of the components fail, one can imagine that the n - j operating components are placed between n - j + 1 urns. The number of components positioned before the first operating component is the number of balls in urn 1; the number between the first and second operating components is the number of balls in urn 2, etc. With j failed components, if each of the ums has fewer than k - 1 balls, then the system will operate. There are N(j, n - j + 1 ; k - 1) arrangements that satisfy the condition. Since for each j each arrangement has probability B-JqJ, it follows that RLv(k;n;p) is as given in (2.2). Moreover, N(j, r; m) can be expressed in a recursive form as

N(j,r;m)=~(:)N(j-mi,

r-i;m-1),

m>_2 ,

(2.3)

i=O

N(j,r;1)=

I

(«), O«_j
0,r

j>r

.

For the special case k = 2, [n+1/2] ~ (

RL(2, n;p) = ~-~

)

n - j + 1 pn-jqj J

(2.4)

.

Eq. (2.4) was derived by Chiang and Niu (1981) by considering the number of ways that, o f j failures and ( n - j) successes, at least one success can be placed between every two failures. Hwang (1982), derived two sets of recursive equations for the reliability of the system even when components are not necessarily identical, by two different arguments. Let Ei denote the event that component i is the last operating one, and F~ denote the event that the system first falls at component i. Clearly E i ' s a r e disjoint events and exactly one of the events En-k+1,.. ,E,, taust occur for the system to operator. Thus we have

RLF(k;n;p~, ..,pn) = ~

P(Ei)RLF(k;i- 1;pl,... ,Pi-1)

i~n-k+l

=

Pi

RLF(k;i- 1;pl,.

i=n-k+ l

I qj

,Pi-1) ""

(2.5) with the boundary condition

RLv(k;n;pl,...,pn) = 1 for n < k . For identical components Eq. (2.5) becomes RLF(k; n;p)

= ~ i=n-k+l

pqn - i R

LF(k;

t" - -

1;p)

.

(2.6)

244

N. A. Mokhlis

A better a p p r o a c h is to w o r k with F~. In the definition of F~, i is the smallest subscript such that the k consecutive c o m p o n e n t s i - k + 1, i - k + 2 , . . , i - 1, i fail. In particular, if i > k, then F / i m p l i e s that c o m p o n e n t (i - k) is operating. Since Fi's are disjoint events and one of them must occur for the system to fail, we have //

1 -- RLF (k; n ; P l , - . . ,

Ph) = Z P(Fi)RLF (k; i -

k - 1; P l , . . . ,

Pi-k-1 )

i=k n

i

= ~-~RLv(k;i-k-1;pl,...,Pi-k-~)pi-k i=k

II

«J

(2.7)

j=i-k+l

with b o u n d a r y conditions

po-l,

RLF(k;n;pl,...,pù) = I

forn
.

It is clear that 1 -

RLF(k;

n;pl,... ,Ph)

n--]

i

~__RLF(k;i-k-1;p,,

qj

..,pi k-1)pi-k H

i=k

j-i-k+ l

+RLF(k;n -- k -

1;pl,...

qj

,pn-k-1)Pn-k r I j-n-k+l

1 -RE(k;n-

1;pl,...,pn-1)

+ RLF(k;n- k - 1;pl,...,pn-k-1)Pn-k

qj ,

rI j=n-k+l

R>e(k;

n;pl,...

,p,~) = RE(k; n - 1 ; p l , . . . ,P,,-I)

- RLF(k;n - k - 1;pl,...,pn-k-1)pn-k

~I

qj .

(2.8)

j=n-k+l

F o r identical c o m p o n e n t s

1 --RLF(k;n;p) = I1 - R L ( k ; n -

1;p)]

+RL(k;n-k-

1;p)pq k .

(2.9)

Lambiris and Papastavridis (1985) derived an expression for the numbers NO'; r; k - 1), r _> 0 in Eq. (2.2), by using their generating function

9(t)=\l-tJ

(a=~o(2)

(i=~o (2.10)

=~ß(:)(r+J-k2-1)(-l)atJ j=k;~a=0 j -- k2

245

Consecutive k-out-of-n systems If we set in (2.10), r = n - j ÷ 1, the coefficient of

tJ is

~'+'( )( j_k),B(-1) ~ n - j +~ l n'«

N(j;n-j+l,k-1)=

;~ .

2=0

(2.11)

Substitution of (2.11) into (2.2) yields

ù~~<~ù;ù,=B"~'

(,,-,+ i ) ( : -)

(-1)~p"-Jq i

I

Interchanging the two summations and adding a zero term with 2 = 0 and j = n + 1, we get n+l

(ù1) 2n ~1(:

2=0

-,~k

)

(n-j+

1)!

.

j=2k (2.12)

For computing the inner sum of (2.12), the following expression is used:

'(Pf+q) n-;'k- lq2k~-'~~(~-)~k'~rln}*" -d'jB-j+lk)~

(2.13)

j=2k Differentiating both sides of (2.13) 2-times and setting t = 1, we get

~(7 j=,~k =p2q2k

(n~J +1

(2.14)

(n -- ~k)! (n-2k-•)!

~.p2-1q2k ~-

(n -- ~k)! (n - R k -

(2-1))!

"

Substituting (2.14) into (2.12) we get

RLF(k;n;p)=~-~~(n--22k)(--l)2(pqk)2 2=0

(2.15)

- qk ~~=o( n - 2k - k ) (- i )X(pqk)2 . Chao and Lin (1984) employed the concept of taboo probability to find a general closed-form formula for the reliability of linear consecutive-k-out-of: F systems with identical components. They defined a finite-state Markov chain Y1, Y 2 , . - , such that the reliability in (2.2) can be represented as a taboo probability, and thus can be written as the nth power of a sub-stochastic matrix. For the given Bernoulli sequence X'I,X2,.. ,Xn, let Y// = (X/»X}__l,...,X/+kq_i) ,

i= 1,2,..,n-k+

1 .

246

N. A. M o k h l i s

The Y1, Y 2 , . . , form a stationary M a r k o v chain with 2 ~ states. For example, if k = 2, the transition matrix of the four states {(1, 1), (1,0), (0, 1), (0, 0)} Markov chain is

/'2 =

I~q0~ ~~ 0° i]

For the linear consecutive 2-out-of-n: F, the system fails if and only if one of the B equals (0, 0). Hence, (0, 0) could be treated as the taboo set, H = {(0, 0)}. The reliability of the system RLv(2;n;p) is just the probability that Yi ~ H for i=l,2,..,n-1 Let Q2 be the 3 x 3 matrix obtained by deleting the last row and last column of Pc then RLV(2; n;p) = (p2,pq,pq)Q~-2(1, 1, 1) T

(2.16)

where T means transpose. The general expression for RLF(k; n;p) is

RLF(k;n;p)

U k ~c -22 , TI k _

(2.17)

,

where lk is 1 × (2 k - 1) row vector (1, 1 , . . , 1), ak is 1 x (2 k - 1) row vector (ak,1,.., a<j,..., ak,2>l) in which akj = pnkjqk-nk«, (n2,1, n2,2, n2,3, n2,4) = (2, 1, 1,0), ~ nk-lj+l nkj = I. nk 1,j

for k _> 3 and for k _> 3 and

1 < j _< 2 k-l, 2 k-1 q- 1 _< j _< 2 k ,

Qk is (2 k - 1) x (2 k - 1) matrix obtained by deleting both the last column of the 2 k x 2 k matrix

pk(i,j)=

{

P q

0 pk(i -- 2k-l,j)

for for for for

1 < i < 2 k-l, j=2i1, 1 < i < 2 k-l, j = 2i, 1 < i < 2k-l, j e 2 i - 1,2i, 2 k 1 + 1 < i < 2 k, 1 _<j_<2 k .

F u and Hu (1987) and Chao and F u (1989, 1991) developed a simple formula for the general case of s-independent but not necessarily identically distributed components RL(k;n;pl,...

,Ph) = IIo

(n / Mi

k,i=l

uT

,

(2.18)

/

where II0 is 1 x ( k + 1) vector ( 1 , 0 , 0 , . . , 0 ) , uis 1 × ( k + 1) vector (1, 1 , . . , 1,0) and 114, is the transition (k + 1) × (k + 1) matrix

Consecutive k-out-of-n systems

i~~qio...o~l Pi

M~ =

247

0

qi

...

0

0 0

0 0

... ...

0 0

If all the components have the same failure probability then (2.18) reduces to RL(k; n; p(n) ) = rloM"u r Fu (1986b) studied the reliability of the system when the failure probability of component i (i = l, 2 , . . , n) depends on the number of consecutive failures immediately preceding this component; i.e. qi = q(m), m : 0, 1 , . . , k - 1. He interpreted this type of dependency as ( k - 1)-step Markov dependence. An example of this dependency is a system for transporting oll from point A to point B by n pressure pumps. Pumps are equally spaced between point A and point B. Each pump is able to transport the oil a distance of k pumps. Therefore, the system falls if and only if consecutive-k-out-of-n pumps fall. In this case, it is most likely that the component failure probabilities are dependent even if all of the pumps are made with the same quality. The reason for this is that the failure probability of component i depends only on the number of consecutive failures immediately preceding pump i. For example, if pumps (i - k + l) to (i - l) fail and the system still functions, then the pump i has to work very hard to raise the oil pressure to the fixed level so that the oll can travel a distance of k pumps. He introduced recursive equations for system reliability using the following notation:

F(i;k; n) probability that a consecutive k-out-of-n: F system fails at component i F*(i;k; n) probability that a consecutive k-out-of-n: F system falls before and including component i R(m; k; i) probability that the system is working at component i, while components i - m + 1,i - m , . . . ,i are failures With initial conditions

F(i;k;n)-:O

fori--0,1,..,k-1,

k-1 F(k; k; n) = I I q(m) ,

(2.19)

m=O

R(0; k; 0) --1,

R(0;k;1)=p(0),

i I

R(i;k;i)=Hq(j) for i 1 , . . , k - I , j=0 R(m;k;i)=O for i < m < _ k - 1 . The recursive equations are:

(2.2o)

N. A. Mokhlis

248

F(i;k;n)=

q(j) Z p ( m ) R ( m ; k ; i - k - 1 )

fork+l
,

m=0

(2.21) k-1

R(O; k; i) = Zp(m)R(m; k;i - 1), m--O

R(g;k;i) =

q(l') ~-~~p(m)R(m;k;i-g- 1) \j=0

m=0

and i = l , 2 , . . , n ,

for 1 < g < k - 1 i

F*(i;k;n)=~F(l';k;n)

fori= 1,2,..,n

.

j-O

Clearly, the reliability of the system, which is a function m = 0, 1 , . . , k - 1 is given as

of p(m),

k-I

RLF(k; n;p(m)) = Z R(m; k; n) ,

(2.22)

m=0

where

R(m;k;n)=

(Ti

)~'

q(j) Z p ( r ) R ( r ; k ; n - m - 1 )

\j=0

(2.23)

r=0

with initial conditions given by (2.20). When the dependence disappears and component i has failure probability qi, then [I~_ö1q(/') is replaced by [I~-i-k+l qj and p(m) is replaced by pi-k, in (2.21). Also F(i; k; n) becomes

F(i;k;n)=

qj p i - k Z R ( m ; k ; i - k - 1 ) =i-k+ 1

m=0

= (/=i_kII+lqj)Æi-kRLF(k;i--k--1;pl,"

,Pi-k-1)

thus we ger

i=0

=i-k+l

x RLv(k;i- k - 1;pl,.

,Pi-k-1)

with the given initial conditions which yield Eq. (2.7) and consequently Eq. (2.8).

Consecutive k-out-of-n systems

249

Again if the components are identical and no dependence exists, i.e. p~ = p for all i = 1 , 2 , . . , n we obtain Eq. (2.9). Papastavridis and Lambiris (1987) studied the reliability of the system under a different Markov model. They assumed that the probability that component i fails depends upon the state of component (i - 1) but it dose not depend upon the state of the other components. Recalling the definition of)(/, state of component i, X~ = 0 or 1 as component i fails or works, they defined pi,0 as the probability that component i works, given that the preceding component is failed i.e., Pi,o = P r { X i =

llXi-l =O}

i=2,..,n

and qi,o =

1 -Pi,o

Pi,1 =

Pr{Xi = llX~_~ = 1}

Similarly

probability that component i works, given that the preceding component works, for i -- 2 , .., n, and qi,1 = 1 - pi,1

with p0,0 = 1 Under the assumption that the probability that component i fails depends only upon the state of component ( i - 1), it is clear that the sequence of random variables X 1 , X 2 , . . ,An form a Markov chain of first-order, Pr{Xi = ji]Xl = j l , X 2 = j2, . . ,Xn = j n } = P r { X / = jl ]Xn = j n }

for j l , j 2 , . . ,jn = 0 or 1 .

If R L v ( k ; n) denotes the reliability of the system. Then RLv(k; n) = R1 (k; n) + R0(k; n) ,

(2.24)

where Rl(k; n) = Pr{the linear consecutive k-out-of-n: F system works and component n works} and R0(k;n) = Pr{the linear consecutive k-out-of-n: F system works and component n fails} . Conditioning on component ( n - 1), R1 (k; n) could be written as R l ( k ; n ) = pn:oRo(k;n - 1) + p n j R l ( k ; n

- 1) .

N.A. Mokhl~

250

Using Eq. (2.24) yields (2.25)

R1 (k; n) = pn,ORLF(k; n -- 1) ÷ (Ph,1 -- pn,o)R~ (k; n - 1) . It is also clear that n

(2.26)

1 - RL(k; n) = ~ Pr(F/)Rl(k;i - k ) , i--k where F / d e n o t e s the event that the system first fails at component i, and Pf(F/) = qi,oqi-l,0 " " " qi-(Ic-2),oqi-(k-1),l

.

The summation in (2.26) could be written as n 1 Z P r ( ~ ) R 1 (k;i - k) q- (qn,oqn-l,0''" qn-k+2,0)qn-k+l,l" RS (k; n - k) .

i-k

(2.27) Substituting with (2.27) into (2.26) yields

RI~V(k; n) = RLv(k;

n -

1) -

(q~,oqn-l,o"" qn-k+2,0)qn-k+l,1 "RI

(k; n -

k) .

(2.28)

F r o m (2.28) and using (2.25), a recurrence formula is obtained as follows: RLF(k;n) RLF(k;n)

1

for k > n,

1) -~ q
= RLv(k;n

-

× qù-«+l,lpù-k,oRLv(k;n

2.2. Circular consecutive

k-out-of-n:

1)

--k-

for k < n .

(2.29)

F systems

D e r m a n et al. (1982) obtained the reliability of the circular consecutive k-out-ofn: F system in terms of the reliability of the linear system. Let N and N' denote, respectively, the number of failed components until the first working orte is reached counting, respectively, clockwise and counterclockwise from any chosen point on the circle between two components. It is clear that Pr(N-----j)=Pr(N'=j)=pqJ, =qn,

j=0,1,..,n-1 j=n

Considering N and N' to be independent, then

.

Consecutive k-out-of-n systems

251

i

P r ( N + N ' : i) = Z

P r ( N = i - j) Pr(N' = j)

j=0 i

= Zpqi-J.pqJ=(i+

l)p2q i,

i=O,...,n-

2 .

j=0

Given N + N ' = i, there is a run o f i consecutive failed c o m p o n e n t s flanked by successes on b o t h sides. Hence, the remainder of the system acts as an n - i - 2 c o m p o n e n t linear system. Thus k-1

RcF(k; n;p) = p2 Z

(i + 1)q2RLF(k; n -- i -- 2;p) ,

(2.30)

i=0

where RLv(k; n - i - 2;p) could be c o m p u t e d f r o m Eq. (2.2). Extending the a r g u m e n t in (2.30) H w a n g (1982) obtained an expression for R c v ( k ; n;pl, . . ,Ph) as

RcF(k;n;pl,...,pù) =

Z s-l+n-g
I-Iqi i=1

e I-I qJ " ,/ \

j=g+l

× RLF(P«+I,p«+2,''' , p e - 1 ; k )

,

(2.31)

where s is the first functioning c o m p o n e n t and g is the last functioning c o m p o nent. N o t e that (s - 1 + n - g) is the n u m b e r o f failing c o m p o n e n t s between comp o n e n t s and c o m p o n e n t g (clockwise from s to g). This n u m b e r must be less than k for system to work. Furthermore, there must be no k consecutive failing c o m p o n e n t s on the line between c o m p o n e n t s + 1 and c o m p o n e n t g - 1 (clockwise f r o m s + 1 to ~ - 1). Lambiris and Papastavridis (1985) derived an exact formula for the system reliability o f the circular consecutive k-out-of-n: F system with i.i.d components. It is given by RLv(n;n;p)=~(n22k)(-l)~(pq~)

;~

2=0

+ k

(_l),Z(pqk),~+l _ qn ;.=0

if n > k .

2 (2.32)

The circular system can be represented in the form o f a linear system, taking in consideration that the two ends o f the scheine close to f o r m a circle. Therefore the system operates if the total n u m b e r o f 0's at the two ends, as well as those in substrings between the l's do not exceed k - 1.

N.A. Mokhlis

252

If there exist i failed components, out of which j are at the two ends and ai denotes all the possible placements of the 0's we have Rcr:(k; n;p) = ~

aipn-i q i

for n > k

(2.33)

i=0

where k-1

ai----~ (j+ 1)N(i-j,n-

i - 1 , k - 1) .

j=0

Since there are

(~+~: 1) l+j ways to place the j balls at the two ends and there are N(i - j; n - i - 1; k - 1) for placing the remaining i - j balls without k consecutive balls. Clearly,

ai ~- I~1(i ) -{- 1~2(i) , where k-1

~l(i) = Z N ( i - j ; n -

i - 1 ; k - 1)

j=0

and k 1

~t2(i) = ~-~~jN(i-j;n - i - l ; k j=0

l)

Summing the generating functions of ~/a (i) and ~2(i) and performing similar expansions as in (2.10) and (2.11) Lambiris and Papastavridis (1985) obtained an expression for ai given by

~(~~ ~=o Z ~

k

;~k) (-1) n-

-1

n-2k-k-i

4=0

i - 2k - k

(-1)Xq" .

(2.34)

Substituting with (2.34) into (2.33), interchanging of summations and adding a term with 2 = 0, n = i which is equal to q", the following expression is obtained: ~n

Rc(k;n;p) =

2,

Z 2=0 i=0

n-1 n-2-1

.... k~

Z

,~=0 i=0

. . . . qn .

(2.35)

253

Consecutive k-out-of-n systems

The two inner sums in (2.35) can be computed, with the use of the expressions (pt + q)~-x» and (pt + q)n-~»+k-1 respectively, so the expression in (2.32) is obtained. Antonopoulou and Papastavridis (1987) announced an O(n. k) recursive algorithm for computing the reliability of a circular consecutive k-out-of-n: F system. They obtained the following recursive formula: R c F ( k , n ; p l , . . . ,pn) 1

for n < k,

1 -j~=lqy

for n = k,

l

[ 1

&

n

,~+i-1

for n = k + 1,

= ] pnRLv(k;n - 1;pl,... ,Ph-l) |

[ 1,

+ qùRcv(k;n - 1;pl,... ,Pù-I)

i=0 ~j=l

/

~kj=n-k+i+l

x RLv(k; n - k - 2;pi+a,...

/I

Pn-k+i

,Pn-k+i-1)

for n > k + l

,

(2.36) where (n + i = i). This formula is similar to (2.31) in that it permits the recursive computation of the reliability of a circular consecutive k-out-of-n: F system via the knowledge of reliability of circular and linear systems with fewer components. The summarion in (2.31) contains k(k + 1)/2 terms while the summation in (2.36) contains k terms. The computations of (2.31) and (2.36) require O(n. k2) time and O(n-k) time, respectively. Also Wu and Chen (1992) described an O(n. k) algorithm for evaluating the reliability of a circular consecutive k-out-of-n: F system. Hwang (1993) proved that the recurrence Eq. (2.36) can be solved in O(n. k) time. Wu and Chen (1993) proposed a method to improve upon the O(n. k 2) algorithm in (2.31) which is written as R c ( k ; n ; p l , . . . ,p~)= Z s--1 g--n-k+s

×

[IqipsRL(k;gi=1

1 - s - 1 + 1;p~+l,... ,Pp-1)Pe

i=2 j=n-k 2+i [,m=l in order to derive an O(n • k) algorithm.

qi

i=g+l

m--j+2

N. A. Mokhlis

254

2.3. R e l a y e d s y s t e m s

H w a n g (1988) introduced the notion of relayed consecutive k-out-of-n: F lines. H e introduced the reliability of relayed consecutive k-out-of-n: F systems as functions of the reliability of linear consecutive k-out-of-n: F systems. T h a t is Ru(k; n ; p i , . .. ,Ph) = p l R L ( k ; n - 1 ; / ) 2 , . . ,Ph) and Rb(k; n ; p l , . . .

,Ph) = p l p n R L ( k ; n - 2 ; p 2 , . . ,Ph-l)

,

where Ru(k; n ; p l , . . . ,ph) and Rb(k; n ; p i , . . . ,pn) are, respectively, the reliability of a unipolar and a bipolar relayed consecutive k-out-of-n: F systems with c o m p o nent reliabilities pl ,P2, • • • ,P~Mokhlis and Mohamed (1999) obtained recursive equations for Ru (k; n; Pa, • • •, P~) and Rb (k; n; P l , . . • ~Pn ) as follows: Ru(k;n;pl,...

,p~)

1

l

for n = 0,

Pl

for 0 < n < k,

R u ( k ; n - 1 ; p l , . . . ,P~-I) - Ru(k;n - k - 1;pl,... ,P~-k-1)P~-k

for n > k . j=n k+l

(2.37)

Eqs. (2.37) are obtained B ( k ; n ; p l , . . . ,pn), where fu(k;n;pl,

by

deriving

the

unreliability

of

the

system

. . . ,p~)

for n = 0, for 0 < n < k,

ql

fu(k;n-

I°

1 ; p l , . . . ,ph-l)

-~- (1 - f u ( k ;

n - k -

l;pl,...

,Pn-k-1))Pn-k

for n > k . j--n -k+ 1

(2.38)

The first and second parts of (2.38) are obvious. F o r n > k, according to the definition of a unipolar relayed consecutive k-out-of-n: F system, system failure happens if and only if either one and only one of the following mutually exclusive and exhaustive events occurs: (a) the first c o m p o n e n t fails with probability q~, or (b) the first c o m p o n e n t does not fail, and the system first fails at the /th component, k + l < i < n .

Consecutive k-out-of-n systems

255

This happens when k consecutive c o m p o n e n t s i - k + 1, i - k + 2 , . . . , i - 1, i fail with probability 1-I)_~-k+l qJ; k + 1 < i < n and the (i - k)th c o m p o n e n t does not fail with probability pi-k. This means that c o m p o n e n t s f r o m the first up to the ( i - k - 1)th f o r m a unipolar relayed consecutive k - o u t - o f - ( / - k - 1): F system having the first unit functioning and no k consecutive failures with probability 1 - f u ( k ; i - k - 1 ; p l , . . . ,Pi-k-1). The probability of this event is i

( 1 - fu(k;i-k-1;pl,...,pi-k-1))pi-k

H

qJ "

j=i-k+l

i=k+l

S u m m i n g the probabilities of (a) and (b) we get fu(k;

n;pt,... ,p~) i

n--I

(1 - f u ( k ; i - k -

= ql ÷ E

1;pl,... ,pi-k-l))Pi-k

+ (1

H qJ

j=i k + l

i-k+l

-fu(k;n-k-

1;pl,...,pn-k-1))pn k I I

j=n-k+l

qJ '

which can be rewritten as

fu(k;n;pl,... ,Ph) = f u ( k ; n - 1 ; p l , . . . ,Ph 1) II

+(1-fu(k;n-k-1;pl,...,pn-»-l))Pn-k

qJ "

j=n-k+l

Since Ru(k; n;pl,... ,pù) = 1 - f u ( k ; n;p~,.. ,Pn) using (2.38) we ger (2.37). W h e n the reliability of each c o m p o n e n t is the same, i.e., pi = P for 1 k . (2.39)

Arguing in a similar m a n n e r a recurslve equation for tained

Rb(k;n;pl,... ,Ph) is ob-

Rb(k; n;p~, .. ,p~)

[ I

0

for n = 0 ,

Pl

for n = 1,

plpn Rb(k;n- 1 ; p l , . . . , p ~ 2,P~)

for 1 < n _ ~ k 4 - 1 ,

--Rb(k;n--k--1;pl,...,p~-k-1)p~j=~_k fi qy

forn>k+l

.

N. A. Mokhlis

256 Again if Pi = P for 1 k + l

1,

for 1 < n < k + l , . (2.40)

2.4. Two-dimensionaI consecutive k-out-of-n: F systems As defined before a linear or a circular connected X-out-of-(n, m): F lattice system fails when at least one subset X of connected failed components occurs which includes failed components connected in the meaning of connected X. We shall discuss the following two forms of X: (i) (r, s) matrix, r _< n , s _< m, and (il) (r, s) matrix or (s, r) matrix, r, s _< n, m, r ¢ s.

L Reliability ofsome connected (r, s)-out-of (n, m): F lattice systems The exact reliability of some lattice systems are obtained by Boehme et al. (1992). They obtained the exact reliability formulas of linear and circular-connected(ij)-out-of-(n,m): F lattice systems with identical components, where (i,j) = (1, 1), (n, m), (1,s), (r, 1), (n,s) and (r,m). (a) Reliability of linear and circular connected (1,1)-out-of -(n,m): F lattice systems: Linear and circular connected (1, 1)-out-of-(n, m): F lattice systems fails if and only if at least one c o m p o n e n t falls. Thus, the system is equivalent to a onedimensional series system. That is R~v((1, 1); (n,m);p) = p ù m

7 = L and C ,

(2.41)

where L stands for linear and C stands for circular. (b) Reliability of linear and eireular connected-(n, m)-out-of-(n, m): F lattice systems: The system either linear or circular fails whenever all its components fail. This means that the system is equivalent to a one-dimensional parallel system. Thus we have

R~E((n,m); (n,m);p) = 1 - q'~,

7 = L and C .

(2.42)

(c) Reliability of linear and circular connected (l,s)-out-of-(n,m)." F lattice systems: A linear (circular)-connected (1, s)-out-of-(n, m): F lattice system fails if and only if at least one row (circle) with s consecutive failed components occurs. Since c o m p o n e n t failures are s-independent, then row (circle) failures are also s-independent. Thus, we can consider the rows (circles) as new components. The system of rows (circles) fails if and only if at least one row (circle) fails. Noticing that each row (circle) is a one-dimensional linear (circular)-consecutive s-out-of-m: F system, we get

Consecutive k-out-of-n systems R~F((1,s); (n,m);p) = (RTF(s;m;p)) n,

257

7 = L and C ,

(2.43)

where R~F(s;m;p) is obtained from Eqs. (2.15) and (2.32) for 7 = L and C, respectively by taking k = s and n = m. (d) Reliability of linear and circular-connected (r,1)-out-of-(n,m): F lattice system: A linear (circular)-connected-(r, 1)-out-of-(n, m): F lattice system fails if and only if at least one column (ray) with r consecutive failed components occurs. Since component failures are s-independent, then column (ray) failures are sindependent too. The columns (rays) are considered as new components. The system of columns (rays) fails if and only if at least one column (ray) fails. Clearly, each column (ray) is a one-dimensional linear-consecutive r-out-of-n: F system. Hence, RTF((r , 1); (n,m);p)= (RLF(r;n;p)) m,

Y = L and C ,

(2.44)

where RLF(r; n;p) is obtained by substituting in Eq. (2.15) with k = r.

(e) Reliability of linear and circular connected-(n,s)-out-of-(n,m): F lattice systems: A linear (circular)-connected-(n, s)-out-of-(n, m): F lattice system fails if and only if at least s columns (rays) each of which includes n failed component occur. Since component failures are s-independent, then column (ray) failures are also sindependent. The columns (rays) of the system can be considered as new components each with failure probability qn and reliability 1 - qn. Then the system is considered as a one-dimensional linear (circular)-consecutive-s-out-of-m: F system with component reliability 1 - q~. Thus

R~F((n,s);(n,m);p)=R,pF(s;m;1-qn),

7=L

and C ,

(2.45)

where R~v(s; m; 1 - q " ) can be obtained from Eqs. (2.15) and (2.32) respectively, bytakingp=l-q~,q = qn k = s a n d n = m.

(0 Reliability of linear and circular connected (r, m)-out-@ (n,m): F lattice systems: A linear (circular)-connected-(r, m)-out-of-(n, m): F lattice system fails if and only if at least r consecutive rows (circles) each with m failed components occur. Since component failures are s-independent, then row (circle) failures are also s-independent: The row (circles) of the system are considered as new components each with failure probability qm and reliability 1-q m. Thus the system is considered as a one-dimensional linear-consecutive-r-out-of-n: F system with reliability

RTF((r,m);(n,m);p)=RLF(r;n;1--qm),

7=L

and C ,

(2.46)

where RLv(r; n; 1 -- qm) is obtained from Eq. (2.15) by replacing p by 1-q m and k by r.

II. Reliability of some connected-(r, s)-or-(s, r)-out-of-(n, m) : F lattice systems Boehme et al. (1992) showed that the reliability of a linear (circular) connected(1, 2)-or-(2, 1)-out-of-(2, 2): F lattice system equals the reliability of a one-dimensional circular-consecutive-2-out-of-4: F system, since both systems fail whenever at least two consecutive components fall. Thus

258

N. A. Mokhlis

R~F((1,2)-or-(2, 1),(2,2)) =RcF(2,4;p),

~ = L and C .

They obtained the reliability of a linear (circular) connected-(1, 2)-or-(2, 1)out-of-(3, 2): F lattice system and the reliability of a linear-connected-(1, 2)or-(2, 1)-out-of-(2, 3): F lattice system depending on the reliability of a one-dimensional linear consecutive 2-out-of 5: F lattice system. That is, Ryv((1,2)-or-(2, 1), (3, 2)) = RyF((1,2)-0>(2, 1), (2, 3)) = pRLF(2; 5;p) + qp3 = p6 + 5pSq + 6p4q2 +p3q3 + qp3 . The reliability of a circular-connected-(l, 2)-or-(2, 1)-out-of-(2, 3): F lattice system is obtained by searching all the failure stares of the system. Thus, Rcv((1,2)-or-(2, 1), (2, 3)) = 1 - (9p4q 2 q- 20p3q 3 + 15p2q4 + 6pq 5 + q6) . The reliability of a linear connected-(1, 2)-or-(2, 1)-out-of-(3, 3): F lattice system is obtained by relating it to the reliability of a one-dimensional circularconsecutive 2-out-of-8: F system. That is RLv((1,2)-or-(2, 1), (3, 3)) = pRcv(2, 8;p) +p3q . Mokhlis et al. (1998) applied a different approach for obtaining exact formulas for the reliability of the more general systems, linear (circular)-connected-(1, 2)or-(2, 1)-out-of-(n, 3): F lattice systems. Their approach depends on counting the operating states of the system. The reliability of a linear-connected-(1, 2)-o>(2, 1)-out-of-(n, 3): F system is given by the following recursive formula: RLF((1,2)-or-(2, 1), (n, 3)) n+l [j/2]

= ~-~~Zfiij(+i-ip 2j (i-1)RLv((1,2)-or-(2 , 1), (n --j, 3)) ,

(2.47)

j - 1 i=0

where RLz((1,2)-or-(2, 1), (-1, 3)) = p 3 B01= 1, f i 0 j = 3 ' U -2 for l < j _ < n + l

(2.48) ,

j-I

Bij=

~

C~mqm(j) for 1 2i+l,

0~2i =

O~

0{2i 1

=

1,

Yx =

2x+l + (-1)x+12 3

qmÜ') = ( j - m + 2)2J-m-3,

1 < i < [j/21 ,

(2.51)

'

ra<j-1 (2.52)

q2i(J) = 0 ,

qj_l(j)= 1 •

The recursive form in (2.47) is obtained by considering all operating states of the system. Operating states of any row are represented by 111,011, 101, 110, 010. A n y of the operating stares of the linear connected-(1, 2)-or-(2, 1)-out-of-(n, 3): F lattice system must have one or m o r e of the previous forms arranged in such a m a n n e r that there are no consecutive zeros appearing in any column. If the row 111 appears at the j t h row for the first time, then RLF((1,2)-or-(2, 1), (n, 3)) = R~v((1 , 2)-or-(2, 1), (J', 3))

XRLF((1,2)-or-(2,1),(n--j,

3)),

1 <j
l ,

where R~F ((1, 2)-or-(2, 1),(j, 3)) is the reliability of a linear-connected-(1, 2)-o> (2, 1)-out-of-(j, 3): F lattice system, for which 111 appears at t h e f l h row for the first time. F o r j = n + 1, the r o w 111 appears in the imaginary (n + 1)th row, i.e., it does not appear. It is clear that R~F((1 , 2)-or-(2, 1), Ü', 3)) = fiijp2j-(i-1)q j+i-' , where fiij is the n u m b e r of configurations of Cj, where Cj is a (L 3) matrix of binary numbers, such that only the j t h row is 111, i rows are 010 and j-i-1 rows with only one 0, where no connected 0's a p p e a r in any column. F o r each j we must have 0 < i < ~/'/2]. F o r determining fit/for 0 < i < [//2], 1 < j <_ n + 1, it is clear that each (/., 3) matrix defined above must contain a Bm-matrix, 2i - 1 < m < j - 1, where a Bmmatrix is an (m, 3)-matrix of binary numbers such that i rows are 010, with the first and last rows are 010, and the rest m - i rows each with only one 0, where no 0's a p p e a r in any column. Thus we obtain (2.49), where am is the n u m b e r of configurations of B , , and qm(J) is the n u m b e r of ways in which the rest (j - m - 1) rows each with only one 0 can be arranged. Obviously for i = 0 (2.48) holds, C~mand qm(J) are given, respectively, by (2.51) and (2.52). c~m is derived by defining new binary matrices Ax contained in Bin, 0 < x < n - 4, where Ax is an (x + 4, 3)-matrix of binary numbers, such that only the first and last rows are 010, where the rest (x + 2) rows each have only one 0, where no connected 0's appear

N. A. Mokhlis

260

in any column, 1 < x < n - 4, while A0 is (3,3)-matrix such that the first and last rows are 010, while the second is 101. The number of configurations of Ax is Yx equals to (2x+l + (-1)*+12)/3. Since Bm contains i rows 010, then Bm will consist ( i - 1) - rAós and Ax~,Ax;,..,A~r such that xl +x2 + . . . + x t = m - (2i+ (r - 1)), where xs ¢ 0 and 1 < r < i - 1. The Ax's can be distributed in ( i~~ ) ways t h u s c~m is given by (2.49). Note that ~2i

1 ~~-

0

=

while c~2i= 0, since there is no configuration B2» The reliability of a circular-connected-(1, 2)-or-(2, 1)-out-of-(n, 3): F lattice system is given by the following equation: RcF((1,2)-or-(2, 1), (n, 3)) n+l

= ~yjqj-lp2j+lRcF((1,2)-or-(2,

1); (n --j, 3)) ,

(2.53)

j--I

where 7j = 3 . 2 j - 2

for j _> 2, 71 = 1

(2.54)

and RCF((1,2)-or-(2, 1), (--1,3)) = p - 3 The operating states of one circle with three components are 11 l, 01 l, 101 and 110. Thus, for a system with n circles and three rays the operating states of the system taust consist of one or more of the previous forms arranged in such a manner that there are no connected zeros appearing in any ray. The determination of the reliability depends on the position of the first 111 circle in the system. If the jth circle is the first circle with 111 that appears in the system, then the reliability is RcF((1,2)-0>(2, 1), (n, 3)) = R~F ((1,2)-or-(2, 1), (j', 3))

×RcF((1,2)-or-(2,1),(n--j,

3)),

1 <j
l ,

where R~F((1 , 2)-or-(2, 1), (j, 3)) is the reliability o f a circular-connected-(1, 2)-or(2, 1)-out-of-(j, 3): F lattice system, for which 111 appears for the first time at the jth circle. If 7j is the number of configurations o f j circles, such that only the jth circle is 111, other j - 1 circles each have only one zero, such that no connected zeros appear in any ray, then obviously R~F((1 , 2)-or-(2, 1), (j, 3))

=

2 j ¢ - l p 2j+l,

where 7j is given by (2.54). Thus (2.53) holds.

1 < j <_ n ÷ 1 ,

Consecutive k-out-of-n systems

261

2.5. Consecutive-kz-out-of-n: G systems Kuo et al. (1990) studied the relationship between the consecutive-k-out-of-n: F and G systems in the one-dimensional case. They showed that a consecutive kzout-of-n: G system is a mirror image of a consecutive k-out-of-n: F system. This relationship is expressed by the following lemma. LEMMA. If the reliability of component i in orte type of consecutive-kz-out-of-n system (e.g. F) is equal to the unreliability of component i in the other type of consecutive-kz-out-of-n system (e.g. G) for all i, and if both types of systems have the same kz and n as well, then the reliability of one type of system is equal to the unreliability of the other type of system. Thus linear and circular consecutive kout-of-n: G and linear and circular consecutive kz-out-of-n: F systems are mirror images of each other. By this lemma and the well-developed results for linear and circular consecutive kz-out-of-n: F systems given by (2.8) and (2.36), the reliability of linear and circular consecutive kz-out-of-n: G system are given, respectively, as follows: RLG(k;n;pl,... ,Ph) = RLG(k;n -- 1;pl,... ,Ph-l) + [1 - R L G ( k ; n - k -

1;pl,...,pn-k-1)]qn k

~-[ Pj j-n-k+l

and RCG(k; n; pl , . . . ,Ph) o

[

for n < k, for n = k,

/ ~/Bqs/-kl]p,

//:] I

/ j:l \j~i

/ /

forn=k+l, i=1

: I/ i=l~ [«j «n-k+iI(i~pjl (h PJI L \s=l / \g=ù-~+s I /

1 t

+[1-RLO(/<;n-k-2;p;+~,..,pù ~+,_2)]

qùRLo(k;n-l;pl~...

,Ph-l)

+ p ù R c G ( k ; n - 1;pl,..- ,Pù-I)]

for n > k + l

.

Zuo (1993) mentioned that linear and circular connected (r, s)-out-of-(n, m): F and G lattice systems are also mirror images of each other. Mokhlis et al. (1998) showed that this relation could be extended to the general connected X-out-of-(n, m): F and G lattice systems. This is clear from the fact that the minimal cuts of the F lattice system are themselves the minimal paths of the G

N. A. Mokhlis

262

lattice system, then the reliability of one type of lattice systems equals the failure probability of the other. Using this relation and (2.41)-(2.47) and (2.53) they obtained the reliability of some connected X-out-of-(n, m): G lattice systems: (a) Reliability of linear and circular-connected-(1, 1)-out-of-(n, m): G lattice systems:

R ~ ~ ( ( 1 , 1 ) , ( n , m ) ) = l - q nm, 7 = L and C . (b) Reliability of linear and circular-connected-(n, m)-out-of-(n, m): G lattice systems:

R~G((n,m),(n,m))=pnm, 7 = L and C . (c) Reliability of linear and circular-connected-(1, s)-out-of-(n, m): G lattice systems: RTG((1,S), (n,m)) = 1 -- ~v~G(s,m;p)]m, where m

- js

)

7 = L and C ,

m(

(_qp,)j _ pS Z

J

j=o

)

m - i s . - s (_qpS)j

j=o

J (2.55)

and

[1( m

PCG(S;m;P)=

)/-«/ m

-

js

if0<m<s, j

J / I +,~ (m-j,-,-1 ~i=o \ I,

j=o \

J

(_qp/+l

pm

i f m _>s .

/ (2.56)

(d) Reliability of linear and circular-connected-(r, 1)-out-of-(n, m): G lattice systems:

R~G((r, 1), (n, m)) -- l -- [PLG(r, n;p)] m,

7 = L and C ,

where pLa(r, n;p) is obtained from (2.55). (e) Reliability of linear and circular-connected-(n, s)-out-of-(n, m): G lattice systems.

R,G((n,s), (n,m)) = 1 --p~G(s,m;pn),

7 = L and C ,

where p~o(s,m;p n) are obtained for 7 = L and C from (2.55) and (2.56), respectively.

Consecutive k-out-of-n systems

263

(f) Reliability of linear and circular-connected-(r, m)-out-of-(n, m): G lattice systems:

R~G((r,m),(n,m)) = 1 --PLG(r;n;pm), ? = L and C , where pLG(r; n;pm) is obtained from (2.55). (g) Reliability of linear-connected-(1, 2)-or-(2, D-out-of-@, 3): G lattice systems: RLG((1,2)-or-(2, 1), (n, 3)) = 1 --p>G((1,2)-or-(2, 1), (n, 3)) , where PLG((1,2)-or-(2, 1), (n, 3)) n+l [/'/2]

= ~ Z fliJpl+i-lq2j (i-1)PLG((I' 2)-or-(2, 1), (n -- j, 3)), j = l i=0

pLG((a, 2)-o>(2, 1), (--1,3)) = q-3 and fl« as in (2.48) (2.50). (h) Reliability of circular-connected-(1,2)-or-(2,1)-out-of-(n,3): G lattice systems: RCG((1,2)-or-(2, 1), (n, 3)) = 1 --PcG((1,2)-or-(2, 1), (n --j, 3)) , where pCG((1,2)-or-(2, 1), (n, 3)) n+l

= Z YJpj-lq2j+lpcG((I' 2)-or-(2, 1), (n -- j, 3)) j=l

7 j = 3 " 2 j-2

for j_>2, 71 = 1

and PCG((1,2)-or-(2, 1), (--1, 3)) = q-3

2.6. Algorithm for computing the reliability Bollinger (1982, 1986) introduced a direct computation and an algorithm for calculating a table of coefficients for the failure probability polynomials associated with a linear consecutive k-out-of-n: F system with i.i.d components, depending on the properties of the list of the 2n binary strings. Following the same idea E1Sayed (1998) introduced an algorithm for calculating the coefficients for the failure probability of a (1, 2)-or-(2, 1)-out-of-(n, 2): F lattice system.

N. A. Mokhlis

264

Khamis and Mokhlis (1997) derived an algorithm for computing the reliability of linear and circular connected- (1, 2)-or-(2, 1)-out-of-(m, n): F lattice systems. In 1998 they derived an algorithm for enumerating the reliability of linear and circular connected-(r, s)-out-of-(n, m): F lattice systems as well as the reliability of linear and circular connected-(r, s)-or-(s, r)-out-of-(n, m): F lattice systems with i.i.d components.

3. Approximation formulas and bounds

3.1. One-dimensional consecutive k-out-of-n: F systems Chao and Lin (1984) examined the limiting behavior of RLF(k; n;p). From (2.17) it is clear that for fixedp, RLF(k; n;p) --+ 0 as n ~ oc. Also, irk _> 2 a n d p ~ 1, for example, with n(1 - p ) -+ 2, RE (k; n;p) ---, 1. Neither case is interesting for system design purposes. They showed that the rate of convergence depends on k. They proved that for 1 < k < 4 lim RLF(k;n;p) = exp(--nqk),

n---+OQ

provided that nq k is bounded .

(3.1)

Recall the stationary Markov chain Y1, Y 2 , . . , where Y/=(Xi,~/+l,...,X/+k_l) ,

i = l,2, . . , n - k + l ,

where X/is the state of component i, X~ = 1 or 0 if the component is functioning or failed, respectively. If S, is the number of times that Y,- enters the taboo set, then Sù converges to Poisson with rate nq k. Chao and Lin (1984) conjectured that (3.1) holds for k > 4. Fu (1985) proved that (3.1) holds for k > 1. The technique for proving this result is based on evaluating upper and lower bounds of RLF(k; n;p). For every small 6 > 0, there exists no(Ô) such that for all n > no(6)

(

2k 1

-

-

,

(1+6)2k+, ~-~o(,)

~

U+7i77~

"~

-)

>- eLf(k; n;p) >-

where

q = 2n-~/k,

{

k

no(f) =

n*(6) n

ifn*(6) < k, ifk_< n*(6) < n, ifn*(6) > n,

(

1-

~ ) ~ k+1 '

(3.a)

265

Consecutive k-out-of-n systems

Fu and Hu (1987) studied the reliability of linear consecutive k-out-of-n: F system when the component failure states have ( k - 1 ) step Markov dependence. This means that the failure probability qi and reliability pi of component i (i = 1 , 2 , . . , n ) depends on the number of m consecutive failures immediately preceding this component, i.e., qi = q(m), Pi = p(m) and

q(m) + p(m) = 1 Recalling (2.22) and (2.23) we have RLF(k;n;p(m)) = S

q(j)

m=0\j=0

~-~~p(r)R(r,k,n - m - 1) ,

r=0

where R(m; k, i) is the probability that the system is working at component i with ending m consecutive failures, and is given by

R(m; k; i)

(m~)~~ q(j)

p(r)R(r; k; i - m - 1 ) ,

1 < m < k -1

r=O

and k-1

R(0;k;i) = Z p ( r ) R ( r

; k ; i - 1) .

r--0

The following theorem stares that if the failure probabilities m = 0, 1 , . . , k - 1 are functions o f n and tend to zero with rates

q(m) = 2mn -I/k,

2m>0

q(m),

form=0,1,..,k-1

then the reliability of the system tends to a given constant.

THEOREM. If q(m)=2mn

z/k, m = 0 , 1 , . . , k - 1

,

(3.3)

then RLF(k; n; p(m) ) ~ exp I --n k-1 I~_oq(m )

(3.4)

and lim

n---+o~

RLF(k;n;pl,...,pn) = exp -

2m

•

(3.5)

For )~m = 2, m = 0, 1 , . . , k - 1, the dependence disappears and (3.5) is reduced to (3.1). Many researchers have proposed lower and upper bounds for the reliability of linear and circular consecutive-k-out-of-n-systems. For example, Chiang and Niu

N. A. Mokhlis

266

obtained lower and upper bounds for linear consecutive-k-out-of-n: F system with identical components operating s-independently given by

(3.6)

(1 - qk)n-k+l < RLv(k; n;p) < (1 - qk)[~/k]

Derman et al. (1982) generalized (3.6) to the linear and circular consecutive kout-of-n: F systems with independent but not necessarily identically distributed components failure probabilities. They showed that R~F(k; n;pl,... ,p~) for 7 = L or C satisfy

Il 1- H i= 1

«~

j=i

/

-< R,~(~;ù;p~,..,pù),

(3.7)

where

g= { n- k +

for7=cf°rT=L'

and

Æ2(N)

(3.8)

RyF(k; n;pl,... ,p~) <_ 1 - E(N2-----~

where N is the number of failed minimal cuts. It is easy to see that ¢ n-k+l i+k 1

E(N) =

B

IIq«

i=1

i=1

for

7 = L,

j=i i+k- 1

FIqj

for7=C

j=i

with q~+~ = qj and

E(N2) = Z E(Ii) + Z i

Z E(IiIj) ,

iTLj

where Ii is the indicator of the event of having all components of minimal cut {i, i + 1 , . . , i + k - 1} failed, i = 1 , . . , g, g = n - k + 1 for 7 = L and g = n for 7=C. Although the upper bound in (3.8) is straightforward, it is messy to compute. The calculations are simple in the circular case when all components are equally reliable and is given by

Rcv(k;n;p) <_ 1 - A / B

,

where

A = nq k,

B= l + (n- 2k-1)q k+2q(1-qk-1)/p

.

Consecutive k-out-of-n systems

267

Using the Stein-Chen method, given by Arratia et al. (1989), Chrysaphinou and Papastavridis (1990) proved that

IRL~(k;n;p) -

exp[-(n - k + 1)qk]] _< (2k - 1)q k + 2(k - 1)q .

(3.9)

Barbour et al. (1991) also using the Chen-Stein method gave the improved inequality IRLv(k; n;p) - exp[-p(n - k + 1)q~]] < (2kp - 1)q k .

(3.10)

Chao et al. (1995) showed numerically that the bounds (3.9) and (3.10) generated by the Stei~Chen method performed poorly, especially for small n. They suggested that this poor performance is due to the fact that the Stein-Chen method uses only the first two moments of the process, but ignores the Markov structure of the reliability system. Papastavridis (1987) obtained closed form lower and upper bounds for linear and circular consecutive k-out-of-n: F systems with identical s-independent components under the restriction that component failure probability is less than k / ( k + 1). His method is based on analyzing the roots of the denominator of the generating functions. The lower and upper bounds for RLv(k; n;p) are, respectively, b m n+l -

e

and

a M n+l

+ e

,

where m=

1

pqk

M=

1--pqk

(1 - q~)~' m k _ qk a~

M k _ qÆ

m k - (k + 1)pq Æ'

b z

M k - (k + 1)pq k

and 2 ( k - 1)q "+2 ex

pik + (k + 1)q] '

The lower and upper bounds for RcF(k; n;p), respectively, are (1-pqk)n-(k-1)q

n and

(1-pqk)n+(k-1)q n .

Mokhlis and Mohamed (1999) obtained approximate formulas for the reliability of unipolar Ru(k; n;p) and the reliability of bipolar, Rb(k; n;p), relayed consecutive k-out-of-n: F systems with identical components. The approximation formulas are obtained using the generating functions of the reliability of such systems, and the result in Feller (1968) for obtaining the coefficients of the generating functions. The approximation formulas for Ru(k; n;p) and Ru(k; n;p) are given by

268

N. A. Mokhlis

1 -

qso

n

Ru(k;n;p) ~ k 5 - ~7~soSO

and Rb(k;n;p) ~ p(1 --_qso) s-(ù-l) k + 1 - kso o

respectively, where so is the smallest root in absolute value of the polynomial V(s) = 1 - s + pqksk+l

The errors in the approximation formulas for Ru(k; n;p) and Rb(k; n;p) are less than el m

2 ( k - 1)q n+l (k + 1)q +k

(3.11)

and

e2-

2 ( k - 1)pq n ( k + 1)q+~ '

(3.12)

respectively. The bounds for Ru(k;n;p) and Rb(k;n;p) are given by < Ru(k;n;p) < ur 1" + e l

(3.13)

Br2(n-l) _ e2 < Rb(k; n;p) < pur1 (n 1) + e2 ,

(3.14)

Lr~ ~ - e l

and

respectively, where rl = (1 _ p q k ) 1, L--

1 - qq (k + 1) - kF1 '

I,I--

1 - qr2 ( k ÷ 1) - kr2

el and e2 are given, respectively, by (3.11) and (3.12) and q < k / ( k + 1) Numerically, Mokhlis and Mohamed (1999) showed that the lower bounds in (3.13) and (3.14) performed better than the upper bounds as compared with the exact reliability computed using (2.39) and (2.40), respectively.

3.2. Two-dimensional consecutive k-out-of-n: F systems

Koutras et al. (1993) approximated the reliability, RLv ((r, r), (n, n));pl, • • • ,Ph), of a linear-connected-(r, r)-out-of-(n, n): F lattice system by an exponential term as given in Arratia et al. (1989).

Consecutive k-out-of-n systems

The bounds for RLF((r, r), (n,n));pl,... inequality:

,Pn) are

269

obtained from the following

//LF((~, r), (n,n));pl,... ,p,,) -- e-~~[

x)[(2r-l)2qr2+4( ~i~-~~qr2-iJ-1)]

_<(1-e

(3.15)

where n-r+l n-r+1 i=1

c~»jfor

i,jc~ij

j=l

i,j = 1 , . . ,n - r + 1 are the minimal cuts of the system, i.e. ~ij={(i+x-l,j+y-1):x,y=l,...,r}, i,j=l,2,..,n-r+l

and q is the failure probability of the worst component of the system. Mokhlis et al. (1997, 1999) using Poisson distribution, obtained an approximation formula for the reliability of a more general two-dimensional systems, linear and circular-connected-(r, s)-out-of-(n, m): F lattice systems, as weil as linear and circular connected-(r, s)-or-(s, r)-out-of-(n, m): F lattice systems. The reliability of the system is approximated by an exponential term using the Chen-Stein method in Arratia et al. (1989). The reliabilities of linear and circular-connected-(r,s)-out-of-(n,m): F lattice systems, respectively, are

RTF((r,s), (n,m); Pl,i,... ,Pù,m)~ exp(--2~)

,

(3.16)

where n-r+l fl

2,= Z Zq~'A') i=1

f°r•=L°rC

'

(3.17)

j=l

q~iA~)= H qe,g Œij(7) are the minimal cuts of defined as

(3.18)

7-connected-(r,s)-out-of-(n,m):

eij(7)={(i+x-l,j+y-1):x=l,...,r;y=l,...,s}

F lattice systems

,

(3.19)

where i = 1 , 2 , . . ,n - r + 1;j = 1 , 2 , . . ,f~ -s+l

f~=

{m m

forT=L, forT=C .

(3.20)

The minimal cuts of 7-eonnected-(r, s)-or-(s, r)-out-of-(n, m): F lattice systems have either the form «q(7) in (3.19) or

N. A. Mokhlis

270

t//j(7 ) = { ( i + x - l , j + y - 1 ) : x = where i = 1 , 2 , . . , n - s +

g~=

{

1,..,s;y=l,...,r}

1 ; j = 1 , 2 , . . , g y , with

m-r+l m

for 7 = L , for 7 = C .

(3.21)

So the reliability is given by !

R~F((r, «)-or-(«, r), (n, m);pl,1,

(3.22)

. . p~~.,~) ~ exp(--27) ,

where n-r+i f~ n-s+1 g,/ f l ; = Z ~q~'J(') + ~ Z « " , J ( ' ) i=1 j=I

for T = L o r C

(3.23)

i=1 j = l

For q~,j(~) is obtained from (3.18) and q~~j(7) =

H

(3.24)

qe,g •

Generalizing (3.15), M okhlis et al. (1997, 1999) obtained bounds for the general systems; linear and circular-connected-(r, s)-out-of-(n, m): F lattice systems as well as linear and c i r c u l a r - c o n n e c t e d - ( r , s ) - o r - ( s , r ) - o u t - o f - ( n , m ) : F lattice systems. For the bounds the following inequalities hold:

L1.;
for ~~= L or C

(3.25)

and LII7 < RTF( (r , s ) - o r - ( s , r), (/7, m); P l , 1 , ' '

,Pn,m) <_ Ul•t

for • = L or C (3.26)

where L17 = e -x~ - (1 - e-'~,)M1,

ul~ = e -~~~+ (1 - e-X')M1,

M1 = (2r - 1)(2s - 1)q rs + 4 L'I, = e-X; - (1 - e-;~;)M'l, M~ - - [ ( 2 r - 1 ) ( 2 s -

+4

q~S-yz _ 1 , u'17 = e-¢~; + (1 - e-¢';)M~,

1)+ (r+«-

~«~~-»+ Z y=l z=l

y=l

1)2]q ~s

Z «r~~_l z=l

q = max,_<,_<ùqü is the failure probability of the worst component of the system, 2~ and 2~ for 7 = L or C are given, respectively, by (3.17) and (3.20).

271

Consecutive k-out-of-n systems

Clearly, taking s = r and m = n in inequality (3.25), we obtain (3.15). The bounds in (3.25) and (3.26) are efficient for large component reliabilities, i.e. when q is small, M1 and M~ become very small. The well-known inequality I -I [ p 1 i- icCj I) I ](1 j

j

(3.27)

given by Barlow and Proschan (1975), holds for any system, where R is the reliability function of the system, Cj and Aj are, respectively, the minimal cuts and minimal paths of the system. Making use of this inequality and determining the minimal cuts of the general linear and circular connected-(r,s)-out-of-(n,m): F and (r,s)-or-(s,r)-out-of(n,m): F lattice systems, Mokhlis et al. (1997, 1999) obtained lower bounds for such systems. These lower bounds are, respectively, n-r+1 f~

L27

I-I H (1 - q~«(~)) for 7 = L or C

L~, =

n]5~1 L H (1 - q««(~)) L i=1 j=l

(3.28)

i=1 j=l

and

n~l I-[g~ (1 L i=1 j=l

- qùi/,))

1

,

(3.29)

where q~«(~),f»g»q~iAT)are given, respectively, by (3.18), (3.20), (3.21) and (3.24). From the properties of the exponential function and (3.16), (3.17), (3.22) and (3.23) the reliability of the two-dimensional systems could be approximated as

R~v((r,s), (n, m); P1,1,.. ,Pù,m)~ L27 and

R~v((r,s)-or-(s, r), for sma11 values of qij.

(n, m); pJ,1,.. ,Ph,m) ~ L~~

Knowing the failure probability of the worst component of the system, rough simple lower bounds could be obtained as L3~ = exp[-(n - r + 1)f~q r']

for 7 = L or C

(3.30)

and L~7=exp([-(n-r+l)f~+(n-s+l)g~lq

's)

forT=LorC

(3.31)

where q = max,
272

N. A. M o k h l i s

Malinowski and Preuss (1996) obtained bounds for 7-connected-(r,s)-out-of(n, m): F lattice systems. In deriving these bounds they used the properties of sindependent binary random variables representing the states of the components of the system. The bounds are given as follows: L47 =

max

Pr{AT(i, r)}, I I Pr{B~(j, s)} L i=1

i=1

RTF((r»s), (n,m); P t , , . . ,Pù,m) < min

Pr{AT((i- 1)r + 1, r)}, H P r { B ~ ( ( j - 1)s + 1, s)} [.i=1

= u2~ ,

i=1

(3.32) where f~ is given in (3.20), A~,(i,r) is the event that a 7-connected-(r,s)-outof-(n,m): F subsystem formed by rows or circles i, i + 1 , . . , i + r - 1 does not contain an (r,s) matrix of failed components, B~(j,s) is the event that a ~-connected-(r,s)-out-of-(n,m): F subsystem formed by columns or rays j, j • 1 , . . , j ® (s - 1) does not contain an (r, s) matrix of failed component, • is an operator such that

j®t=[(j+t-1)modm]+l,

j, t E ( 1 , m ) , oo+r--1

Pr{AT(~o,r)} =R~F(s;n; Po),I,...,Po),~),

Po»,j= 1 -

II (1 -p««),

g=co

o)+s- 1

Pr{BL(CO ,

s)} =

R L F ( r ; n; O-l,•j,... , O'n,c~),

ai,~=l-

rI

(1-pi,«),

g=co

o)~(s-1) Pr{Bc(e),s)} = RLv(r; n; rl,o~, . . , zn,o~),

~,,~=1- g=o Il (1-p~,«)

For systems with identical components, Salvia and Lasher (1990) introduced a lower bound for the reliability of linear connected-(r,r)-out-of-(n,n): F lattice system. The bound depends on the reliability of the one-dimensional consecutiver-out-of-n: F system. It is given by

~i(:)

Z

[1 -- RLF(r; n; p)]i[RLF(r; n; p)]~-i .

i=0

However, Mokhlis et al. (1997) extended this result to the case of the general connected-(r,s)-out-of-(n, m): F lattice system to obtain

R~F((r,s), (m,n);p) > max(Ls~,L6~ )

for 7 = L or C ,

(3.33)

where

L»~--- ~ i=0

I1 - R~F(«; m; p)] i [R~F(s; m; p)]ù-i,

(3.34)

Consecutive k-out-of-n systems

L6, = ~

[1 -

273

(3.35)

R,F(r; n; p)]J[RTF(r; n; p)]m-j

i=0

and R,~F(k;n;p) for 7 = L or C are given by (2.15) and (2.32), respectively. Mokhlis et al. (1997, 1999) derived upper bounds for the two-dimensional linear (circular) systems with identical components by dividing the systems either horizontally or vertically into matrices (subsystems) forming cuts of the system. They showed that the reliability of a 7-connected by (r,s)-out-of-(n, m): F lattice system satisfy

R,/F((r,s),

(n,m); p) _< u3~ for 7 = L

or C ,

where

u37 = min[(RyF( (r, «), (r, m); p))[n/d,

(RLF((r, s), (n, s);

p) )[m/«]]

for 7 = L o r

C ,

(3.36)

where R,/F((r, s); (r, m);p) and RLF((r, s); (n, s);p) are to be calculated using (2.45) and (2.46), respectively. Note that this upper bound is equivalent to u27 when pi = p. The reliability of linear and circular-connected-(r,s)-or-(s, r)-out-of-(n, m): F lattice systems satisfy

R~F((r, s)-or-(s, r),

•

!

!

(m, n);p) < mln(u2~ , u37)

for 7 = L or C ,

(3.37)

where u~~ = min[(R~F((r, «), (r, m);

p)[n/d,

(R~F((S,r), (s, m);

p))[n/sl]

for 7 = L

or C ,

(3.38)

and

u~~ = min[(RLF ((S, r),

(n, r); p)[m/r] (RLF ((r, S), (tl, S); p))[m/s]], for 7 = L o r

C ,

(3.3s)

where R~F((r,s), (r,m); p) and R~F((s,r), (s, m); p) are calculated using (2.45), while RLF ((S, r), (n, r); p) and RLF ((r, s), (n, s); p) are calculated using (2.46). For the sake of comparison, the bounds presented in this subsection are calculated for linear and circular connected-(4, 3)-out-of-(6, 8): F lattice systems in Tables 1 and 2, respectively. For simplicity, take Pi.j = P. From the numerical results we see that there is no significant difference between the different lower bounds. However, the difference appears for small p, after the fourth decimal place. The best (largest) lower bound is obtained by (3.32), while the best (least) upper bound is obtained by (3.25).

274

N. A. Mokhlis

Table 1 Lower and upper bounds on reliability of linear-connected-(4, 3)-out-of-(6, 8): F lattice system p

L1L

L4L

L 2 L ~ e 2La

L6L

glL

N3L

0.95 0.90 0.85 0.80 0.75

1 1 0.99999999 0.99999992 0.99999884

1 1 0.99999999 0.99999992 0.99999893

1 1 0.99999999 0.99999992 0.99999893

1 0.99999999 0.99999985 0.99999603 0.99994972

1 1 0.99999999 0.99999992 0.99999901

1 0.99999999 0.99999985 0.99999603 0.99994972

a

Where )~~is as (3.17) for 7 = L or C

Table 2 Lower and upper bounds on reliability of circular-connected-(4, 3)-out-of-(6, 8): F lattice system p

L1c

L4c

L2c ~ e 2ca

L6C

UlC

bt3c

0.95 0.90 0.85 0.80 0.75

1 i 0.99999999 0.99999989 0.99999845

1 1 0.99999999 0.99999992 0.99999893

1 1 0.99999999 0.99999990 0.99999856

1 0.99999999 0.99999985 0.99999603 0.99994972

1 1 0.99999999 0.99999990 0.99999868

1 1 0.99999999 0.99999996 0.99999952

a Where 2.~ is as (3.17) for ), = L or C Numerical calculations for the bounds o f reliability o f linear and circular connected-(4, 3)-or-(3, 4)-out-of-(6, 8): F lattice systems are presented in Tables 3 and 4 for different values of p. Tables 5 and O c o m p a r e between the b o u n d s for reliability in (3.26), (3.29) and (3.37) and the exact reliability, using (2.47) and (2.53), for linear and circular-connected-(1, 2)-or-(2, 1)-out-of-(5, 3): F lattice systems. It is clear that e ~'~'is the best lower bound, 7 = L or C. F o r ~-connected-(4,3)or-(3, 4)-out-of-(6, 8): F lattice system the best upper b o u n d s are u~i~ for p > 0.6 and u~7 f o r p < 0.6. F o r 7-connected-(1,2)-or-(2, 1)-out-of-(5, 3): F lattice systems the best upper bounds are u'l~ f o r p _> 0.95, f o r p < 0.95 the best upper b o u n d s are i and U~c for linear and circular systems, respectively. U3L F r o m what is mentioned here above, we conclude that best lower and upper bounds depend on the values o f p, r, s, n and m. It is clear that the reliability o f an F system is greater than that o f a circular F system. Also, we see that the reliability o f (r, s)-out-of-(n, m): F system is greater than that o f (r, s)-or-(s, r)-out-of-(n, m): F system. This is because, the n u m b e r o f minimal cuts of circular systems is greater than that o f linear systems, as well as the n u m b e r o f minimal cuts o f (r, s)-or-(s, r)-out-of-(n, m) is greater than that of (r, s)-out-of-(n, m): F systems. 3.3. Consecutive k-out-of-n: G systems

Using the relation that consecutive k-out-of-n: G and consecutive k-out-of-n: F systems are mirror images o f each other K u o et al. (1990) derived bounds for the one-dimensional linear and circular consecutive k-out-of-n: G systems.

275

C o n s e c u t i v e k-out-oj~n s y s t e m s

v

~L ©

H

ù

? Y

3

~5 ©

z

Il

©

& ©

N . A . Mokhlis

276

Il r~

.=

0

9

? Y

ù.-,

~eq

3ù

~5

II

-8 ùt ©

t-'q

8 ¢.~

8 M

~

Consecutive k-out-of-n systems

tt~

©

H ,A

ù~

©

? ,..y Y ee~

Il

ù

©

277

278

N. A. Mokhlis

o@ ¢'~

o

~~

oo

ddd

©

t~

gù.ù

t~

r~

Consecutive k-out-of-n systems

279

The lower bounds are g~=l-

H

1-

PJ

H

for•=LorC.

j=(i-l)k+l

i=1

The upper bounds are g (

~,= l - II

i=1

i+k- 1 )

1 - II pJ j=i

for7

LorC

,

/

where

g={n-k+l n

for 7 = L, for 7 = C .

The bounds for the reliability of the two-dimensional: F systems could by used by means of the mirror image relation, to obtain bounds for the corresponding G systems. References Antonopoulou, I. and S. Papastavridis (1987). Fast recursive algorithm to evaluate the reliability of a circular consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-36, 83-84. Arratia, R., L. Goldstein and L. Gordon (1989). Two moments suffice for Poisson approximations: The Chen-Stein method. Ann. Prob. 17, 9-25. Barbour, A. D., L. Holst and S. Janson (1991). Poisson Approximation. Oxford University Press. Barlow, R. E. and F. Proschan (1975). Statistical Theory of Reliability and Life Testing. Holt, New York. Boehme, T. K., A. Kossow and W. Press (1992). A generalization of consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. 41, 451~457. Bollinger, R. C. (1982). Direct computation for consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. R-31, 44~446. Bollinger, R. C. (1986). An algorithm for direct computation in consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. R-35, 611 612. Bollinger, R. C. and A. A. Salvia (1982). Consecutive-k-out-of-n: F networks. IEEE Trans. Reliab. R-31, 53 56. Chao, M. T. and J. C. Fu (1989). A limit theorem of certain repairable systems. Ann. Ins. Stat. Math. 41, 809-818. Chao, M. T. and J. C. Fu (1991). The reliability of large series systems under Markov structure. Adv. Appl. Prob. 23, 894-908. Chao, M. T., J. C. Fu and M. V. Koutras (1995). Survey of reliability studies of consecutive-k-out-ofn: F and related systems. IEEE Trans. Reliab. 44, 120-127. Chiang, D. T. and R. Chiang (t986). Relayed communication via consecutive-k-out-of -n: F system. IEEE Trans. Reliab. R-35, 65 67. Chiang, D. T. and S. Niu (1981). Reliability of consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-30, 87 89. Chao, M. T. and G. D. Lin (1984). Economical design of large consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. R-33, 411-413. Chrysaphinou, O. and S. Papastavridis (1990). Limit distribution for a consecutive-k-out-of-n: F systems. Adv. Appl. Probab. 22, 491493.

280

N. A. Mokhlis

Derman, C., G. J. Liberman and S. Ross (1982). On the consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-31, 57-63. E1Sayed, E. M. (1998). Algorithm for reliability of-(1, 2)-or-(2, 1)-out-of-(n, 2): F system. J. Egypt. Math. Soc. 6, 169-173. Feller, W. (1968). An Introduetion to Probability Theory and Its Applications (3rd ed.). Vol. 1, Wiley. Fu, J. C. (1985). Reliability of a large consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-34, 127-130. Fu, J. C. (1986a). Bounds for reliability of large consecutive-k-out-of-n: F system with unequal component probabilities. IEEE Trans. Reliab. R-35, 316 319. Fu, J. C. (1986b). Reliability of consecutive-k-out-of-n: F system with (k-1) step Markov dependence. IEEE Trans. Reliab. R-35, 602-606. Fu, J. C. and B. Hu (1987). On reliability of a large consecutive-k-out-of-n: F system with (k - 1)-step Markov dependence. IEEE Trans. Reliab. 36, 75-77. Hwang, F. K. (1982). Fast solutions for consecutive-k-out-of-n: F related systems. IEEE Trans. Reliab. R-31, 447448. Hwang, F. K. (1988). Relayed consecutive-k-out-of-n: F lines. IEEE Trans. Reliab. 37, 512-514. Hwang, F. K. (1993). An O(k n)-time algorithm for computing the reliability of a circular consecutivek-out-of-n: F system. IEEE Trans. Reliab. 42, 161-162. Khamis, S. M. and N. A. Mokhlis (1997) An algorithm for computing the reliability of connected(1, 2)-or-(2, 1)-out-of-(m, n): F lattice system. Congressus Numerantuim 127, 143-154. Kontoleon, J. M. (1980). Reliability determination of a r-successive-out-of-n: F system. IEEE Trans. Reliab. R-29, 437. Koutras, M. V., G. K. Papadopoulos and S. G. Papastavridis (1993). Reliability of 2-dimensional consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. 42, 658-661. Kuo, W., W. Zhang and M. Zuo (1990). A consecutive-k-out-of-n: G system: The mirror image of a consecutive-k-out-of-n: F system. IEEE Trans. Reliab. 39, 244253. Lambiris, M. and S. Papastavridis (1985). Exact reliability formulas for linear and circular consecutive-k-out-of-n: F related systems. IEEE Trans. Reliab. R-34, 124-126. Malinowski, J. and W. Preuss (1996). Lower and upper bounds for the reliability of connected-(r, s)out-of-(m, n): F lattice systems. IEEE Trans. Reliab. 45, 156-160. Mokhlis, N. A. and A. S. Mohamed (1999). Bounds for reliability of relayed consecutive-k-out-of-n: F system. J. Egypt. Math. Soc. 7(2), 225 237. Mokhlis, N. A. and S. M. Khamis (1998). An exact enumeration of the reliability of some linear and circular connected-X-out-of-(n, m): F lattice system. Congressus Numerantium 135, 93 118. Mokhlis, N. A., E. M. E1Sayed and G. Youssef (1997). Bounds on reliability of connected-(r, s)-ont-of(n, m): F lattice systems. In The 32-nd Annual Conference on Statistics and Computer Science Operation Research. Vol. 32, pp 19-34. I.S.S.R., Cairo University. Mokhlis, N. A., E. M. E1Sayed and G. Youssef (1998). Reliability of a connected-(1, 2)-or-(2, 1)-outof-(n, 3): F lattice system. J. Egypt. Math. Soc. 6, 175 183. Mokhlis, N. A., E. M. E1Sayed and G. Youssef (1999). Reliability of consecutive-k-out-of-n systems. M.Sc. Thesis. Papastavridis, S. (1987). Upper and lower bounds for the reliability of consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-35, 607 610. Papastavridis, S. and M. Lambiris (1987). Reliability of a consecutive-k-out-of-n: F system, for Markov-dependent components. IEEE Trans. Reliab. R-36, 78-79. Salvia, A. A. and W. C. Lasher (1990). 2-dimensional consecutive-k-out-of-n: F models. IEEE Trans. Reliab. 39, 382 385. Wu, J. S. and R. J. Chen (1992). An O(k n) algorithm for a circular consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-41, 303 305. Wu, J. S. and R. J. Chen (1993). Efficient algorithm for reliability of a circular consecutive-k-out-of-n: F system. IEEE Trans. Reliab. 42, 163-164. Zuo, M. J. (1993). Reliability and design of two-dimensional consecutive-k-out-of-n system. IEEE Trans. Reliab. 42, 488-490.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserve&

| •

lk &

Exact Reliability and Lifetime of Consecutive Systems

Sigeo Aki

1. Introducfion

A linear consecutive-k-out-of-n: F system is a system of n components in sequence where the system fails if and only if k consecutive components fail. The system was introduced by Kontoleon (1980). The reliability of the system has been investigated by many researchers (e.g., Chiang and Niu, 1981; Derman et al., 1982; Shanthikumar, 1982; Lambiris and Papastavridis, 1985; Hwang, 1986; Papastavridis and Hadzichritos, 1988; Peköz and Ross, 1995). Excellent surveys were presented by Hirano (1994) and Chao et al. (1995). For i = 1 , . . , n, let X/be a random variable such that X / = 1 (= 0) if the ith component is functioning (resp., fails). When X1,X2,..,X~ are independent identically distributed (i.i.d.) random variables with P ( ~ = l ) = p and P(Xi = 0) = q(= 1 - p ) , the reliability of the system is given by

rk)--(n-k(r+l))(-1)qk)

R(p) = Z ( - 1 ) r ( p q k ) r r=0

.

F

Many papers have been published in reliability journals and in most papers the reliability value of each consecutive system was calculated under various conditions mainly by means of techniques of enumerative combinatorics. In view of practical situations, some generalizations of the system have been attempted. For example, some dependence structures between components have been taken into consideration (Fu, 1986; Papastavridis and Lambiris, 1987; Ge and Wang, 1990) and more complex systems such as m-consecutive-k-out-of-n: F system (Papastavridis, 1990), consecutive-k-out-of-r-from-n: F system (Griffith, 1986; Sfakianakis et al., 1992) and some extensions to two-dimensional systems (Salvia and Lasher, 1990; Ksir, 1992; Boehme et al., 1992; Yamamoto and Miyakawa, 1995) have been investigated. As the systems become more complex, it seems more difficult to calculate their exact reliability values by means of usual techniques of enumerative combinatorics. However, we can deal with the problem successfully by using new powerful tools such as Markov chain imbedding method and the method of conditional 281

282

S. Aki

probability generating functions, which have been recently developed in statistical distribution theory of runs. Statistical distribution theory of runs has also developed extensively over the past two decades. In particular, the exact distribution of the number of runs of a specified length in various random sequences and that of the number of trials until the first occurrence of a run of a specified length in a sequence of trials have been investigated and derived by using the above new tools even if the sequence of trials is dependent or multivariate. Actually, the reliability value of each consecutive system means the probability that a run of failed components of a specified length does not exist in the sequence of components. Therefore, we can use some results in distribution theory in order to calculate the reliability values of much more complex consecutive systems. Here, we briefly overview some approaches to the study of discrete distributions of runs, which are substantially related to the reliability study of consecutive systems. Of course, the standard method is enumerative combinatorics also in the study of statistical discrete distribution theory. The method is the following: to start with, all the typical sequences are considered, and then, they are split into subsequences which can be interpreted successfully. The method is useful when trials are independent or nearly independent (e.g., when the sequence of trials is a Markov chain). However, when dependence structure of trials becomes complex (e.g., when the sequence of trials is a higher-order Markov chain), it becomes difficult even to write down all the typical sequences (see Aki et al., 1996). The Markov chain imbedding method is to imbed the sequence of trials (or enumeration process of runs or patterns) into a Markov chain with an appropriate state space. Then the desired probability of the number of runs or patterns can be obtained by multiplying the transition probability matrices. The method was introduced by Fu and Koutras (1994) and developed by Fu (1996), Koutras and Alexandrou (1995, 1997a, b) and Koutras (1997). The method of conditional probability generating functions (p.g.f.) can be explained as follows: to begin with, all the different situations which are possible to occur are considered, and the relations between conditional p.g.f.'s given the situations are examined, and finally, the system of equations which represent the relations is solved. The method was introduced in statistical distribution theory of runs by Ebneshahrashoob and Sobel (1990) and used in various situations by Hirano and Aki (1993), Aki and Hirano (1995), Uchida and Aki (1995), Balakrishnan et al. (1997), Hirano et al. (1997) and Han and Aki (1998). In Section 2, we explain how to use the method of conditional p.g.f.'s. When the relations between conditional p.g.f.'s are linear, the method and the Markov chain imbedding method are essentially equivalent. Markov chain imbedding approaches for reliability study of usual consecutive systems were investigated very well by Koutras (1996). Hence, in order to emphasize a merit of the method of conditional p.g.f.'s we deal with an example of a consecutive system on a directed tree, where relations between the conditional p.g.f.'s are not necessarily linear. In Section 3, we give some numerical examples of the reliability of consecutive systems on directed trees. Algorithms for computing the p.g.f.'s of the number of

Exact reliability and lifetime of conseeutive systems

283

failure runs of a specified length k are also given. By using the algorithms and computer algebra systems, we can obtain the exact p.g.f.'s. After expanding the p.g.f, with respect to the variable t (if necessary), we can easily calculate the exact reliability of the k-consecutive system by substituting t = 0 into the p.g.f. In order to calculate the reliability of the m-consecutive-k-out-of-n: F system on a directed tree, we only sum up the coefficients of tJ of the p.g.f, for j = 0, 1 , . . . , m - 1. In Section 4, we discuss lifetime distributions of consecutive systems. After giving a very general result, we review lifetime distributions of some systems systematically. In particular, we focus on derivation of the mixing weights by which the lifetime distribution of a consecutive system is represented as a mixture of distributions of order statistics of lifetimes of the components.

2. Reliability of consecutive systems on directed trees Let us consider a consecutive system on the directed tree in Figure 1. The root of the tree is vl and all the edges are directed away from the root. Each vertex is assumed to be a component of a system, which fails if and only if three consecutive components along the direction fail. For simplicity, we assume also that all the components independently fail with probability q (= 1 - p ) . By using the method of conditional p.g.f.'s, we can derive the exact distribution of the number of non-overlapping failure runs of length three along the direction on the directed tree and calculate the reliability value of the system. In the i.i.d, case, the following result holds generally. We fix any vertex v. Let T~ be the subtree which consists of the vertex v as the root of Tu and of all the descendants of v. In the path of all the ancestors of v from the root to the parent of v, assume that a failure tun of length g along the direction is observed at the parent of v. If the vertex v is the root, we conventionally define that a failure run of length 0 is observed at the parent of v, though the root does not have its parent. 731

U4

U5

~, U6

\

/ /\ 739

7310

7311

U12

V13

U8

'/)14

Fig. 1. An example of a directed tree.

s. Aki

284

Then we denote by 4(/),g;t) the p.g.f, of the conditional distribution of the number of non-overlapping failure runs of length k along the direction in the subtree T~. If the vertex v has c(v) children/)(1), v ( 2 ) , . . , v(c(/))), the numbers of failure runs of length k in T~(1), T~(2),. • •, and T~(c(~)) are conditionally independent given that all the ancestors from the root to the vertex/) are observed. Hence, by considering one-step ahead from the vertex v, we obtain the following proposition. PROPOSITION 1. If all the components independently fail with probability q(= 1 - p ) , the conditional p.g.f.'s satisfy the recurrence relations:

4(v,g;t) = ~ lPY[~(--~Iq~(v(j),o;t)+ qtI]:(=vlqS(v(j'),0;t) /

(p+qt

ifc(v) > 0 and g < k ifc(v) > 0 and g = k ifc(v) =0 and g < k ifc(v)=0 and g = k -

1, 1, 1, 1.

(1) REMARK. In this section we assume that all the components fail independently with a common probability. However, more general results can be derived. Aki (1999) treated the problem when the joint distribution of the components is a directed Markov distribution (see, e.g., Lauritzen, 1996). He obtained the exact distribution of the number of non-overlapping runs on directed Markov trees. In fact, Proposition 1 is a special case of the result of Theorem 2.1 of Aki (1999). However, it should be noted that the result of Proposition 1 is written very simple because of the assumption. By using Proposition 1, we shall derive the p.g.f, of the number of non-overlapping failure runs of length three along the direction on the directed tree given in Figure 1. For simplicity, it is assumed that all the components fail independently with probability q (= 1 - p ) . Let 4(t) be the p.g.f. Then, 4(t) = 4(Vl, 0; t) holds from the definition. By applying (1) to the root/)1, we obtäin

4(/)~, o; t) = p4(/)2, o; t)~(V3, O;/)4(V4~ O; t) + q4(/)2, 1; t)~(~3, l; t)~(~4,1; t) . Next, by applying (1) to every child of the root Vl, we see that

4(~2, o; t) = p4(/)», o; t) + q4(/)», 1; t), 4(/)2, 1; t) ~-~P445, 0; t) + q4(/)5, 2; t), 4(/)3, 0; t) = P446,0; t) -k q446, 1; t), 4(/)3, 1; t) = P4(Vo, 0; t) q- q4(v6, 2; t) , and

~(V4,0; t) ~ p 4 ( v % 0; t)4(V8,0; t) + q4(v7, 1;t)4(V8, 1; t), 4(v4, 1; t) = pO(v7, O; t)4(v8, O; t) + q4(v7, 2; t)4(vs, 2; t) .

285

Exact reliabiIity and lifetime of consecutive systems

Since the vertices v9,Vl0, Vll,Vl2, Vl3 and /)14 do not have a child, we see from Proposition 1 for i = 9, 1 0 , . . , 14, Ó(vi, O;t)= l,

O(vi, 1 ; t ) = l,

é(vi,2;t)=p+qt

.

Then, by applying Proposition 1 to the vertices v», v6, v7 and va, we obtain

qS(vs, 1; t) = p + q(p ÷ qt) 2, ~b(v6, 0; t) = 1, qS(v6, 1; t) = p q- q(p + qt) 2, ~b(v7, 0; t) = 1, qS(v7, 1;t) = p + q ( p + q t ) , é(vs, 0; t) = 1, ~b(v8, 1; t) = p + q(p + qt),

qS(vs, 0; t) = 1,

qb(vs, 2; t) = p + qt, qS(v6, 2; t) = p + qt, qS(v7, 2; t) = p + qt, OS(va,2; t) = p + qt .

Therefore, we obtain ~b(t) = p~o + q{p + q(p ÷ qt)2}12~ + q{p + q(p + qt)} 2] + q{p + q(p + «t)}2{p + q(p + qt) 2} . The reliability value of the consecutive system is the coefficient of to in the polynomial qS(t). Hence, we can write the reliability value of the consecutive system as p(p + q(p + qp2))2(p + q(p + qp)2) + q(p + qp)2(p + qp2) . As a corollary of Proposition 1 we can deal with a linear consecutive-k-out-of-n: F system. We can regard the special tree given in Figure 2 as the system. According to the result of Proposition 1, we see that the p.g.f.'s of the conditional distributions of the number of non-overlapping failure runs of length k satisfy the linear recurrence relations: [p4)(vi+l,0;t)+qcB(vi+l,g+l;t) O(vi,g;t)= ~pß(vi+l,0;t)+qt~(Vi+l,O;t)

l"

kp+qt

ifi
and and and and

g
.

We can treat much more complex consecutive systems with dependent structures (see Aki, 1999). Even if the system becomes complex, we can derive the p.g.f.'s only by some algebraic manupulations for polynomials (e.g., addition, multiplication, expansion, extraction of coefficients, etc.). Therefore, use of computer algebra systems is very useful.

Vl @

V2 .0

?)3 ~I

Vn-2 '''

•

Vn--1 Re

Fig. 2. Linearconsecutive-k-out-oßn:Fsystem.

Vn :~

286

S. Aki

3. Algorithms for exact reliability and numerical examples In this section we illustrate how to calculate reliability values of consecutive systems on directed trees by using computer algebra. As we saw in the previous section, Proposition 1 can be used easily by hand to calculate reliability values of consecutive systems if the systems are not so large. EXAMPLE 3.1. Let us consider a consecutive system on a directed tree given in Figure 3. The system has 41 components which are put on the vertices of the directed tree one by one. We assume that the components fall independently with probability q(= 1 - p ) ; i.e., every component has reliability value p. Let k be a positive integer and suppose that the system fails if and only i f k consecutive components along the direction fail. The problem we have to consider is how to use Proposition 1 recursively in computer algebra systems. One of the feasible methods for that is to input the tree structure by giving an array of link-lists from every vertex to the list of its children. Then we can give an algorithm for calculation of the p.g.f, of the number of non-overlapping failure runs of length k based on Proposition 1 as follows. Recursive procedure PGF(v, g; t).

Inputs: cv, v E V: an array of link-lists from every vertex to its children of the directed tree (e.g. % = @2, v3, v4), % = (rs, v6), ..., c~4~ = ( ) for the directed tree given in Figure 3). k: the length of a failure run which fails the system. p: the common reliability value of individual components. v: a vertex in the directed tree. & the current length of failure run observed at the parent of v.

'U1 B

«

'

L,

/~6

~

229

I

~)7

L,\.,

/~8 ,~9

=.L~41

~~...o,~4o

"k Vl~(q7 __~.~ V39

~~1 ~~1~-O~ / ~V14~Vi5 ~ - ; 3 8

j ~ Otl~Og31

V21

.~~v

~

24

~~~~

g32 Ot~g33 "~O~34 ~ 3 5 Fig.3. Anexampleof a consecutivesystemon a directedtree.

Exact relh~bility and lifetime of consecutive systems

287

Output: qS(v,g; t): the p.g.f, of the conditional distribution o f the n u m b e r of non-overlapping failure runs of length k along the direction in the subtree Tv.

Algorithm proeedure PGF(v, g; t) Let q = 1 - p . Let

ch =

the length of the list c~.

If

(ch =

0 and g < k - 1), then return 1.

If

(ch =

0 and g = k - 1), then return Co +

qt).

I f (g < k - 1), then return{PI~~~l

PGF(c~[]],O;t)+ q]~~h 11~=1 eaF(coO.],g + l;t)}

If (g = k - 1), then return

ch {pl-U x *2=1 PGF(cvO'], O; t) q- qt Hk=l

PGF(cvO'], O; t) }.

end By using the algorithm and c o m p u t e r algebra systems, we can obtain the exact p.g.f. After expanding the p.g.f, with respect to the variable t (if necessary), we can easily calculate the exact reliability of k-consecutive system by substituting t = 0 into the p.g.f. In order to calculate the reliability o f m-consecutive-k-out-of-n: F system on a directed tree, we only sum up the coefficients o f tJ of the p.g.f, for j=0,1,..,m1. F o r k = 1,2, 3, 4, we let R(k) be the reliability o f the consecutive-k-out-of-41: F system on the directed tree given in Figure 3. Then we obtain the following exact reliability values o f the system as the function o f reliability o f individual components p. G r a p h s of R(1),R(2),R(3) and R(4) are given in Figure 4.

R(1)

=1o41 ,

R(2)

= -767/) 37 q- 6653p 23 ÷ 4143p 36 - 3348/) 22 + 43185p 27 - 14425p 35 - 53723p 33 + 60462p 32 - 21135p 26 + 348p 19 + 52874p 3° + 10496p 24 + 79p 18 _ 4p 39 -1-- 2045p 21 + 33699p 34 + _ 706@ 28 + 83p 38 ,

5 / ) 17 _

288/920

--

38580p 29

15506/) 25 - 557171o31

S. Aki

288

I

I

R2 R R 43

0.8

. .-.-. .

.

i

I

.

.

°,,

.

" ....

=

. ""

' ,

0.6

0°4

ù

/

0.2

,

il

,I "

° ''

°" "'

0.2 0.4 0.6 0.8 p (Reliability Value of Each Component) Fig. 4. Reliability of the consecutive-k-out-of-41: F systems on the directed tree with 41 components given in Figure 3.

R(3)

= 7 6 4 p 37 + 9 9 5 7 6 7 p 23 - 6 3 8 6 p 36 - 524128/) 22 + 1 5 9 9 9 p 16 - 2 4 3 5 7 1 @ 27 + 3 7 1 8 @ 35 + 517612/) 33 - 1 2 8 5 0 0 2 p 32 + 3 8 5 0 5 9 5 p 26 + 8 4 8 5 2 p 19 - 3 3 0 6 3 0 6 p 3° + 7 2 p 1° + 3 2 8 p 11 _ 218/) 12 _ 1877p 13 + 1 0 1 2 9 p 15 + 1 5 2 4 6 @ 24 - 5 4 2 6 2 p 18 + 14p 9 + 2/) 39 + 1 1 4 7 1 1 p 2° + 2 8 7 6 8 9 3 p 29 - 3 3 0 6 p 14 - 1 2 2 5 2 0 p 21 - 1 5 9 4 2 7 p 34 - 30444/) 17 - 2 5 6 2 1 4 1 p 25 + 2 4 1 6 9 0 7 p 31 - 5 8 2 5 0 @ 28 - 57/) 38 ,

and

R(4)

= - 2 1 0 1 1 3 p 23 -r- 1 0 7 5 3 @ 22 + 3 1 2 0 1 p 16 - 3 0 3 3 0 p 27 q_p32 + 7 8 2 8 0 p 26 + 2 2 6 7 0 p 19 + 268/) 3° + 6 p 4 + 3p 5 + 10p 6 + 3 6 5 p 1° _ 477/) 11 + 9 4 3 p 12 ÷ 2449/) I3 q- 1 1 5 8 9 p 15 + 2 1 5 5 9 7 p 24 + 6 8 1 6 2 p 18 + 18/) 9 - 85028/) 2° - 1847p 29 -- 13555/) 14 ÷ 2 8 9 2 9 p 21 + 39p 7 -- 8 3 7 1 3 p I7 -- 2 1 3 p 8 -- 1 5 1 5 3 7 p 25 -- 2 @ 31 + 8 7 7 @ 28 .

F o r k = 1,2, 3 , 4 , w e ler R 2 ( k ) b e t h e r e l i a b i l i t y o f t h e 2 - c o n s e c u t i v e - k - o u t - o f n: F s y s t e m o n t h e d i r e c t e d tree. T h e n w e o b t a i n t h e f o l l o w i n g e x a c t r e l i a b i l i t y v a l u e s o f t h e s y s t e m as t h e f u n c t i o n o f r e l i a b i l i t y o f i n d i v i d u a l c o m p o n e n t s p. G r a p h s o f R 2 ( 1 ) , R 2 ( 2 ) , R 2 ( 3 ) a n d R 2 ( 4 ) a r e g i v e n i n F i g u r e 5.

R2(1)

= - 4 0 p 41 + 4 1 p 4° .

Exact reliabilityand lifetime of consecutivesystems

k.l=--.3. . I

k=l k=2

0.8

[

-....

I

289

,,'''r'-

."

."""

'

..'"

."

.."

0.6

0.4

0.2

0.2 0.4 0.6 0.8 p (Reliability Value of Each Component) Fig. 5. Reliability of the 2-consecutive-k-out-of-41: F systems on the directed tree with 41 components given in Figure 3.

R2(2)

= 1 9 3 3 3 p 37 - 2 9 9 6 8 8 / ) 24 - 7 8 7 3 7 6 p 27 - 8 3 8 7 9 p 25 -

1 9 6 2 5 0 8 p 3°

- 2 0 3 4 6 7 1 p 32 + 9 1 0 5 9 7 p 26 - 4482671o 28 + 2 1 0 2 7 3 3 p 31 + 1 5 4 1 4 2 0 / ) 29 - 8 9 2 4 0 5 p 34 + 1 5 7 0 9 3 5 p 33 - 2 2 8 3 p 38 _ 3 p 4o + 1 4 3 p 39 + 3 6 1 4 2 1 p 35 + 68/) 16 _ 2 1 8 7 / ) 19 - 2 5 0 4 6 p 2° -

1 0 6 7 0 p 21 + 9 1 8 p 17 + 1 2 2 2 2 7 / ) 22 + 2 7 2 7 p 18

+ 1 8 4 8 5 / ) 23 -

R2(3)

102023/936

.

= 1 2 8 0 p 1° - 2 2 4 6 / ) 11 - 3 4 9 9 / ) 12 - 1 3 2 6 6 / ) 13 + 6 7 p 8 + 8 4 1 2 8 p 15 + 3 8 6 3 8 / ) 14 - 1 2 2 5 2 / ) 37 - 4 2 3 3 8 6 2 9 p 24 - 3 5 5 7 9 1 l p 27 + 7 2 6 4 1 4 9 7 p 25 + 8 1 7 3 0 2 9 3 p 3° + 2 5 2 0 5 5 3 0 p 32 - 5 6 6 8 5 8 8 l p 26 + 6 6 3 9 7 2 5 3 p 28 - 5 1 8 8 2 1 0 5 p 31 - 9 3 7 1 3 6 1 0 p 29 + 2 7 9 6 8 1 9 p 34 - 9 5 2 5 1 1 2 / ) 33 -? 3 9 p 9 + 8 9 7 p 38 - 3 1 p 39 - 6 2 8 6 3 5 p 35 - 1 3 3 4 6 9 p 16 + 3 4 2 8 0 3 p 19 - 7 7 3 6 3 8 p 2° - 5 2 4 6 6 4 0 / ) 21 - 5 3 1 9 7 4 p ~7 + 1 0 8 9 9 7 4 5 p 22 + 1 0 5 3 5 6 3 / ) 18 + 3 7 5 1 5 0 7 / ) 23 + 1 0 4 8 4 0 p 36 .

R2(4)

= 21o3 + 1 9 p 4 - 46/) 5 + 2 4 5 / ) 6 - 2 9 6 p 1° - 7 8 5 4 p 11 + 328931o 12 - 55062/913 -

1 6 1 p 8 + 4 7 2 4 3 5 p 15

- 6 5 9 3 5 p 14 - 5 9 1 p 7 - 3 9 8 5 6 1 5 / ) 24 + 4 0 2 4 7 0 p 27

S. Aki

290

+ 240533@ 2 5 - 3101p 3 ° - l lp s 2 - ll20161p 26 - 10995@ 28 ÷ 270p 31 + 22138/)29 ÷ 1638,o9 - 90885@ 16 - 598795p 19 - 341440p 2° + 2548759p 2~ + 755879ff 7 - 4531745p 22 + 80869p ts + 5006671p 2s We have seen that the reliability value can be calculated exactly for the consecutive system on the directed tree which has 41 components. Though Example 3.1 may be sufficient to show the possibility of calculation of reliability values of consecutive systems of practical size, we give here examples which have rauch more components. EXAMPLE 3.2. AS a special case of directed trees, we consider a complete m-ary tree of length (n - 1). In the tree, every vertex except leaves has just m children. The length of a directed tree means the maximum of the lengths of the paths from the root to leaves. Hence, the complete m-ary tree of length ( n - 1 ) has (m n - 1 ) / ( m - 1) vertices. Two simple examples are given in Figure 6. We consider the k-consecutive system on the m-ary tree of length (n - 1), that is, the system fails if and only if k consecutive components fail along the direction. In the complete m-ary tree of length (n - 1), all the vertices can be simply classified into n categories by their generations: only the root belongs to the first generation; m children of the root belong to the second generation,..., and m n - 1 leaves belong to the nth generation. If vi and vj belong to the same category, then T~, and T~j have the same structure. Hence, since 4(vi, g; t) = (~(vj, g; t), we denote by ~b(u,g, t) the conditional p.g.f, if they belong to the (n - u + 1)-st generation. Then, from Proposition 1, we obtain the following recurrence relations:

( p ( ~ ( u - 1,0;t)) m + q ( 0 ( u - 1,g+ 1;t)) mif u > 1 and g < k - 1, (o(u,g;t)=Jp(4(u-l,O;t)) +qt((b(u-l,O;t)) i f u > l and g = k - 1 ,

/1

if u = l if u = l

[,p+qt

and g < k - 1 , and g = k - 1 .

(2)

•

A/• •

•

•

•

•

•

•

Complete binary tree of length 3

•

Complete trinary tree of length 2

Fig. 6. Two examples of complete m-ary tree (m = 2, 3).

E x a c t reliability a n d l i f e t i m e o f consecutive s y s t e m s

291

In particular, when m = 1, the k-consecutive system on the complete m-ary tree of length (n - 1) reduces to the linear consecutive-k-out-of-n: F system and from the formula (2) we obtain the linear recurrence relations of the conditional p.g.f.'s

(p4)(u- 1 , 0 ; t ) + q 4 ( u - 1,g+ 1;t) O(u,6;t)=)p~(u-l,0;t)+qt~(u-l,O;t )

11 tp+qt

if u > 1 and 6 < k - 1, i f u > l and g=k-1, ifu=l and6
(3) Further, if we set qS,(t) -- ~b(n, 0; t), then from the above linear relations (3), we obtain

{ én(t)

~bn(t)

k-1 • 1

ifn
.

This linear recurrence relations for ~n(t) was solved by Aki and Hirano (1988, Theorem 2.1) and 4)n(t) is written as

Z

~

m = 0 t71+2n2+...+kn~,=n-m

n]+n:+...+nk qn n l ~ H2 ~ •

l+pt

.

~ l~lk

We give here an algorithm for calculation of the p.g.f, of the number of nonoverlapping failure runs of length k on complete m-ary tree of length (n - 1) based on the formula (2). Recursive procedure PGF2(u, 6; t)

Inputs: k: the length of a failure run which fails the system. p: the common reliability value of individual components. m: the common number of children of each vertex except leaves. u: 1+ length of complete m-ary (sub)tree. 6: the current length of failure run observed at the patent of the root of the complete m-ary (sub)tree.

Output: q~(u,6;t): the p.g.f, of the conditional distribution of the number of non-overlapping failure runs of length k along the direction in the complete m-ary (sub)tree of length ( u - 1).

Algorithm procedure PGF2(u, g; t) Let q = 1 - p. If (u = 1 and g < k - 1), then return 1. If (u -- 1 and g = k - 1), then return (p +

qt).

S. Aki

292 If(u>l

andg
then return {p(PGF2(u - 1,0; t)) m + q(PGF2(u - 1, g + 1; t))m}. If(u> 1 andg=k-1), then return {p(PGF2(u - 1,0; t)) m + qt(PGF2(u - 1,0; t))m}. end By using the above algorithm we calculate here the exact reliability of consecutive-k-out-of-32767: F systems on complete binary tree of length 14 for k = 5, 6, 7 and that of consecutive-k-out-of-29524: F systems on complete trinary tree of length 9 for k = 5, 6, 7. The exact reliability values of the systems can be easily obtained by substituting t = 0 into the p.g.f, of the distribution of the number of failure runs of length k along the direction. However, the exact formulas are very long and may not be suitable for publication. Therefore, we give here the following graphs of the reliability of the system as functions of the common reliability value of the individual components. 4. Lifetime of conseeutive systems Lifetime distributions of linear consecutive-k-out-of-n: F systems were discussed by Derman et al. (1982), Shantikumar (1985), Chen and Hwang (1985), Iyer (1990, 1992) and Aki and Hirano (1996), under the assumption that the component lifetimes are i.i.d, or "nearly" i.i.d. If we suppose that the component lifetimes are independently identically distributed with a distribution function G(t), the distribution function F(t) of the system lifetime LT of the linear consecutive-k-out-of-n: F system can be written as F(t) = 1 - R(1 - G(t)), where R(p) was given in Section 1. Further, the distribution function F(t) of the system can be written as a mixture of the distribution of order statistics of n component lifetimes (See Figure 7). In general, without the assumption of independence of component lifetimes, the hext theorem given by Aki and Hirano (1997) holds for usual systems which have n components satisfying the following assumptions A1 and A2. We are assuming that the component lifetimes 4 1 , . . , in are random variables defined on a probability space (f/, Y , P). Let LT be the lifetime of the system. A1. The system fails when some of n components fail, i.e., for every co E f/ there exists a number j = j(co) such that LT(co) = ~j(o~)(co). A2. The (random) number j(co) is a function of ranks of ~1(co), 42(0»),.., d,n(o»), i.e., the number j(oo) is determined by the order of failure of the components. In order to simplify the notations we identify the components {C1, C 2 , . . , C,,} with the set {1, 2 , . . , n}. A subset of { 1 , 2 , . . , n} is called a cutset if the system fails when the corresponding components of the subset fall. A subsequence of { 1 , 2 , .., n} is called a cutsequence if it is a permutation of a cutset. A cutsequence (al, az,..., a/c) is said to be minimal if the subsequence (al, aa,..., ak-i) is not a

Exact reliabilityand lifetime of consecutivesystems I

I

I

I

I

I

I

293

1.'

k=5 -k=6 k=7 --

0.8 0.6 0.4 0.2

0

0.5

f

0.55

I

)

0.6 0.65 0.7 0.75 0.8 0.85 0.9 p (Reliability Value of Each Component)

0.95

1

Fig. 7. Reliability of the k-consecutive systems on the complete binary tree of iength 14 with 32767 components.

cutsequence. Let J ù be the permutation group on { 1 , 2 , . . , n } . For every ~ = ( r c ( 1 ) , . . , ~ ( n ) ) Œ Y , , there exists an integer j uniquely such that (rc(1),..,r@')) is a minimal cutsequence. The integer j is called the minimal cutnumber of ~ and it is denoted by m(rc). TH~OREM 1. If the distribution of the component lifetimes (41~..., 4,) has a joint

density g(sl,..., s,), the distribution of the lifetime of the system has the following density." f(s) =

ds~(,_l)

ds~(m(~)+l) • • rCC~C~n ~. S~(m(~))

S~(n-2)

ds~(,) Sx(n-l)

X~0 s~(m(~)) . [ s~(3) fsr(2) dsrc(m(=)-l) "" ./0 ds~(2) J0 dsg(1)g(Sl'

...,Sn)}s~(~(~))= s

Based on the result of Theorem 1, Aki and Hirano (1997) derived by using a computer algebra system some exact lifetime distributions of linear consecutive systems whose component lifetimes have two kinds of dependence structures. The first dependence structure was introduced by Kamps (1995) based on sequential order statistics. The second one was a simple model such that the hazard rate of each component takes two possible constant values according to the stare of the left adjacent component (Aki and Hirano, 1997) (see Figure 8). If we assume independence of component lifetimes in addition to the assumptions A1 and A2, we can derive from Theorem 1 some important results systemätically. If 4 1 , 4 2 , . . , 4, are independently distributed and 4i has cdf Fi and pdf f., then the density of the system lifetime LT is written by

294

S. Aki

I

I

I

I

I

I

.1''

0.8

0.6

0.4

0.2

0 0.5

I ~ 0.55

I

0.6 0.65 0.7 0.75 0.8 0.85 0.9 p (Reliability Value of Each Component)

I 0.95

Fig. 8. Reliability of the k-consecutive systems on the complete trinary tree of length 9 with 29524 components.

f ( s ) = ~c~'. ~_ù f~(m(=))(S) ( m ( ~1) - 1)[ F~(1)(s) "'F~(m(~) 1)(s) 1

x (n -m(~))! (1 -F~(m(~)+l)(s))"" (1 -F~(n)(s)) .

(4)

This result is easily derived by repeating integration by parts in the result of Theorem 1. Further, if the minimal cutnumber m(rc) = r for every ~ (this means that the system is an r-out-of-n: F system), then the density of the lifetime of the system can be written as 1 1 ~ F~(1)(s)...F~(,_l)(s)f~(~)(s) f ( s ) = (r - 1)! (n - r)! ~j,, × (~ - F~(r.~)(~))... (1 - F~(~)(~)) +

F1 (s), 1

1

( r - 1)! ( n - r)!

~(s)

Fl(S),

fl(~),

•.*? '1

+

P,,(~)

1 -Fl(S),

1 - en(~)

1 - El ( s ) ,

1 - F,

(~)

Exact reliability and lifetime of consecutive systems

295

This formula representing the density of the rth order statistic of 41,-- •, ~, agrees with V a u g h a n and Venables (1972) (see also Balakrishnan and Rao, 1998). W h e n 4 1 , 4 ; , . . , 4n are independent identically distributed r a n d o m variables with c d f Fo and p d f fo, the p d f of the system lifetime of the linear consecutivek-out-of-n: F system can be written f r o m (4) as (j - 1)[ (n

j=l

- F°(s))n-Jf°(s)

'

where v ( j ) - n u m b e r o f { ~ z E 5 g , ] m ( r c ) = j } . H w a n g (1991) gave the explicit f o r m u l a for the n u m b e r o f minimal cutsequences of length m as

,

rmk=(n_m+l).(m_l)IZ(_l)i

i=o

'

_ m[ Z ( _ I )

(

/ n-m+1

i=0

i

( n - m + 2 )( )(: - :) . i

-

By using this result we can write v(j) = rj,k(n - j ) !

(n -j)! n!

1

)

'

n>_m>k>

coj,k-=rj,~,

n-la" n -m+

l,

7=In/k]

.

and further by setting

'

we have B

f(s)

~oj,k(/_

n[

1)!(n

_ j)!FJo-l (s)(1 - Fo(s) )" Jfo(s) .

This equation shows that the distribution of the system lifetime of the consecutive-k-out-of-n: F system is written as a finite mixture of the distributions of order statistics of the c o m p o n e n t lifetimes in the i.i.d, case. Based on the result Aki and H i r a n o (1996) studied estimation p r o b l e m s of the linear consecutive systems. Finally, let us consider the lifetime of consecutive systems on a directed tree T with the set of vertices V. A subset W ( c V ) is called an i-cutset if and only if W contains just i elements and the system fails if the corresponding c o m p o n e n t s of the subset fail. A subsequence of elements of V o f l e n g t h i is called an i-cutsequence ifit is a p e r m u t a t i o n of an i-cutset. An i-cutsequence (vl, v2,..., vi) is said to be minimal if the subsequence (/)1, V2,--,/)i-1) is not an (i - 1)-cutsequence. Let 4 ~ , . . , 4N be the lifetimes of the components. Ler us assume that ~1, - - -, ~N are independent and identically distributed with a cumulative distribution function G(t). Let 4(a) -< • • • _< ~-(N)be the order statistics of 41, • • •, ~N and let G(i) (t) be the cumulative distribution function of ~(i). Then, f r o m T h e o r e m 1.1 of Aki and H i r a n o (1996), there exist constants COl,...,CON (c& _> 0 for i = 1 , . . , N and COl + . . . + tON = 1) such that the distribution function F(t) of the lifetime of the system can be written as F(t) = ~Ni= ~ coiG(i)(t). Here, coi is given explicitly as coi = ri,k(N - i)[/N[, where ri,k is the n u m b e r of minimal i-cutsequences of the system. Let bi,k be the n u m b e r of

S. Aki

296

subsets B of V such that IBI = i and B is not a cutset of the system, where IBI denotes the cardinality of B. Then, the number of minimal i-cutsequences can be written as r,,k = (N - i -k 1)((i - l)!)bi_l, k -- i]bi,k • We give a way of calculating bi,k for i = 0, 1 , . . , N. We take a vertex v arbitrarily. Suppose that the vertex v has a(v) ancesters v l, v 2 , . . , v a(u) with pa(vi) = t7+1 for j = 1 , 2 , . . , a(v) - 1, where pa(v) denotes the patent of v. Assume also that the vertex v has c(v) children vl, v 2 , . . , v«(~). F o r i = 1 , 2 , . . , m i n { ( k - 1), a(v)}, we let Pi - {v 1, v 2 , . . , v i} and let Po - Ó. We define b (i) -

numberof{Mc

V~] [ M l = m

and M U P,. is not a cutset of the system} . Let t]~i) (t) be the generating function of ~~,m,~(i)namely, r/!i) (t) = ~mV~0 ~v,m~ t~(i) ,m. Aki (1999) gave the following theorem. 2. The generating functions tl~i) (t) f o r i = O, 1 , . . , k - 1 satisfy the recurrence relations; THEOREM

(0) Il! i) (t) = ]

(0) (i+1)

+ttlvl 1, ( t + 1)

(0) (i+1)

(t)rl~2

. (/-kl)f..

(t)...rt~+l ~,)

) . . . . ,,v«/~ (0)/ (t) t/!k-1)(t ) = {,~(0)(t)l~02)(t ,,v~ 1

ifc(v)>0 if c ( v ) = 0

and 0 _ O, if c(v) = 0 .

I f we expand tl!~) (t) f o r the root Vl with respect to t, the coefficient o f t i is the number bi,k.

Here, we give an algorithm for calculation of the generating function t/!e) (t). Recursive procedure GF(v, g; t)

Inputs: c~, v E V: an array of link-lists from every vertex to its children of the directed tree. k: the length of a failure run which fails the system. v: a vertex in the directed tree. g: the current length of failure tun observed at the patent of v.

Output: t/!e) (t): the generating function.

Algorithm proeedure GF(v, g; t) Let ch = the length of the list c~. If (ch = 0 and g < k -

1), then return (t + 1).

297

Exact reliability and lifetime of consecutive systems

If(ch=0andg=k-1), I f (g < k then r e t u r n I f (g = k -

thenreturnl.

1),

{1-I~hlGF(Cv)'I,O;I ) ÷ t II;~=, GF(«+], e + 1; t)}. 1),

h GF(c~~/'], O; t)}. then r e t u r n {I-I~=i end By using the algorithm a n d a c o m p u t e r algebra system for the directed tree given in Figure 3, we o b t a i n the generating f u n c t i o n t/~~)(t) for k = 2, 3, 4 as 1 + 41t + 780t 2 + 9156t 3 + 74382t 4 + 444592t 5 + 2029139t 6 + 7244029t 7 + 20557311t s + 46876463t 9 + 129643037t l~ + 158129534t 12 + 86490173t l° ÷ 156881253t 13 + 126299046t ~4 + 82154215t 15 + 42899192t ~6 + 5814402t 18 + 1464217t 19 + 1 7 8 1 9 6 8 7 S + 277430t 2° + 38051t 21 + 3545t 22 + 199t 23 +

5/24

1 + 4 1 t + 820t 2 + 10623t 3 + 99911t 4 + 725149t 5 + 4 2 1 6 7 1 8 t 6 + 2 0 1 4 1 1 8 0 f + 80411736t 8 + 271713086t 9 + 1945744395t 11 + 4170276203t 12 + 784170897t m ÷ 7744693779t 13 ÷ 12485286741t 14 ÷ 17484613655t 15 ÷ 21264350088t 16 ÷ 2048368411St 18 + 16144885346t 19 + 22431907350t 17 ÷ 10943577822t 2° ÷ 6351011533t 21 + 3138832292t 22 ÷ 1312717762t 23 ÷ 461015041t 24 ÷ 134668361t 25 ÷ 32323909t 26 + 6272054t 27 + 961561t 28 + 112542t 29 + 9504t 3° ÷ 520t 31 + 14t 32 , and 1 + 41t -- 820t 2 + 10660t 3 + 101239t 4 + 748286t 5 + 4477020t 6 + 22263759t 7 + 93761874t 8 + 339018455t 9 + 2915131084t 11 + 7028202155t 12 + 1063266508t 1° + 14967784908t 13 + 28253297804t 14 + 47386872551t 15 ÷ 70742309738t 16 + 111611167463t 18 + 118030740102t 19 + 94105308119t 17 + 111255209855t 2° ÷ 93399729296t 21 ÷ 69751416938t 22 ÷ 46263197225t 23 ÷ 27194138262t 24 ÷ 14128729018t 25

298

S. Aki

+ 6 4 6 6 0 0 7 6 4 6 t a6 + 2 5 9 5 2 7 2 6 4 4 t 27 + 9 0 8 4 5 6 8 8 4 t 28 + 2 7 5 3 0 4 4 2 2 t 29 ÷ 7 1 5 2 8 8 3 0 t 3° + 1 5 7 2 5 1 3 3 t 31 + 2 8 7 2 4 1 5 t 32 + 4 2 4 7 5 3 t 33 + 4 8 8 9 9 t 34 + 4 1 1 4 t 3» + 2 2 5 t 36 + 6t 37 ,

respectively. From the coefficients of the above generating functions, we easily obtain the number of minimal i-cutsequences and hence we can see the lifetime distributions of the consecutive systems. Some graphs of the density functions of the lifetime of the consecutive-3-out-of-41: F system on the directed tree given in Figure 3 with exponential components are given in Aki (1999).

References Aki, S. (1999). Distributions of runs and consecutive systems on directed trees. Ann. Inst. Star. Math. 51, 1-15. Aki, S., N. Balakrishnan and S. G. Mohanty (1996). Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials. Ann. Inst. Star. Math. 48, 773-787. Aki, S. and K. Hirano (1988). Some characteristics of the binomial distribution of order k and related distributions. In Statistical Theorey and Data Analysis II, Proceedings o f the Second Pacißc Area Statistical Conference, pp. 211-222 (Ed. K. Matusita). North-Holland, Amsterdam. Aki, S. and K. Hirano (1995). Joint distributions of numbers of success-runs and failures until the first consecutive k successes. Arm. Inst. Stat. Math. 47, 225-235. Aki, S. and K. Hirano (1996). Lifetime distribution and estimation problems of consecutive-k-out-ofn: F systems. Ann. Inst. Stat. Math. 48, 185-199. Aki, S. and K. Hirano (1997). Lifetime distribution of consecutive-k-out-of-n: F systems. Nonlinear Anal. Theory, Meth. Appl. 30, 555 562. Balakrislanan, N., S. G. Mohanty and S. Aki (1997). Start-up demonstration tests under Markov dependence model with corrective actions. Ann. Inst. Stat. Math. 49, 155-169. Balakrishnan, N. and C. R. Rao (1998). Order statistics: An introduction. In Handbook of Statistics 16, Order Statistics: Theory and Methods, pp. 1-24 (Eds. N. Balakrishnan and C. R. Rao). Boehme, T. K., A. Kossow and W. Preuss (1992). A generalization of consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. 41, 451~457. Chao, M. T., J. C. Fu and M. V. Koutras (1995). Survey of reliability studies of consecutive-k-out-ofn: F and related systems. IEEE Trans. Reliab. 40, 120-127. Chen, R. W. and F. K. Hwang (1985). Failure distributions of consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. 34, 338 341. Chiang, D. and S. Niu (1981). Reliability of a consecutive-k-out-of-n: F system. IEEE Trans. Reliab. 30, 87-89. Derman, D., G. Lieberman and S. Ross (1982). On the consecutive-k-out-of-n: F system. IEEE Trans. Reliab. 31, 57-63. Ebneshahrashoob, M. and M. Sobel (1990). Sooner and later problems for Bernoulli trials: frequency and tun quotas. Star. Prob. Lett. 9, 5 11. Fu, J. C. (1986). Reliability of consecutive-k-out-of-n: F systems with (k - 1)-step Markov dependence. IEEE Trans. Reliab. 35, 602 606. Fu, J. C. (1996). Distribution theory of runs and patterns associated witla a sequence of multi-state trials. Statistica Sinica 6, 957474. Fu, J. C. and M. V. Koutras (1994). Distribution theory of runs: A Markov chain approach. J. Am. Stat. Assoc. 89, 1050 1058.

Exact reliability and lifetime o f consecutive systems

299

Ge, G. and L. Wang (1990). Exact reliability formula for consecutive-k-out-of-n: F systems with homogeneous Markov dependence. IEEE Trans. Reliab. 39, 600-602. Griffith, W. S. (1986). On consecutive-k-out-of-n: F systems and their generalizations. In Reliablity and Quality Control, pp. 157-165 (Ed. A. P. Basu). North-HolIand, Amsterdam. Han, Q. and S. Aki (1998). Formulae and recursions for the joint distributions of success runs of several lengths in a two-state Markov chain. Stat. Prob. Lett. 40, 203-214. Hirano, K. (1994). Consecutive-k-out-of-n: F Systems. Proc. Inst. Stat. Math. 42, 45 öl (in Japanese). Hirano, K. and S. Aki (1993). On number of occurrences of success runs of specified length in a two-state Markov chain. Statistica Siniea 3, 313-320. Hirano, K., S. Aki and M. Uchida (1997). Distributions of Numbers of success-runs until the first consecutive k successes in higher order Markov dependent trials. In Advances in Combinatorial Methods and Applications to Probability and Statisties, pp. 401-410 (Ed. N. Balakrishnan). Birkkhäuser, Boston. Hwang, F. K. (1986). Simplified reliabilities for consecutive-k-out-of-n: F systems. S I A M J~ Alg. Disc. Math. 7, 258-264. Hwang, F. K. (199 i). An explicit solution for the number of minimal p-cutsequences in a consecutivek-out-of-n: F system. IEEE Trans. Reliab. 40, 553-554. Iyer, S. (1990). Distribution to time to failure of consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. 39, 97-101. Iyer, S. (1992). Distribution of the lifetime of consecutive-k-within-m-out-of-n: F systems. IEEE Trans. Reliab. 41, 448-450. Kamps, U. (1995). A concept of generalized order statistics. J. Stat. Plann. Inferenee 48, 1-23. Kontoleon, J. (1980). Reliability determination of a r-successive-out-of-n: F system. IEEE Trans. Reliab. 29, 437. Koutras, M. V. (1996). On a Markov chain approach for the study of reliability structures. J. Appl. Prob. 33, 357-367. Koutras, M. V. (1997). Waiting time distributions associated with runs of fixed length in two-state Markov chains. Ann. Inst. Stat. Math. 49, 123-139. Koutras, M. V. and V. A. Alexandrou (1995). Runs, scans and urn mode1 distributions: A unified Markov chain approach. Ann. Inst. Stat. Math. 47, 743 766. Koutras, M. V. and V. A. Alexandrou (1997a). Non-parametric randomness tests based on success runs of fixed length. Stat. Prob. Lett. 32, 393-404. Koutras, M. V. and V. A. Alexandrou (1997b). Sooner waiting time problems in a sequence of trinary trials. J. Appl. Prob. 34, 593-609. Ksir, B. (1992). Comment on: 2-dimensional consecutive-k-out-of-n: F models. IEEE Trans. Reliab. 41, 575. Lambiris, M. and S. Papastavridis (1985). Exact reliability formulas for linear and circular consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. 34, 124-126. Lauritzen, S. L. (1996). Graphical Models. Clarendon Press, Oxford. Papastavridis, S. (1990). m-consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. 39, 386-388. Papastavridis, S. and I. Hadzichritos (1988). Formulas for the reliability of a consecutive-k-out-of-n: F system. J. Appl. Prob. 26, 772 779. Papastavridis, S. and M. Lambiris (1987). Reliability of a consecutive-k-out-of-n: F system for Markov-dependent components. IEEE Trans. Reliab. 36, 78-79. Peköz, E. A. and S. M. Ross (1995). A simple derivation of exact reliability formulas for linear and circular consecutive-k-out-of-n: F systems. J. Appl. Prob. 32, 554-557. Salvia, A. A. and W. C. Lasher (1990). Two-dimensional consecutive-k-out-of-n: F models. IEEE Trans. Reliab. 39, 382 385. Sfakianakis, M., S. Kounias and A. Hillaris (1992). Reliability of a consecutive-k-out-of-r-from-n: F system. IEEE Trans. Reliab. 41, 442-447. Shanthikumar, G. (1982). Recursive algorithm to evaluate the reliability of a consecutive-k-out-of-n: F system. IEEE Trans. Reliab. 31, 442-443.

300

S. Aki

Uchida, M. and S. Aki (1995). Sooner and later waiting time problems in two-state Markov chain. Ann. Inst. Star. Math. 47, 415~433. Vaughan, R. J. and W. N. Venables (1972). Permanent expressions for order statistics densities. J. R. Star. Soc. Ser. B 34, 308-310. Yamamoto, H. and M. Miyakawa (1995). Reliability of a linear connected-(r, s)-out-of-(m, n): F lattice system. IEEE Trans. Reliab. 44, 333 336.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statisti«s, © 2001 Elsevier Science B.V. All rights reserve&

Sequential k-out-of-n

Vol. 20

Systems

E. Cramer and U. Kamps

Notation Xr,n

x(:), v,(') FX,F, Fi f x , f ,f~ Y Exp(#, vQ)

r(r,o) ~(#,~) EX VarX

Cov (x, Y) O~j, ~ij

7U

r(~) B(r,s) F MLE UMVUE BLUE iid w.r.t.

IFR (DFR) IFRA (DFRA) NBU (NWU)

random variables rth (ordinary) order statistic in the sample X 1 , . . ,Xn rth sequential order statistic (cumulative) distribution function (of X) density function (of X) location-scale family of distributions two-parameter exponential distribution with location parameter # and scale parameter v~ (= expected value) gamma distribution with parameters r and normal distribution with expected value # and covariance matrix Z expectation of random variable X variance of random variable X covariance of random variables X and Y model parameters of some sequential k-out-of- n system

(ni - j + 1) ~q gamma function (at s) beta function (at r and s) hypergeometric function maximum likelihood estimator uniformly minimum variance unbiased estimator best linear unbiased estimator independent and identically distributed with respect to increasing (decreasing) failure rate increasing (decreasing) failure rate average new better (worse) than used

st.

<

stochastic ordering

f.r.

<

failure rate ordering

301

10

I z..

302 Lr.

_<

E. Cramer and U. Kamps

likelihood ratio ordering

t

_<

tail ordering

1. Introduction 1.1. k-out-of-n systems

In the probabilistic modelling and analysis of redundant systems, k-out-of-n systems play an important role, since k-out-of-n structures often appear in technical systems or subsystems. A k-out-of-n system consists of n components which start working simultaneously. The system operates as long as k components function, and it falls if n - k + 1 or more components fall. Hence, there are some redundant components in order to raise the reliability of a system. Since all components start working at the same time, this kind of redundancy is called active redundancy. Parallel and series systems are particular cases of k-out-of-n systems corresponding to k = 1 and k = n, respectively. Other examples of systems with k-outof-n structure are presented in, e.g., Meeker and Escobar (1998). For instance, an aircraft with four engines will not crash if at least two of them are functioning, or a satellite will have enough power to send signals if not more than four out of its 10 batteries are discharged. There is a broad literature on the analysis of several variants of k-out-of-n systems. For monographs dealing with this topic we refer, e.g., to Barlow and Proschan (1981), Hoyland and Rausand (1994) and to Meeker and Escobar (1998). Technical systems are often represented by so-called reliability block diagrams. A k-out-of-n system can be illustrated as a parallel system of series systems, in which, however, all components appear several times. Hence, although dealing with independent components, the series systems viewed as subsystems are no longer independent. In the case of the above mentioned 2-out-of-4 system of aircraft engines, the respective diagram is shown in Figure 1, where the engines No. 1, 2, 3 and 4 appear three times each.

Fig. 1. Reliabilityblock diagram of a 2-out-of-4system.

Sequential k-out-of-n systems

303

If the lifelengths of the components ard described by the random variables X 1 , . . ,Xù, then the lifelength of an (n - r + 1)-out-of-n system is represented by the corresponding rth order statistic Xr,ù, 1 < r < n. In order to make use of the variety of results in the area of order statistics from independent and identically distributed (iid) random variables with respect to (w.r.t.) probabilistic analysis and related statistical inference, the components ard often supposed to be of the same kind with iid lifelengths. For excellent expositions on order statistics we refer to David (1981), Arnold et dl. (1992) and to the recent Handbooks of Statistics, Vol. 16 and 17, edited by Balakrishnan and Rao. Results on statistical inference in k-out-of-n systems can also be found in the analysis of type II censoring which is described by order statistics (cf. Lawless, 1982). However, in the conventional modeling of these structures by order statistics from an iid sample it is supposed that the failure of any component does not affect the remaining ones. The assumption that the breakdown of some component does not influence the components at work will generally not be fulfilled in practice. In some systems, a component failure will more or less strongly influence the remaining components. For example, the breakdown of an aircraft's engine will increase the load put on the remaining engines such that their lifetimes tend to be shorter. Thus, a mord flexible model, which is therefore mord applicable to practical situations, taust take some dependence among the system components into account. Sequential order statistics have been introduced in Kamps (1995a) as an extension of (ordinary) order statistics in order to model sequential k-out-of-n systems, where the failures of components possibly affect remaining ones. This can be thought of as a damage caused by failures or, as mentioned before, as an increased stress pur on the active components. The model of sequential order statistics is flexible in the sense that, after the failure of some component, the distribution of the residual lifetime of the components may change (cf. Kamps, 1995a, Chapter 1.1; Cramer and Kamps, 1996). We suppose that, after dach failure, the remaining components possess a possibly different failure rate than before; i.d., the underlying failure rate of the remaining components is adjusted according to the number of preceding failures. For illustration, we consider the above 2-out-of-4 system modeling the reliability of an aircraft w.r.t its engines. In Figure 2 the situation is considered as a sequential 2-out-of-4 system with absolutely continuous lifelength distributions F1,F2,F3, respective density functions f l , f 2 , f 3 and failure times r(1) r(2) y(3) which will be called sequential order statistics. In Figure 3 the distribution function Ford of an ordinary 2-out-of-4 system with underlying standard exponential distribution and that of a sequential 2-out-of-4 system based on exponential distributions F1, F2, F3 with expectations 1, 1/3 and 1/6 (Fseq) ard compared. The distribution functions ard given by (cf. (1.1) and Theorem 2.5): For d (t) = 1 - e 2t (6 - 8e-*+ 3e-2t), Fseq(t) = 1 - 1 Ö1e _4t(27 - 32e -5' + 15e-8*),

t >_ 0

304

E. Cramer and U. Kamps

failure no. i

failure time

failure rate remaining components after the ith failure

NSNN NN []

xP X! 2) X ! 3) Failure time of the system:

N [] []

X!3)

Fig. 2. Sequential 2-out-of-4 system.

1.00 . ,.,..,,... .......

0.75

..-'

0.50

- -

/~seq

........

/?ord

0.25

0

1

Fig. 3. Distribution functions of the lifelengths of a sequential 2-out-of-4 system (Fseq) and of an ordinary 2-out-of-4 system (Fora). Figure 3 illustrates that the lifetime of the ordinary k-out-of-n system is stochastically larger than that of the sequential one. In the following section, sequential order statistics are introduced intuitively in view of the above example, whereas in Section 2.1 we define them formally and present some results on this structure. 1.2. Sequential k-out-of-n systems

As outlined in Section 1.1, sequential order statistics serve to describe sequential k-out-of-n systems in order to model the influence of some component failures on the lifelength distributions of the remaining components. Thus, the lifetime of a sequential ( n - r + 1)-out-of-n system is modeled by the rth sequential order statistic X,(r), 1 < r < n. We introduce sequential order statistics first intuitively by means of a triangular scheine of random variables. In line r we consider n - r + 1 random vari-

Sequential k-out-of-n systems

305

ables indicating that r - 1 components previously failed. A formal definition of sequential order statistics is given in Section 2.1. Let F1 . . . . ,Fù be continuous distribution functions and z! l) < z ! 2) . < (n-l) ,.n- J,n-Iù • ~- Z1, 2 be real numbers. Consider a triangular scheme (see Figure 4)

(z(;)) ] ]l<_i<_n,l<~
B(.)

r~/" (i 1) - ~, tz,,ù_i+~)

1

<

_(0)

i < n, Zl,n+ l

=

--00

1 - Fi ~,z1,n_i+2) (i--1)

which is F~ truncated on the left at the occurrence time Z
ables with survival function (1-F/(.))/(1-E-(zl'i~_ll.+2)). These random variables represent the lifetimes of the remaining n - i + 1 components after the ( i - 1)th failure. A precise description in the general set-up as well as the joint density function of the first r sequential order statistics based on F a , F 2 , . . are shown in Kamps (1995a). Ordinary order statistics describing (ordinary) k-out-of-n systems are contained in the model of sequential order statistics as a particular case by putting F1 . . . . . Fn. In this sense, ordinary k-out-of-n systems can be viewed as special sequential k-out-of-n systems.

x! ~)

<

Z} n

Z~ n

z(nl) 1

X.(2)

(

Z} 2)

Z~ 2)

Z(~I

Z (I)

,-a /~I(')

,~(.)_ ,~(&)) 1 - F2(z~I ) )

:

X,(~-1) X ! n)

<

Z~ ~)

F,,(.) - F,~(z~,%-')) i - - Fn (Z~y 1))

l" line m i n i m a

X ! 1) __< , - . __< X ! n) Fig. 4. Triangular scheme for sequential order statistics.

306

E. Cramer and U. Kamps

1.3. D a t a situation and outline

In what follows we restrict ourselves to a particular choice of the involved distribution functions F 1 , . . , F n , namely Fr=l-(1-F)

~r,

l
,

(1.1)

where F is an absolutely continuous distribution function and ~ l , . . . , ~ n are positive real numbers. The motivation for this choice and further comments are postponed to Section 2.1. The results on statistical inference with sequential k-out-of-n systems will be based on s independent observations of possibly differently structured systems, i.e., we have independent observations of (nj - ri + 1)-out-of-ni systems,

1 < ri < ni, 1 < i < s

with ri (dependent) observations each, with model parameters (~ij)l<_i<_«,l<_j<_ri,and with corresponding sequential order statistics (X~iC/))l
=---=~r

versus the alternative A : B i • j,

1 <_ i , j < r, suchthatc~i¢c~j ,

in order to decide whether the model of sequential order statistics is the appropriate orte in a given situation with k-out-of-n systems. For illustration we provide a simulation study. Most of the results on statistical inference with sequential order statistics are established within a location-scale family ~ of distributions,

Sequential k-out-of-n systems

307

Table 1 Statistical inference with sequential order statistics Section

Model parameters ( c ~ q ) i «

Distribution function F

4

~gj = c9, 1 < i < s MLEs, UMVUEs of cq, c~2,.., and their properties

Known

5

cqy = ~j, 1 < i < s test procedures for H : gl .... ~ ~ r versus A : 3 i ß j with o:i ¢ o:j

Known

7

c~ij=cg, l < i < s MLEs of aj/O MLEs of # and aj/O

F E Y , # known

8

Known

F E Y , # known MLE of ~ and its properties F -- Exp(p, ~), # known MLE, UMVUE, BLUE of

9

Known

FE~ MLEs of # and 0 and their properties F - Exp(#, ~) MLEs, UMVUEs, BLUEs of # and

10

Known

F(1),F (2) E Y , common location parameter #, MLEs of ~91,02 (# known) MLEs of #, 01 and 02 and their

11

Known

F 0 ) , F (2) E MLE, U M V U E of P ( X < Y) for

FEY

properties

known/unknown and their properties

which is introduced and analyzed in Section 6. Some distribution function F c has the form

F(t)=l-exp{ for a

g(t)-o--/~},t>_g-l(#), #~ IR, #>0

suitable function g. For g ( t ) = t we obtain two-parameter exponential distributions as an important particular case, which are denoted by Exp(#,0), throughout. Further examples of included distributions are Weibull, generalized Pareto, Pareto, Lomax, uniform, Burr XII and Pearson I distributions. In Sections 7 11 we are concerned with different situations and aims in the area of estimation of model and distribution parameters based on observations of sequential order statistics. In order to get a brief overview about the main topics and the structure of the present survey the reader is referred to Table 1 and to Figure 5. In Section 12 we are finally concerned with aging properties and partial ordering of sequential order statistics.

E. Cramer and U. Kamps

308

m.

o o ~ o

B "R m ~4

;~i

,-4

o~ ~¢~~ ~~ ~o

Nrä ~

,e ~õ'Z..

~o

~

:~

o

/ i

/

õ

0

~ ~ o.o

0

o

~~ °~ ©

O .~ -~

~l~õ .o .~

.~o~~

ù~

81 ~

309

Sequential k-out-of-n systems

2. Sequential order statistics In Section 1.2 sequential order statistics were introduced intuitively. Here, we define them properly and present some distribution theoretical results. Moreover, their relation to other models of ordered r a n d o m variables is demonstrated.

2.1. Definition and properties of sequential order statistics We consider a sequential (n - r + 1)-out-of-n system where the lifelength distribution of the remaining c o m p o n e n t s in the system m a y change after each failure of the components. I f we observe t h e / t h failure at time x, the remaining components are n o w supposed to have a possibly different lifelength distribution as indicated in Figure 4. This one is truncated on the left at x to ensure realizations arranged in ascending order of magnitude. In the definition of sequential order statistics we start with some triangular scheme of r a n d o m variables where t h e / t h line contains n - i + 1 r a n d o m variables with distribution function F~, 1 < i < n (cf. K a m p s 1995a, p. 27). (i) DEFINITION 2 1 Let (Y )l
.

.

J

,

J

-

distribution functions with F~ 1(1) <_ ... <_ F£-1 (1).

Ler X) 1) = YS1), 1 _< j _< n, X,(1) = m i n { X ( 1 ) , . . ,X,(1)}, and for 2 < i < n let ~ Then x)i)=Fi-l(Fi(Yj(i))(1-Fi(X~i-1))+Fi(x(*i-1))), X(*i) = min{X~/), • • • ,"ù-i+lJ-Y(') the random variables 2(,(1),.. ,X,(~) are called sequential order statistics (based on

F1,.. ,F,). (i-l) Given the realization zl.,_i+ 2 of the m i n i m u m in line i - 1, i.e., the occurrence (i 1) o f the (i - 1)th failure in 'the system at time zl,,_i+2, we obtain the conditional distribution of the r a n d o m variables X(i) for 1 < i < n. Since y(i) and AT,(,i-1) are independent, we have

(

~(t) - ei(s)'~

_ F~(t) - Fi(s) _ Gi(tls),

say .

1 - ~(s)

Thus we turn back to the intuitive a p p r o a c h in Section 1.2. In the triangular scheine (w7 (j i ) ~I I < i < n , l < j < n - i + l o f r a n d o m variables in Figure 4, the ~( z (ji ) ) Jl<_j<_,-i+l are iid according to

G i ( , I z l i , -n l ~ + 2 ) , 1 < i < "

n, Z (0) 1,n--1 = --OO, i.e., the next failure time is

modeled as the m i n i m u m in the sample Z} ~)

" " " ~ Z n(i) -i+l

of iid r a n d o m variables

(i-l)

with distribution function Gi(.IZl,n_i+2). F r o m Definition 2.1 we derive the joint density function of sequential order statistics 2(,(1),.. ,Xff ), r < n, based on F 1 , . . ,Fù (cf. K a m p s 1995a, p. 29). THEOREM 2.2. Let F ~ , . . , F ~

be absolutely continuous distribution functions with respective density functions f l , . . . ,f~. The joint density function of the

310

E. Cramer and U. K a m p s

first r sequential order statistics given by

X,(1),..

, X (r) based on these distributions

is

f x ! l ) ' " ' ~ ( f l (Xl , . . . ,Xr) n!

(~

/1-

-(n-r)! \i=1 tl--~ (n--r)[

Fi(xi) .~ n i

X ~

~) f(s)) (1-~(x~))"-~f~(xr)

\l-F/(xi_l)

i-F/(xi_l)'

r
,

where - o c = xo < Xl <_ • • • <_ xt.

Obviously, ordinary order statistics are contained in this model in the distribution theoretical sense. Choosing F1 . . . . . Fù = F, say, we obtain the joint density function of the order statistics Xl,n,. • •, Xr,n based on an iid sample from F with density function f , 1 < r < n:

fx, flQ"(Xl,...,Xr) - (n r)~! ......

f(x,) (1 -F(xr))" r, Xl ~ . . . ~ X r •

The marginal density function of the rth order statistic Xr,~ is given by

~",o ~~~--~(7)~ ~ '~~~~,-

~~.,~

~~~/~ ~ ~~/~~.

It is directly seen that sequential order statistics form a M a r k o v chain with transition probabilities

~~~~~~ » ,~~~_l~ ~-~~ _- (i- ~~~,~~~_~+l F~(s)J

,

2
In what follows we restrict ourselves to the particular choice (1.1) of the involved distribution functions F 1 , . . , F n , namely F/= 1-(l-F)

~i,

1
with an absolutely continuous distribution function F and positive real numbers el,. •., c~n. This choice of the distribution functions leads to the hazard function c~i+lf/(1 - F) of each component at work on level i + 1, i.e., after t h e / t h failure, such that the parameters «1, e 2 , . . model the influence of a failure on the remaining components. Aside from the interpretation in terms of hazard rates, this limitation is reasonable in order to reduce the uncertainty in the model to the parameters Œl,c~2,.. and the distribution function F. Moreover, the question arises as to whether we can obtain the distribution theory of sequential order statistics and their properties by analogy with ordinary order statistics, which have been extensively investigated in the literature (cf. David, 1981; Arnold et al., 1992; Balakrishnan and Rao, 1998a, b). F r o m this theoretical point of view, the model of sequential order statistics in the general setting of Definition 2.1 turns out to be

Sequential k-out-of-n systems

311

t o o extensive in o r d e r to establish a n a l o g o u s p r o p e r t i e s as f o u n d in the case o f o r d i n a r y o r d e r statistics. A l t h o u g h this setting seems to be very restrictive, m a n y m o d e l s o f o r d e r e d r a n d o m variables are included in the d i s t r i b u t i o n theoretical sense which is subject m a t t e r o f Section 2.2. T a b l e 2 gives some e x a m p l e s o f w e l l k n o w n m o d e l s a l o n g with the respective choices o f the p a r a m e t e r s c j, 1 _< j < n. F o r m o r e details a n d further m o d e l s we refer to K a m p s (1995a, 1999). The j o i n t density f u n c t i o n o f sequential o r d e r statistics then reads as follows. COROLLARY 2.3. L e t X2(1) , • • ,X(,n) be sequential order statistics based on F1, . . . , F~ with (1.1), which subsequently are called sequential order statistics based on F (and ~1, • • •, ~~ > 0). L e t F be absolutely continuous with density f u n c t i o n f . Then the (1) (~) j o i n t density f u n c t i o n o f the f i r s t r sequential order statistics X , , . . ,X2 is given by

f

(x~,..,x~)

(~ ~)!

~: j=l

×

(1 -- F ( x j ) ) m J f ( x j )

F-l(0+)<Xl with mj = (n - j +

(1 - F(xr))et~(n-r+l)-lf(xr),

<'"<xr
r<_n ,

1)~j - (/'t - j ) ~ j + l - 1, 1 < j <

n - 1.

C h o o s i n g r = n a n d cq . . . . c~~ in C o r o l l a r y (2.3), we o b t a i n the j o i n t density function o f the o r d e r statistics Xl,~ _< • • • _< Xù,ù based on iid r a n d o m variables X 1 , . . ,X~ with d i s t r i b u t i o n f u n c t i o n 1 - (1 - F ) ~1. A n extremely useful result for sequential o r d e r statistics states t h a t weighted successive differences o f sequential o r d e r statistics f r o m an e x p o n e n t i a l distribution (with l o c a t i o n p a r a m e t e r zero) are a g a i n iid a c c o r d i n g to the respective e x p o n e n t i a l d i s t r i b u t i o n (cf. K a m p s 1995a, T h e o r e m 1.3.3.5, C r a m e r a n d K a m p s Table 2 Models of ordered random variables and their correspondence

Sequential order statistics Generalized order statistics Ordinary order statistics Progressive type II censored order statistics Record values

O:r (1 < r < n - - 1 ) o:n=k

m~ (i < r < n - 1 )

~?r (1 < r < n - - 1 )

c~r

Œn

( n - - r + 1)~~ -(n - r)~r+l -- 1

(n-- r + 1)~,.

1

0

n--r+l

Rn+l

Rr (C I N 0 )

N-r+l-E~~_~Ri - n - r + 1 +~7=~Ri

1

-1

1

fln

flr - fir+l - 1

flr

k+n

n 1

m)

r+z.~g=~

n r+l

1 N r+i ~,"i R~ ù-~+1 1 n-r+l

Pfeifer's records

/~' n-r+l

E. Cramer and U. Kamps

312

1996, Theorem 3.1). This feature will be a key tool in the following sections to derive properties of point estimators. In the particular case of ordinary order statistics, this result on normalized spacings is well known and due to Sukhatme (1937) (cf. Rényi, 1953). THEOREM 2.4. Let X ( , U , . . . , X (n) be sequential order statistics based on F with F(x) = 1 - e -x, x >_ O. Then the random variables n«lX(, l)

and

(n-j+

l)ej(X,~)-X,~-a)),

2 <_j<_n ,

are iid according to F. It will be seen in Section 2.2 that sequential order statistics can be viewed as generalized order statistics which are introduced and treated in Kamps (1995a). W.r.t. marginal distributions, e.g., one-dimensional marginal distributions, it turned out that simple and useful expressions result if the choice of parameters is restricted. The distribution theory in the general case along the lines of that of ordinary order statistics requires some efforts and advanced tools and will be subject matter of a forthcoming paper by the authors. If we assume, however, that the parameters ( n - r + 1)er, 1 < r < n, are pairwise different, i.e., (n-i+l)~i¢(n-j+l)c~j

foralll<_i,j
ißj,

(2.2)

we obtain tractable expressions by applying Theorem 2.4. In the following we establish expressions for marginal densities and distribution functions. Moreover, a recurrence relation for the moments of sequential order statistics is presented which is well known in the particular case of ordinary order statistics. In view of the following section on models of ordered random variables and the concept of generalized order statistics, the results follow from those in Kamps and Cramer (1999). By using Theorem 2.4 and a result of Likeß (1967) on the distribution of weighted sums of independent, exponentially distributed random variables together with a transformation argument, we arrive at TrIEOnEM 2.5. Let XO), .. ,X(, ~) be sequential order statistics based on an absolutely continuous distribution function F with density function f , and let assumption (2.2) be fulfilled. Then the one-dimensional marginal density and distribution functions are given by

and F x~rl (x) = 1 -

y~ /=1

l
Sequential

where 7i

=

(n

ai =

--

k-out-of-n

systems

313

i + 1)cq, 1 < i < n, and

B 1

- , j=l ~J ~i

l
.

For ordinary order statistics X 1 , , . . , X , ~ based on F, we find the simple • -1 expressions ai = ( - 1 ) r - i ( ( •t - 1 ) ! ( r - ' ~)!) , 1' < z. < r, and f xr.ù as given in (2.1), l
1 < r < n-

1 ,

which can be expressed in terms of density and distribution functions as well. Utilizing the representation of the marginal density functions of sequential order statistics we extend this relation to sequential order statistics. It is stated in terms of density functions. THEOREM 2.6. Ler X(,1), .. ,X(, ") be sequential order statistics based on an absolutely continuous distribution f u n c t i o n F with model p a r a m e t e r s c q , . . . , Œ ~ , let y.(1),.., y,(~-l) be sequential order statistics based on F with model p a r a m e t e r s 0;2,.. , O~n and ler assumption (2.2) be fulfilled. Then (n - r)ctr+lf xS') (x) + (nul - (n - r)Œr+~)f X:r+l)(x) = n ~ l f y:~)(x),

l
.

It has to be noted that the parameterizations of the sequential order statistics on both sides of the above equation differ. Obviously, the identity can be stated in terms of distribution functions and moments, too. Moreover, it may equivalently be written as =

(V/~ 1 -- (F/ -- F)CXrq_I)

×

(x)-

(x

,

1
.

Expressions for two-dimensional marginal density functions and densities of subranges are developed in Kamps and Cramer (1999). Conditional densities turn out to be densities of sequential order statistics based on a transformed distribution, namely, based on the underlying distribution function F truncated on the left. In the case of ordinary order statistics, this representation is well known. These results can be applied to extend a basic characterization of exponential distributions via identical distributions of a subrange and of a corresponding sequential order statistic. In the context of ordinary order statistics Xl,n,... ,X,,n based on F we have (cf. Ahsanullah, 1984; Iwifiska, 1986; Gajek and Gather, 1989; Gather et al.,

E. Cramer and U. Kamps

314

1998, p. 266/7; Kamps, 1998; see also Gather, 1988) that, under assumption, the identity of the distribution of the subrange Xs,n the order statistic Xs r,~-~ as well as the identity EX~,~ -EX~,n characteristic properties of exponential distributions. The result can be extended to sequential order statistics with isfying (2.2), which then reads as follows.

N B U or N W U Xt,n, r < s, and = EX~_~,~_r are parameters sat-

THEOgEM 2.7. Let X(,1), .. ,X(, ~) be sequential order statistics based on an absolutely continuous distribution function F with model parameters c q , . . . , ~, and let (2.2) be fuIfilled. Moreover, let F be strictly inereasing on (0, oc) with F(O) = O, and let F be N B U or N W U . Then F =_ Exp(0, O) for some 0 > 0 (i.e., F(t) = 1 - e x p ( - t / O ) , t _> 0) /ff there exist integers r, s and n, 1 <_ r < s <_ n, such that 1. X(, " ) - X ( f ) ~ Y(«-r) or 2. Ex(, ~) - Ex(, ~) = E Y 2 -~)

assuming that the expected values are finite, where y,(U ...~ y(~-~) are sequential order statistics based on F with model parameters

Cgr+ 1 ~ . . . » a n.

For one-parameter exponential distributions, the distributional identity in condition 1 of Theorem 2.7 is valid for all 1 < r < s < n. More details in the area of ordinary order statistics can be found in Gather et al. (1998). 2.2. Models of ordered random variables The model of sequential order statistics is closely connected to several other models of ordered random variables. In its general form, the model coincides with Pfeifer's record model in the distribution theoretical sense (cf. Kamps, 1995a, p. 29). With the restriction on F/(t) = 1 - (1 - F ( t ) ) ~i for some distribution function F and positive real numbers « 1 , . . , c~n (cf. (1.1)), we are in the model of generalized order statistics as a unified approach to a variety of models of ordered random variables with different interpretations (see Kamps, 1995a, b 1999), such as ordinary order statistics, sequential order statistics with (1.1), order statistics with nonintegral sample size, progressive type II censoring, record values, kth record values, Pfeifer's records with (1.1) and k, records from nonidentical distributions. The concept of generalized order statistics enables a common approach to structural similarities and analogies. Known results in submodels can be subsumed, generalized, and integrated within a general framework. The very particular models of ordinary order statistics and record values are widely used in statistical modeling and inference and especially in reliability theory. Order statistics appear, e.g., in the description of the lifelengths of k-out-of-n systems. However, more models can be effectively applied in reliability theory as well. Record values are closely connected with the occurrence

Sequential k-out-of-n systems

315

times of some corresponding n o n h o m o g e n e o u s Poisson process and are used in so-called shock models. Other record models are m o r e flexible, and therefore m o r e applicable to practical situations. Well-known distributional and inferential properties of ordinary order statistics and record values turn out to be also valid for generalized order statistics (cf. K a m p s , 1995a; K a m p s and Cramer, 1999). Thus, the concept of generalized order statistics provides a large class of models with m a n y interesting and useful properties for b o t h the description and the analysis of practical problems. Progressive type II censored order statistics are applied in life testing (cf. Cohen, 1963; Viveros and Balakrishnan, 1994). The results presented in this survey article apply to other models of ordered r a n d o m variables by choosing the p a r a m e t e r s appropriately. Thus we briefly introduce generalized order statistics and three other models to illustrate their correspondence. Based on an absolutely continuous distribution function F with density function f , the r a n d o m variables X(1, n, r~, k ) , . . ,X(n, n, Fn, k) are called generalized order statistics if they possess a joint density function of the f o r m fX(~,ù,m,~),.«r(,,ù,r~,k)(Xl, = Æ

7j k,j=l

• . . »Xn)

(1 - F(xi))mif(xi)l (1 - F(xn))k-lf(x~)

/

on the cone F - l ( 0 + ) < X 1 ~_ - - - _< X n < F - I ( 1 ) of the n-dimensional Euclidean space IRn, where n >_ 2, k > 0, and rh = ( t a l , . . . ,rnn 1) E ]Rn-1 are p a r a m e t e r s suchthatTr=k+n-r+~~=~mj>0forall 1
316

E. C r a m e r and U. K a m p s

m~ = R~, 1 < r < n - 1, and k = R~ + 1 leading to 7~ = N - r + 1 - ~ i 2 ~ Ri, l 0, Pfeifer's record values can be viewed as generalized order statistics by choosing m~ =/~~ - flr+l - 1, 1 < r < n - 1, and k =/~~ leading to 7~ = fi~, 1 < r < n - 1. Table 2 shows the correspondence between the above-introduced models. Sequential order statistics, which we are concerned with in this survey, are taken as the basic model (cf. Kamps, 1999). Occasionally, assumption (2.2) is imposed on the parameters of sequential order statistics, which is equivalent to assuming pairwise different 7's, i.e., =

7i¢7j

forall 1 <_i,j
ißj

(cf. Section 2.1). For instance, this assumption is no restriction on progressive type II censored order statistics, since 7 r - 7r+1 = m r + 1 (_> 1) in this case; in Pfeifer's record model, however, the parameters/31, • -,/~n have to be chosen pairwise differently. The choice ml . . . . . ran-1 -- - 1 , which has to be excluded here, corresponds to c o m m o n record values. In its generality, the subclass of generalized order statistics defined by pairwise different parameters 71,- • •, 7n is a sort of compromise (w.r.t. handling, representations, efforts in the analysis) between the restriction on ml . . . . . rar-1 (see Kamps, 1995a) w.r.t, the rth generalized order statistic and an arbitrary choice of the parameters. The applicability and the usefulness of our assumption (2.2) are demonstrated in K a m p s and Cramer (1999).

Sequential k-out-of-n systems

317

3. Sampling situation In the following sections we consider the situation of s sequential k-out-of-n systems. Each system is allowed to have a different structure, i.e., we have independent observations of a number of s (ni - ri + 1)-out-of-ni systems,

1 < ri < nj, 1 rs>l

.

For v = 1 , . . . , rl let c~ be the number of ri's with ri ~ v: c~=]{i:ri>_v,

1 _v,

1
.

Obviously, we have cl = s.

4. Maximum likelihood estimators of model parameters for arbitrary distributions 4.1. D i f f e r e n t l y s t r u c t u r e d s y s t e m s .

.

.

.

.

.

(1)

(r)

According to the model descrlptaon in Sectlon 3 and thejolnt denslty f x ; ....~~ of (1) (r) sequential order statistics X~ ~... ,X~ based on a known distribution function F (cf. Corollary 2.3), the likelihood function is given by

L(o~ij,xij; 1

< i < s, 1 <_ j <_ ri)

s

× 1 - [ ( 1 - F (,x

~~%(n'-ri+I)-l((x iri]] J \ iriJ~ ,

(4.1)

i=1 where

m i j = (tl i - j + 1)o:ij -

(ni - j)Œi,j+l -

1,

1 <_ j <_ r i - 1, r i ~ Hi» 1 < i < s.

In this section, we suppose to have identical model parameters al, ~ 2 , . . in all systems, i.e., ~ij = ~lj = aj,

say, (1 <_ j <_ ri),

1 < i < s .

318

E. C r a m e r a n d

U. K a m p s

In T h e o r e m 4.1 (cf. C r a m e r and K a m p s , 1998a, T h e o r e m 7.2.1) explicit expressions of t h e M L E s %, * .. . , %* of ~ 1 , . . , % are stated which turn out to be independent and inverted g a m m a distributed r a n d o m variables. The estimation of the model p a r a m e t e r c~~ is based on c~ observations, 1 < v < tl. THEOREM 4.1. L e t e~j = ~j, 1 < i < s. Given observations (xij)l<j<_, l<j<~, the M L E s o f the model p a r a m e t e r s o~1». . . , O~rl are given by -1 ~1 ~ - - S

0

and

e* = -«~

--

(

«'

Xil

1 - F(xiv)

v+l)

~ni

-1

l°gi_I~l(1-F(xi#-l)J

,

2
.

Expressing the M L E s ~ ~ , . . , e~ in terms of the underlying r a n d o m variables, i.e.,

-1

and ev* = -«~

(~~

(ni - v + l ~' ) ,og

(1 ~<»>~) I--F~)

)

,

2
,

we obtain the following results (cf. C r a m e r and K a m p s , 1998a, T h e o r e m 7.2.2). THEOREM 4.2. The M L E s ~ * l , " . , ~~~ o f ~1, . . , ~,-~ have the f o l l o w i n g properties: 1. ~~, ..,c~~, are j o i n t l y independent. 2. c~~ has an inverted g a m m a distribution with p a r a m e t e r s c~ and (c~c~~) l, i.e.,

,

~v ~°%

~~~)1

\ Cv i=1

V/v

,

1 < v
,

where (Viv)~<~<,~,l
E~~ --

c~

cv

1 ~~' c~ > 1"

--

1; hence

Var~*

~

cv + 2

(cv --

2 G ~2 c~ > 2; 1)2(Cv -- 2) v,

2

M S E ( ~ ~ ) = («~ - - i ) 7 7 , - 2) ~~'

«v > 2,

1 < v < r~ .

Sequential k-out-of-n systems

319

4. The s«atisti« (Œ*l,..., c~*~,) is «ompl«tely suffi«i«nt f o r ( o q , . . . , c~~,). 5. The sequences o f estimators (c~~) are strongly consistent

w.r.t.

Cv --+ O(3(1 < V < F1).

6. c~; is asymptotically normal, 1 < v < rl, i.e., v@7(c~~/~v - 1) ~ Cv ---+(X).

JV'(0, 1) w.r.t.

In part 2 of the the above theorem it is shown that Œ~ is distributed as lITt, where the random variable T~ is gamma distributed with parameters cu and (c~ • c~~)-1, i.e, its density function is given by fr~(t ) -- («vc~~)~~ t«,,_le_«~~#

(7~ --F)!

t _> o

The distribution of c~; is known as inverted gamma distribution, which is used as a prior density in Bayesian analysis (cf. Bain, 1983; Johnson et al., 1994, p. 524).

4.2. Identically structured systems

Let us consider s _> 1 iid observations of some sequential (n - r + 1)-out-of-n system leading to the set of data (xu)~
xa <_ . . . <_ xir, 1 < i < s .

That is, we again suppose knowledge of all the times of failures of components during the lifelength of the system. Let (X*~i))l<_i<_s,l<j<_r

be the corresponding random variables. Then the likelihood function is given by L(O~l,...,C~r;Xij , =

1

1 <j
~~ ~~(~ )(nl r-1 \(n

-

r)!/I

c~j

II(1

-

F(xij))m'f(xij)

j=l s X U(1

--

F(xir))~"(n-r+l)-lf(xir)

= L(oq,...,

C~r,F ) ,

say ,

i=l

(4.2) and we obtain THEOREM 4.3. The M L E s o f o~i,... , :Xr a r e , S ~1 ~ - - - n

(1 0 gi-I~l (ls

F(X(.])))) -1

E. Cramer and U. Kamps

320

and

o~~--

( ~~~«'~~~

s n-j+l

log

1 - - ~ i

B

,

_ _ 2<j
These estimators have the properties shown in Theorem 4.2 with rl replaced by r and cv = s, 1 < v < rl = r. For more details we refer to Cramer and K a m p s (1996, Theorem 3.2). REMARK 4.4. Applying the theorem of Lehmann-Scheffé it is obvious from Theorem 4.2 that (~i,.

- - , ~F 1 ) = S -- 1 ( ~ 1 ' ' " ' ' (~;l) S

is the U M V U E of ( c q , . . . , ~,-1). In the case of ordinary order statistics, similar MLEs result in the following situation (see Sarkar, 1971): The components of a series system are independent and exponentially distributed with possibly different parameters and the MLEs of these parameters are based on independent copies of the respective components under type II censoring.

5. Test procedures for model seleetion Based on the MLEs c~~,.., c~~ given in Theorem 4.3 we propose some test procedures to decide whether the model of sequential order statistics is appropriate for describing some k-out-of-n system. For simplicity we restrict ourselves to identically structured systems. Moreover, as in Section 4 we suppose to have identical model parameters in all observed systems, namely ~ij = c~j, 1 < i < s. Since the utilized estimators are independent, inverted g a m m a distributed r a n d o m variables (cf. Theorem 4.2), the resulting tests turn out to be well known in the context of testing homogeneity of variances from normally distributed populations, e.g., Hartley's and Bartlett's test. A detailed discussion is provided by Rosenbusch (1997). We are interested in testing the hypothesis H : ~1 . . . .

----~r

versus the alternative

A : 3 i T~ j,

i,j E {1, . . , r } ,

suchthat~i~c~j

,

at the level of significance et. The hypothesis H corresponds to an ordinary ( n - r + 1)-out-of-n system, whose lifelength is described by the rth order statistic in a sample of size n, whereas the alternative A represents a (real) sequential ( n - r + 1)-out-of-n system.

321

Sequential k-out-of-n systems

Hence, if the hypothesis is rejected, then the model of sequential order statistics is adequate and ordinary order statistics should not be used to describe the system. The MLEs c @ . . . , ~~ of c q , . . . , c~~ are independent and inverted gamma distributed with parameters s and (s • ej)-1, respectively. In the following tests we use the random variables/?~,..,//~ defined by

p~=s/~~,

1 <j
which are independently gamma distributed with parameters s and c~)-1, respectively. Under the hypothesis H , / ? ~ , . . , / 3 " are identically distributed. If there is some prior information about the possibly common value c~0 of c q , . . . , c~~, then we may be interested in testing the hypothesis /~

: gl

~

''"

~

O~r ~

C~O •

A test based on the range/~:« -/?*1,r can be found in Cramer and Kamps (1996).

5.1. Likelihood

ratio test - Bartlett's

test

The likelihood ratio procedure for testing H against A is based on the statistic

sup L(cq,...,c¢r,F)/sup

Q=

L(~,

..,~r,F) ,

where L ( c q , . . . , c~r,F) is defined in (4.2). The hypothesis H is rejected at the level of significance c~ if Ql/(r~,) < z~ . For ~ - 0.01 and c~= 0.05 and some r and s the respective quantiles z~ can be found in Table 3. The values are taken from the tables given in Harsaae (1969) and Dyer and Keating (1980). Noticing that the supremum in the numerator of Q is attained at O~1

O~r = s r

. . . .

n --j+

1)Aj

(5.1)

,

j=l

where s

A, = - Z

log/1 -

FIxe?Il

i=1

and Aj

Z[I°g(1-F(X~ i=1

1)))_log(I_F(X(,j)))I,

2 <_j
,

E. Cramer and U. Kamps

322

Table 3 Values of c~-quantilesz~ in the likelihood ratio test s

~ = 0.01

1 2 3 4 5 6 7 8 9 10

c~= 0.05

r=2

r=3

r=4

r=2

r=3

r=4

0.0157 0.2843 0.4850 0.6031 0.6783 0.7299 0.7674 0.7958 0.8181 0.8360

0.0219 0.3165 0.5149 0.6282 0.6996 0.7483 0.7835 0.8101 0.8309 0.8476

0.0295 0.3475 0.5430 0.6518 0.7195 0.7654 0.7985 0.8235 0.8429 0.8586

0.0785 0.4780 0.6563 0.7456 0.7984 0.8332 0.8578 0.8761 0.8902 0.9015

0.0758 0.4699 0.6483 0.7387 0.7924 0.8280 0.8532 0.8719 0.8865 0.8980

0.0822 0.4803 0.6559 0.7444 0.7970 0.8317 0.8564 0.8747 0.8890 0.9003

we o b t a i n

Q=

n -j+

1)Aj

n -j+

1)Aj

z

j=j

U n d e r the hypothesis H, Q1/(rs) is distributed as the ratio of geometric a n d arithmetic m e a n of iid g a m m a variables W~,.. W~ with density f u n c t i o n

f~l(w ) _

l

(s- 1)!

ws 1e w

w>0":

It is worth m e n t i o n i n g that Q1/(rs) does n o t depend o n the p a r a m e t e r ~1 0~r = 2, say, a n y more. The statistic - ~ l o g Q appears in the c o m p u t a t i o n of M L E s of the parameters of g a m m a distributions. Tests a n d confidence intervals for these parameters are also based o n Q. However, the d e r i v a t i o n of the exact d i s t r i b u t i o n of Q is very complicated a n d thus chi-square a p p r o x i m a t i o n s are used for large sample sizes (see Bain, 1983). I n our situation, the role of the sample size is t a k e n u p by the p a r a m e t e r r which is usually small. The density f u n c t i o n of Ql/(r~) is shown in Glaser (1976). I n the context of testing h o m o g e n e i t y of variances from n o r m a l p o p u l a t i o n s , the above procedure is k n o w n as Bartlett's test (cf. Glaser, 1982). .

.

.

.

.

Sequential k-out-of-n systems

323

5.2. Test A - Hartley's test Let fi~,r-<""-< fl*« be the order statistics corresponding to /3~,..,fi*. The hypothesis H is now rejected if the ratio fi*1,r//fi*r , r is too small, i.e., if l,r/

r,r

--

where z~ is determined by equating PH(/GI/G

<- z~) : ~

In the area of testing homogeneity of variances in the normal case a test based on this statistic is wellknown as Hartley's test. In order to compute the critical value c, we use a representation of the joint density of the minimum XI« and the maximum Xr« in the sample X 1 , . . ,Xt of nonnegative iid random variables with distribution function F and density function f (cf. David, 1981, p.10) fXl'r'Xr'r(Xl,Xr)

= F(F --

1)(F(xr)

- F(xl))r-2f(xl)f(Xr),

x 1 < xr ,

to obtain the density function of the ratio XI«/Xr,r fXl'r/X"r(t) = r ( r - - 1)

(F(y) - F ( y t ) ) ~ - 2 f ( y ) f ( y t ) y d y ,

t E (0, 1) .

Hence, • • <- z«) = ez~ f ~L/~;«(t)dt PH(fll«/flr,~

= 1

(s r 1)!

l\/_z-c-ói! g - " (zie z ~ t _ e t)

ts

le

t

dt

"

The ratio of the smallest and the largest order statistic from a sample of size r has been considered earlier in the literature. Gumbel and Keeney (1950) call it the extremal quotient and study its asymptotic distribution w.r.t, r for symmetrical, continuous and unlimited parent distributions. They use extreme value theory along with the fact that the minimum and the maximum are asymptotically independent. We do not use this method here since the number of components is usually small. Corresponding tables are given in Gumbel and Pickands III (1967). Muenz and Green (1977) show representations of the distribution functions of arbitrary ratios of order statistics based on an absolutely continuous and strictly increasing distribution function F. In particular, we find P ( X I « / X r « <_ x) = 1 - r

(t - F ( x F -1 (t))) r i dt .

In terms of fl~« and/~;« we obtain the above expression again.

E. Cramer and U. Kamps

324

T a b l e 4 shows critical v a l u e s for ~ = 0.01 a n d c~ = 0.05 a n d several v a l u e s o f r a n d s, w h i c h m a y be f o u n d in I z e n m a n (1976).

5.3. Test B Since, u n d e r the h y p o t h e s i s H , f i ~ , . . , fi~ are iid g a m m a r a n d o m variables, it is n e a r at h a n d to use

j-1 as a test statistic, w h i c h h a s a b e t a d i s t r i b u t i o n o n (0, 1) w i t h p a r a m e t e r s s a n d ( r - 1)s. T h e d e n s i t y o f Br is g i v e n b y

1

fÆr(t) -- B(s, (r - 1)s)

t~-1(1 _

t)(r

1)s 1,

t E (0, 1)

w h e r e B ( a , b ) = fò ta-l(1 - t) ó ~ dt, a , b > 0, d e n o t e s the b e t a f u n c t i o n . T h u s , h y p o t h e s i s H is rejected, if either Br is t o o small or t o o large, i.e., if

B r < z~/2

or

B r > Zl_a/2 ,

w h e r e z~/2 a n d z1-~/2 are d e t e r m i n e d b y the e q u a t i o n s

PH(B~ <_ z«/2) = ~ / 2

and

PH(B~ > zl ~/2) = ~ / 2 .

5.4. Simulation study I n the p r e c e d i n g s u b s e c t i o n s we h a v e p r o p o s e d test p r o c e d u r e s to decide w h e t h e r the m o d e l o f s e q u e n t i a l o r d e r statistics is the a p p r o p r i a t e o n e in a given s i t u a t i o n o f s o m e k - o u t - o f - n system. S u p p o s e t h a t o n e o f these tests is a p p l i e d to s o m e d a t a Table 4 Values of ~-quantiles z« in Test A s

1 2 3 4 5 6 7 8 9 10

~ = 0.01

~ = 0.05

r=2

r=3

r=4

r=2

r=3

r=4

0.0050 0.0432 0.0903 0.1334 0.1710 0.2038 0.2326 0.2581 0.2809 0.3014

0.0022 0.0273 0.0641 0.t006 0.1340 0.1639 0.1906 0.2147 0.2365 0.2564

0.0014 0.0207 0.0522 0.0850 0.1157 0.1439 0.1692 0.1923 0.2137 0.2331

0.0256 0.1041 0.i718 0.2256 0.2690 0.3051 0.3357 0.362t 0.3853 0.4058

0.0114 0.0647 0.1196 0.1666 0.2064 0.2404 0.2698 0.2957 0.3186 0.3391

0.0070 0.0486 0.0963 0.1391 0.1763 0.2088 0.2370 0.2625 0.2849 0.3049

325

Sequential k-out-of-n systems

set where the alternative is accepted. Then the assumption that the failure of some component does not affect the remaining ones cannot be maintained at the level of significance ~. The following simulation study illustrates the power of the above tests. The level of significance is chosen as ~ = 0.05. We generated 10,000 experiments with s = 5, s = 10, and s = 50 iid copies each of a sequential 1-out-of-3 system based on the one-parameter exponential distributions Fj(x) = 1 - e-~Jx, x _> 0, j = 1,2, 3, with model parameters ~i

~2

~3

1 1 1

2 3 1

3 6 4

c~~,e~ and c~; are the M L E s of cq, 0~2 and ~3, respectively. We assume, for comparison, that we observe outcomes of ordinary 1-out-of-3 systems based on a oneparameter exponential distribution with parameter 1/2. In our framework, under the hypothesis, this corresponds to assuming ~1 = ~2 = c~3 = 2. The M L E 2* of 2 is given by 2* = 3/(1/~~ + 1/Œ~ + 1/ag) (cf. Eq. (5.1)). Table 5 shows the empirical means and the empirical variances of the bias corrected versions (UMVUEs) -5s-1 «j. , J = 1,2,3 . . 1 2 . of .the MLEs Œl,~2, Œ3 and 3S3s and 2* as well as the percentage of rejection w.r.t, the likelihood ratio test, Test A and B based on the data of 10,000 simulation steps. The simulation study gives rise to some comments. Obviously, the empirical variances sometimes turn out to be large for large values of c% Hence, the point estimates may be inaccurate and less useful. However, recalling the theoretical variances of the estimators s-1 • j = 1,2, 3, given in Theorem 4.2, i.e., ~-c~j, 1 Var ( s~-~ - c~j)/-

4

s- 2,

j = 1,2.3 , ,

they depend on c~2. Thus, the sample size should be chosen sufficiently large. The power of the tests increases subject to an increasing range of the c~'s as well as to an increasing sample size s. Test B seems to be more powerful than the other tests in the first two situations. However, it is less useful in the third example, which may be due to the fact that the respective test statistic is not symmetric in the il's.

6. A l o c a t i o n - s c a l e f a m i l y o f distributions

In Section 6.1 we introdu¢e a location-scale family @ of distributions, and point out some important properties of sequential order statistics based on some

E. Cramer and U. Kamps

326

Table 5 Simulation results for 1-out-of-3 systems (c~ = 0.05) Bias corrected MLEs (UMVUEs)

Percentage of rejection

i~s-- 1e,

7--2s-1 a,

Likelihood Test A ratio test

mean variance mean variance mean variance

0.994 0.360 0.998 0.123 0.999 0.021

1.989 1.267 1.988 0.469 1.998 0.084

2.983 2.910 3.000 1.098 3.001 0.190

1.654 0.248 1.645 0.115 1.638 0.022

cq = 1, c~2 = 3, c~3 - 6 s = 5 Emp. mean Emp. variance s = 10 Emp. mean Emp. variance s = 50 Emp. mean Emp. variance

0.999 0.324 0.997 0.127 1.000 0.021

2.998 2.922 3.003 1.126 2.995 0.189

6.022 12.080 5.964 4.154 5.991 0.736

2.069 0.494 2.030 0.222 2.006 0.041

1.003 0.330 0.996 0.121 1.000 0.020

0.996 0.318 0.998 0.127 1.000 0.021

4.058 5.503 4.011 1.996 4.002 0.338

1.300 0.182 1.340 0.078 1.335 0.014

Model parameters ~1 = I, a2 = 2, ~3

s = 5 s = 10 s = 50

cq -

1,~a-

s = 5 s = 10 s - 50

Emp. Emp. Emp. Emp. Emp. Emp.

3

3 s -c¢ -~ 1

~s12 *

Test B

25.0

28.1

36.9

55.9

56.8

63.6

99.9

99.9

99.9

64.2

67.3

72.3

94.3

94.8

94.4

100.0

100.0

100.0

46.0

54.7

I6.7

86.3

88.7

26.4

100.0

100.0

77.4

1, c~3 = 4

Emp. Emp. Emp. Emp. Emp. Emp.

mean variance mean variance mean variance

distribution function F C ~. The other subsections contain results on moments o f s e q u e n t i a l o r d e r s t a t i s t i c s b a s e d o n p a r t i c u l a r d i s t r i b u t i o n s , s u c h as W e i b u l l and Pareto distributions.

6.1. Definition and properties In this section we introduce the location-scale family Y b y its d i s t r i b u t i o n f u n c t i o n s F w i t h F(t)=l-exp{

-g(t)-p}

t>g-l(#),

#E

of distributions defined

IR, v ~ > 0

w h e r e F is r e q u i r e d t o b e a b s o | u t e l y c o n t i n u o u s . F o r c o n v e n i e n c e w e a s s u m e g t o b e d i f f e r e n t i a b l e o n ( g - l ( # ) , ~ ) as well as s t r i c t l y i n c r e a s i n g . S o m e i m p o r t a n t m e m b e r s o f t h i s f a m i l y °B- a r e s h o w n i n T a b l e 6. REMARK 6.1. A i m i n g a t e s t i m a t i n g t h e p a r a m e t e r p a n d / o r v~ i n t h e f o l l o w i n g sections, g has to be independent of the respective parameter(s). Hence, in lines 4 a n d 5 o f T a b l e 6, d s h o u l d b e c h o s e n as d = d'/O. F o r W e i b u l l d i s t r i b u t i o n s ( l i n e 2) a n d L o m a x d i s t r i b u t i o n s i n line 6 o f T a b l e 6, t h e f u n c t i o n g is i n d e p e n d e n t o f g

Sequential k-out-of-n systems

AI ~

AI

A[

I~

AI~ AI~ . . . .

327

~ A]

AI ~"

~_ I

A] ~

~ I

I AI

6

2

A]

AI

II

II

AI

v

L~

% AI

~

f~

.~~ ©

~

~~

%

II

I

II

II II

,~

% t~

©

=

~d

8

0

~

+

+

«

A

A

~1~

~,q

]

I

]

I

I

I

~

:::k

~

-~"

~~

A[~

AI

II

I

o

~ ~

~.~ ~ ~-

÷

~1~ ~1TM

~~ -~~~~ «~ ~ ~ ~~

E. Cramer and U. Kamps

328

~1~

II

II

II II

o ,.Q

~

° o

g

~

0

II

~ o

~u~ II

II ~ 0

329

Sequential k-out-of-n systems

and zg; in these cases, the parameters can be estimated simultaneously according to the results in Section 9. Apart from the estimation context, other interesting distributions are contained in Table 6 with va = 1. In line 4 we are led to power function distributions (a > 0, c --- 1/d), and in line 5 we have log-logistic distributions (a > 0, c = -I/d). Based on F E ~-, the joint distribution of X,(1),..,X, (4 is given by the Lebesgue density function fx!il"~r(*~l (xl , . . . ,x~)

z

(,

-Il! ,')!" ~

=

I O~i riI~ll[O;ig (xi) exp{-~(n-i

+ 1)(g(xi)-g(Xi_l)

) }] (6.1)

where g-1 (#) = xo ~ Xl ~ ... ~ Xr" For g(t) = t, this distribution coincides with the distribution of the first r (out of n) order statistics based on a Weinman multivariate exponential distribution (cf. Weinman, 1966; Johnson and Kotz, 1972, p. 268/9; Block, 1975, p. 303), which in Cramer and Kamps (1997b) and Cramer (2001) is referred to as WMEn(#, 0, ~), = ( c q , . . . , c~,). This particular multivariate exponential distribution is an extension of Freund's bivariate exponential distribution (cf. Freund, 1961). Hence, our results contribute to the analysis of Weinman's multivariate exponential distribution. Regarding related works in estimation theory, we are only aware of point estimation results for the parameters c~j given by Weinman (1966) and for the entropy of the distribution (6.1) with g(t) = t (cf. Ahmed and Gokhale, 1989). Putting ~1 . . . . . 0~r = 1 in (6.1), we obtain the joint density function of ordinary order statistics X I # , . . , Xr,n based on iid random variables 321,.. ,X, with distribution Exp (#, zg). For F ~ ~ as an underlying distribution, sequential order statistics can be introduced in a simplified manner as compared with Definition 2.1. Given F ~ ~with parameters # and ~9 we find F-l( t)=g

l(#_Olog(l_t)),

tel0,1)

.

Assuming (1.1), i.e., Fr = 1 - (1 - F) ~r, 1 < r < n, we conclude that Fr c ~- with the same function g and parameters # and 0/c~r, 1 < r < n. This set-up can be interpreted as follows. If r - 1 components have failed, the residual lifetimes of the remaining components are supposed to be distributed according to N c Y with scale parameters 0/~r, 2 < r < n. Let (Y)i/)l
E. Cramerand U. Kamps

330

and (i)

~,~~' =~ ' (~<,~_~+l)+ ~/~, «-'~) - ~),

2 < i < n

,

where Yl,n-j+l denotes the minimum of the iid sample YU),..., yU)j+ 1 (with

distribution function Fj), 1 _<j _< n. Evaluating this recurrence relation we obtain

x(i)

= g-I

~ _t_ E [ g ( j=I

-j+l) -- #] ,

2 < i <

n

.

In particular, choosing F = Exp(/~, d), i.e., g(t) = t, we find

x(i)

i = # -}- X-"ryO) .~..¢t 1,n-j+l - ,ul, j=l

2< i< n .

(6.2)

Hence, the distribution of X,(i) - # is given by the distribution of the convolution of independent minima from exponential distributions with possibly different scale parameters and common location parameter zero. It has to be mentioned that in the one-parameter exponential case Scheuer (1988) introduced a similar model motivated by renewal theory. REMARK 6.2. From (6.1) it is directly seen that if X,(I) , . . , X , (n) are sequential order statistics based on F E ~,~, then g(X,0)),.., g(X,,(n)) form sequential order statistics from a two-parameter exponential distribution. To be more precise, the joint distribution P ( g ( X (1))

~Xl,...,~(X(. n)) ~Xn) =P(X (1)
leads to the joint density function f

(g(xc?l,g<x!~~>_Xl ..

,x~)

=

~;Tn!~[~iexp{

°~i(,v/- i + l ) ( x i

-

xi_,)}]

p = x o <_xl <_... <_x~ , which coincides with the joint density function of sequential order statistics based on Exp(#, tg) (cf. (6.1)). Obviously, we have vice versa that if y,(1),..., 2",(') are sequential order statistics based on Exp(#,O), then the random variables g-l(y,(1)),..,g-l(Y,(~)) are sequential order statistics based on F ~ ~ with function g and parameters # and 0. Hence, Theorem 2.4 yields that n~l(g(zl((1))--/~) and ( n - j +

1)c¢j(9(X2))-g(X,(J-1))),

2 <_j<_n , (6.3)

are iid according to Exp(0, ~).

Sequential k-out-of-n systems

331

In the definition of sequential order statistics (cf. Definition 2.1) as well as in (6.2), the situation of sequential k-out-of-n systems is described in a distribution theoretical way, which means that we do not assign failures to components. This may be done by analogy with Heinrich and Jensen (1995), who present a general, bivariate set-up in the sense of Freund (1961). Starting with a scheme (yj(i))l<_i,j<_nof independent random variables, where the (Yj(i))l_<j_<~ are iid according to Exp(#, v~/~i), 1 < i < n, let X(1) = y~,l), X(2) __ y(2)

which denotes the second order statistic from the sample XJ2) =X,(' ) + (~2) _ #)l{#,/¢y()},

1< j < n .

v(2) ~ - # (cf. (6.2)). This description becomes O b v i o u s l y , w e h a v e X~2 ~ X~ 1~ + ~1,o complicated in the next steps, and thus we proceed as in (6.2), instead.

6.2. Moments of sequential order statistics based on exponential distributions In this section, let the considered sequential order statistics be based on a twoparameter exponential distribution Exp(#, 0) with distribution function

F(t) = l - e x p

{

t-# -~--j,

1

t_>#, t g > 0

i.e., we choose 9(t) = t in ~ . Utilizing (6.2) we directly obtain U) E(YI'n J + l - # ) = # +

Ex~r)=#÷

0~

j=l

1 c~j(n-j+l)'

l
.

j=l

For determining higher-order moments, we can make use of the moment generating function of the Weinman multivariate exponential distribution, i.e., E exp

)

= exp #

tj

1 j=0

(n - j)~j+l

tx

,

k-j+l

q,...,tn>0

(6.4)

(cf. Johnson and Kotz, 1972, p. 269). For instance, the moments of first sequential order statistics are given by

E(x(,ll)v = j=0 \ J /

~#~-J, kn~l)

vCN .

In this connection, recurrence relations are a useful tool in computing the moments. There is an extensive literature on recurrence relations and identities for moments of ordinary order statistics. For a detailed survey on this topic we refer to Balakrishnan and Sultan (1998).

332

E. Cramer and U. Kamps

In the mode1 of sequential order statistics we find the relation E ( X ( r ) ) v __ E ( X ~ r _ l ) ) v __

yO

(n_r4-1)c~

2
E(X(f))~-a,

The equation can be deduced from a relation valid in a certain class of distributions as shown in Cramer and Kamps (2000, Theorem 2.1). Noticing that, in the distribution theoretical sense, the models of sequential order statistics and Pfeifer's records coincide (cf. Kamps, 1995a, p. 55), the recurrence relation can also be taken from Balakrishnan and Ahsanullah (1995). Subject to the restriction (n - j 4-1)~j T/- (n - i + l )~i

foriCj,

l < i,j < n ,

an explicit expression for the moments E(X~r)) ~ is available as a particular case in (6.5) putting a 1. A recurrence relation for the product moments of sequential order statistics is stated in Cramer and Kamps (2000, Theorem 3.1). The covariances of sequential order statistics are obtained by using the above moment generating function ] Cov(X,(J),x(k)) = VarX, ~) = 0 2 Z ( ( n - g 4- 1)c~«)-2,

1 < j < k _< n ,

g=l which will be used in Section 9 to calculate the BLUEs of # and 0 from possibly differently structured sequential (nj - ri 4- 1)-out-of-ni systems. 6.3. M o m e n t s o f sequential order statistics based on Weibull distributions

Weibull distributions are offen considered in lifetime analysis. Due to an additional parameter they may lead to a better fit than exponential distributions. First, we consider a two-parameter Weibull distribution with distribution function F(t)=

1-exp

-

,

t_>0, a > 0 ,

0>0

.

It is denoted by Wei(0, a). Kamps and Cramer (1999) have calculated the moments of order v of sequential order statistics based on two-parameter Wei(O/cq, a)-distributions provided that (n - i + 1)~i • (n - - j + 1)czj for i C j, 1 _ - a , is given by E(X(,r)) v = O "/a

(n - k 4-1)~k

F

4-1

(6.5)

r

× ~aA(n-y+ j=l

1 > 2 -Iv/a+l) ,

333

Sequential k-out-of-n systems

where r

aj=aj(r)=II((n-k+l)ak-(n-j+l)c~j)

-1

1 <j
k=l

kCj

An expression for the product moments E(X(,OX~ )) is developed in Cramer and Kamps (1998b) and presented in Section 8.3. Considering the three-parameter Weibull distribution as in Table 6, line 2, i.e., - / { } , t >_ #U~ F(t) = 1 - exp { t ~~_

a >0 ,

a recurrence relation for the moments results from formula (2.5) in Cramer and Kamps (2000): 0(v+a)

E (X •fr)) v+a -E(X(f-1)) ~+a =

a(fi_--;TT)~

L,;v(r)~~ ~~l~,

j ,

~ >

-a

.

For v = 0 the formula reduces to E(X(r))a - E ( X ( r - l ) ) a

-- (n - r + 1)at "

A recurrence relation for the product moments of the three-parameter Weibull distribution is shown in Cramer and Kamps (2000, Example 3.5).

6.4. Moments of sequential order statistics based on Pearson distributions

(( ~/1~~ )

In Table 6, line 4, distribution functions of the form F(t)=l-(dO(c-ta))

1/~,

tE

c-

,c 1la

a>0,

,

d>0,

v~>0

are introduced including special Pearson I and power function distributions. For these distributions we find the identity E(X~r))v+a

E(x(,r 1))v+a

v+a

(~--~ ( n - j + l ) ~ j

= a d ( ( n - F~- 1)~ r-~~9) \«=1 ( n - - 7 ~ ~ ) ~ j T ~ 9

) E(y*(r))v '

where y.(1),.., y.(•) are sequential order statistics based on E" = 1 - (1 - F ) ~i, 1 < i < n , with O

Bj=c~j+-n-j+l'

l<_j
and

(cf. Cramer and Kamps, 2000, formula (2.5)).

fij--ej,

r+l<_j
E. Cramer and U. Kamps

334

An identity for the product moments of sequential order statistics is stated in Cramer and Kamps (2000, Example 3.5).

6.5. Moments of sequentiaI order statistics based on Pareto and Lomax distributions In Table 6, line 5, distribution functions of the form

F(t)= 1-(dO(t a-c))

1/o,

tC

e+

,oo), a>0,

d>0,

O>O

are introduced including Pareto distributions. Choosing c = 0, the moments of sequentia, order statistics are given by

E(X(f))~ = (dO)-~/a I-I

(n-/+

1)c~,

j:l(n-J+ 1)~j ~' (n-j+

v# l)c~j>--, l < _ j < r < n a

.

(cf. Kamps and Cramer, 1999, (11)). In general, we find the identity E(X,(r)) v+a _ E(X,(,-I))'~+ ~

~+a =ad((n-r+

r-1 / n ~ + ' > ,

o~

,)~zr- vg)j~l(n~ .= J+J)cg-~

'

where y,(1) . . . , y,(n) are sequential order statistics based on F~ = 1 - (1 - F) g, (cf. o >0, l
7. Joint maximum likelihood estimation of model and distribution parameters In this section we eonsider the estimation of model and distribution parameters in the situation of Section 4. The underlying distribution function is specified by F(t)=l-exp{-g(t)

0-#}

as discussed in the preceding section.

t>g

1(#), # E ] R , 0 > 0

Sequential

k-out-of-n

335

systems

7.1. E s t i m a t i o n in the l o c a t i o n - s c a l e f a m i l y

We consider both the case of a known parameter # and that of unknown # and 0. If p is supposed to be known, g may depend on #. By analogy with Theorem 4.1 we obtain THEOREM 7.1 • T h e M L E s o f ~ T h e o r e m 4.1, i.e.,

= T, ~ 1 < v < rl, are g i v e n b y ~~* = T~ with c~v* as in

-1 O~1 =

S

ni[g

) - - [A

\i=!

and -1

-,

)

ni-v+l)[g

1)

)-g

)1

,

2
H e r e , the n e w p a r a m e t e r s ~1, . . . , ~r~ are c o n s i d e r e d i n s t e a d o f O, ~1, . . . , c~r, since there is no f i n i t e M L E

o f O.

Properties of the estimators may be derived along the lines of Theorem 4.2. COROLLARY 7.2. In the p a r t i c u l a r case o f i d e n t i c a l l y s t r u c t u r e d s y s t e m s , i.e., I, 1 z

the M L E s

• • • =

Fs z

I~

nl = • • • = ns = n

and

cv = s ,

o f ~j = ~ j / O , 1 <_ j < r, r e a d

and -1 &,.__ J

S

g

)

)-g

1)

,

2<_j<_r

.

n--j-[-1

Supposing # and ~ to be unknown, we find THEOREM 7.3• T h e M L E s o f p a n d &~ = ~ / 0 , /2" = min g ( X ~ ) ) , 1< i < s

-*

)

~1 = s

ni[g \i=1

)-#*

1 < v < r l , are g i v e n b y

336

E. Cramer and U. Kamps

and -* ~v=-cv

)) - g

ni-v+l)Ig

Moreover, the stati«ti« (/2", O@...,

~,~)

-1)

,

2
ic suffi«ient for (l~,aj/0, ..,

.

~,,/0).

The proofis carried out as in Cramer and Kamps (1996) (see also Lawless, 1982; Epstein, 1957). COROLLARY 7.4. The special case ri = r, ni = n, 1 < i < s, leads to the resuhs stated in Cramer and Kamps (1996). The M L E s read

/~*= min g(X(,])), 15i<_s

~(~~~~~~~~~ 0 -1

n

--/2*

and ~]

_

s

s

n--j+l

-1)

g

))--q

,

2<j
.

Varde (1970) considers the case of a two-component series system where the lifetime distribution is exponential. 7.2. A two-parameter Weibull ease The location-scale family of distributions ~ contains the two-parameter Weibull distributions with distribution funetion F(t)=l-exp

-

,

t>_0, a, 0 > 0 ,

(7.1)

as shown in Table 6, line 3. In contrast to Section 7.1, where the function g (and hence the exponent a) is required to be known and 0 is to be estimated, we here aim at estimäting the parameter a as well. We restriet ourselves to the particular case of identically structured systems with r« = r and n~ = n, 1 < i < s. Hence, we are interested in simultaneous estimation of ~ 1 / 0 , . . , ~~/O and a. The Weibull distribution is frequently used in statistical models, and a variety of papers can be found in the literature dealing with parameter estimation. For maximum likelihood estimation in a type II censoring set-up, i.e., based on ordinary order statistics, we refer to, e.g., Cohen (1965), Harter and Moore (1965), Pike (1966), McCool (1970), Rockette et al. (1974) and Lawless (1982, Chapter 4.1). Usually, the case s = I is considered in the literature.

Sequential k-out-of-n systems

337

It turns out that in general there is no explicit solution for the M L E of a. This problem also arises in the classical case (cf. Lawless, 1982, p. 143 and the references above). However, the existence and the uniqueness of an M L E of a can be proved. THEOREM 7.5. For the Weibull distribution as in (7.1), we find:

1. For s = 1 there is no M L E of a. 2. I f s >_ 2 andmaxl minl
~logxij-

i=l j = l

× Z

( x iaj l o g x i j

(x~j- xiS_l) j=2 k,i=l

--Xi,ja

1 logxi,j-1)

i=1

--

Xatl

Xal 1ogxil = 0 i=1

The M L E s ~~ of ~j = ~j/O, 1 « j <_r, are determined according to Corollary 7.2 with 9(t) = t J* and # = O. Evidently there is no M L E of a in the case s = 1, since then r + 1 parameters have to be estimated based on r observations. 8. Estimation of the scale parameter In Cramer and K a m p s (2001) estimation in the set-up of sequential order statistics based on one-and two-parameter exponential distributions is considered. Three estimation concepts are applied to derive estimators for the location and scale parameters, namely m a x i m u m likelihood estimation, uniformly minimum variance unbiased estimation and best linear unbiased estimation. The model parameters Œil, cq2,..., 1 < i < s, are supposed to be known, and are allowed to be different in the systems. Let the sequential k-out-of-n systems be based on a distribution function out of the class .Y introduced in Section 6. In the first subsection, we derive the M L E of the scale parameter 0 by assuming the location parameter # to be known. Some properties such as consistency and asymptotic normality are presented. In Example 8.3 we illustrate the different modelling of a 3-out-of-4 system by ordinary order statistics and sequential order statistics. Then we discuss the possible consequences if it is erroneously assumed that the impact of failures on the remaining components could be neglected. In Section 8.2 we focus on two-parameter exponential distributions. In addition to the results valid in the class ~-, the Cramér-Rao lower bound can be calculated. The M L E of the scale parameter is seen to coincide with the U M V U E and the BLUE. Finally, we consider two(and three-) parameter Weibull distributions in a parameterization different from the one in Table 6. We present the M L E of the scale parameter and, utilizing

E. Cramer and U. Kamps

338

representations of moments and product moments of sequential order statistics, we obtain the B L U E of 0.

8.1. Estimation of the scale parameter in the family Y By analogy with the results established in Cramer and K a m p s (2001) for twoparameter exponential distributions, the M L E of O is computed based on s possibly differently structured (nj - ri + 1)-out-of-ni systems with a distribution function F out of f r . By considering the log-likelihood function s

ni!

s

l( O, g; (~iy)i,j, (xij)i,j) = Zi:l log (ni 7 ri)! + Z

ri

Z

i=1 j = l

s

Fi

; ~ ~ +:- J + :

log o~i/

l/+/~/+/-~+,»~/l

- RlogO , where x~,0 = g-1 (#), 1 < i < s, we arrive at THEOREM 8.1. The M L E of O is given by

O*

1

s

r~

z ~i~l j~.l(ni-- j 27 l)~ij[g(X,Üi ")) --g(X,(/J-1))]

wirk X (0) = ~

1(/2), 1 < i < s , R = EiS__l

ri.

We make use of the assertions in R e m a r k 6.2 in order to derive properties of the M L E 7". Some of them, which are due to the property (6.3) of successive differences of transformed sequential order statistics, are summarized in the following theorem. It coincides with Theorem 3.3 in Cramer and K a m p s (2001). THEOREM 8.2. In the above situation with ~)* = O*(R), we find that

1. O* ~ F(R, O/R), i.e., O* is a gamma distributed random variable with parameters R and O/R. Its density function is given by

f + (t) - ((R/O)R ~ ~ ~ ! tR ~e-Rt/O,

t > o.

02 Hence, O* 2. F(~~,~k ~~~ j -- ~(Æ+k-1)!(~)k ~~/ , »,~ E N; in particular, EO* = 0 and Var O* -- -y. is an unbiased estimator of ~9. 3. O* is sufficient for O. 4. (O*(R)) R is strongly consistent for #, i.e., #*(R) -+ 0 a.e. w.r.t. R -+ cx~. 5. (O*(R)) R is asymptotically normal, i.e., x / R ( O * ( R ) / O - 1 ) d j v ( 0 , 1) w.r.t.

R---+ oo.

Sequential k-out-of-n systems

339

In the situation of type II censoring, i.e., cqj = 1, ri = r, ni = n, 1 <_ j <_ ri, 1 < i < s, some particular cases appear in the literature. Usually, s = 1 is only considered. In the one sample case we refer to Lawless (1982, p. 102), Johnson et al. (1994, p. 514) for the exponential distribution (g(t) = t, # = 0), to Johnson et al. (1994, p. 656/7) for the Weibull distribution (9(t) = t ~, a > O, # = 0), and to Johnson et al. (1994, p. 593) for the Lomax distribution (g(t) = log((t + d ) / d ) , d > 0, # = 0). In case of the exponential distribution Basu and Singh (1998) present the M L E of 0 for s _> 1. The distributional results of Theorem 8.2 can be utilized to derive confidence intervals for the parameter ~. From the first item of the above theorem we find ~9" ~ ~5{2R,~°2where Zq2 denotes the z2-distribution with q degrees of freedom. Hence, a two-sided confidence interval with level ~ E (0, 1) is given by

[

. 2RO* 2RO* ] )~2R(1 - ~/2)' )~22R(cz/2)J '

where Z}(~) is the c~-quantile of the )~}-distribution. A one-sided confidence interval is given by (

2RO* ] o,~j

In the case of one sample of ordinary order statistics from exponential distributions, the results reduce to the representations given in, e.g., Cohen (1995). For progressive type II censored order statistics we refer to Balakrishnan et al. (1999). EXAMPLE 8.3 (Sequential 3-out-of-4 system). Suppose that the s underlying systems have a common sequential 3-out-of-4 structure with a distribution function F E Y and # = 0. Without loss of generality we assume that cq = 1 and that ~2 is an arbitrary, but known positive real number. Applying the preceding results, the MLE ~9" of 0 is given by 0* = ~ sxil~-, = (4g(X!~)) + 3c~2[g(X(,2))_g(X(,~))]) in terms of the random variables X *0) X *(2) i = 1 , " " ,s. In case of an underlying i 1 i ~ ordinary k-out-of-n structure, i.e., e2 = 1, the M L E 0; of 0 is given by O°* = 2sl i~1 (49(X~~)).= + 3 [9(X~/2))_ 9(X~~))]) To analyze the different impact of the respective modeling of the system, we consider the ratio Q = O;/O*. The distribution of Q is obtained as follows. Since the random variables 4g(X!] )) and 3c~2[g(X}2)) -g(X,(]))] are iid according to F(t) = 1 - exp(-t), t _> 0, for all i = 1 , . . ,s (cf. (6.3)) we obtain

E. Cramer and U. Kamps

340

Q-

y~+~-l-

1

Ns

1

by putting

B = ~ 49(X(,]))

and

Z~ = ~

i=1

3c~2[g(X}~)) - 9(X}¢))] .

i=1

Since Y~ and Zs are independent and gamma distributed with parameters s and 1 we obtain that Zs/(Y~ + Zs) has a standard beta distribution with both parameters equal to s. Therefore the expectation of Q is given by EQ-- 1 - (1 - 1/c~2)/2 = 1 / 2 + 1/(2~a). A plot of EQ as a function of c~2 is shown in Figure 7. Let us suppose that the underlying distribution of the models is Exp(0, v~). In the sequential 3-out-of-4 model, we start with an Exp(0, 0/cq) _= Exp(0, 0)-distribution. After the first failure, the lifetime distribution of the remaining components is given by Exp(0,0/c~2). If «2 is large, the survival time of these components tends to be short. In the classical model, however, we still assume the remaining components to be Exp(0, ~)-distributed. In what follows, let us assume that the observed system is in fact a sequential 3-out-of-4 system with ~1 1 and c~2 > 0. The expectation EQ as a function of c~2 can be interpreted as follows. Both 0* and 0; estimate the lifelength of a component, namely ~, where 0* is an unbiased estimator of 0 in the considered set-up. When applying v~; as an estimator of ~, we are supposing that there is no impact of failures on remaining components. That is, the classical model, described by order statistics, is erroneously considered. The distinct values of the estimators 0; and 0* w.r.t, e2 are =

2.5-

2.0-

1.5-

EQ 1.0-

0.5-

I

I

[

I

I

I

i

2

3

4

5

6

OL2

Fig. 7. Expectation EQ

E(~;/~*) as a function of c~2.

341

Sequential k-out-@n systems

indicated by the ratio Q = 0~/0'. For illustration, a plot of the expectation of Q as a function of a z is shown in Figure 7. After the failure of the first component in our system, the hazard rate of the remaining components has changed to ~2/0. If e2 > 1, then

1(~),

~, = ~ 1 ( g + z ~ ) > ~

~+

Zs = ~ 0

a.e.,

such that 0 is underestimated by 0;. If g2 tends to infinity, E Q tends to 1/2. On the other hand, the expected lifetime of the system is estimated too optimistically if the considered system is erroneously supposed to be an ordinary 3out-of-4 system. This results from the expected lifetime of the respective systems, i.e., O

~9 ~2>10 0 < 4 -t-~=EX2,4 '

EX(*2) = 4 - ~ - ~ ~ 2

where X2,4 denotes the second order statistic in a sample of four iid random variables from Exp(0, 0).

8.2. Estimation o f the scale p a r a m e t e r in the two-parameter exponential distribution

In the sampling situation of the preceding subsection with sequential order statistics from an Exp(#, 0) distribution, the former results apply putting g(t) = t. The M L E O* of O is given by 1 ,7

ri

with X ,(°) = #, 1 < i < s, R = ~i~_1 ri (cf. Theorem 8.1). In the situation of ordinary type II censoring described by ordinary order statistics, i.e., Œ/j = 1, r i ~ r, rti = 1"t, 1 < i < s, 1 <_j <_ r, the representation (8.1) can be found in Lawless (1982, p. 102) and in Johnson et al. (1994, p. 514) in the case of one sample (s = 1). In the particular case c~/j= cq for all j, the estimator 0* can be written as 0*

1

O~i (rli -- r i +

= R i=l

nj#

j=l

~~ ( . ~Ri=l

1)X*(~~) + Z X~) -

-

r~ + l~(X ù', *i(ri) - #) +

( x 2 / - ~)

which, for s = 1 and ~i = 1 for all i, leads to the representation usually found in the literature (see, e.g., Epstein, 1957, Eq. (3), # = 0). The result in terms of progressive type II censoring with s = 1 is given in Cohen (1995) (# = 0).

342

E. Cramer and U. Kamps

The MLE O* attains the Cramér-Rao lower bound (cf. Cramer and Kamps, 2001, Theorem 3.4) which is given by 02/R. Since O* is unbiased (cf. Theorem 8.2) and linear, it coincides with the U M V U E and the BLUE of 0. REMARK 8.4. Choosing appropriate values for the parameters eij, the preceding calculations lead to results for the particular models as pointed out in Section 2.2. Previous results dem with only one sample, i.e., s = 1. For ordinary order statistics we refer to Epstein (1957) and Engelhardt (1995). For progressive type II censoring see Cohen (1995) and Balakrishnan and Sandhu (1995, 1996). In the model of record values, i.e., eij = 1/(ni - j + 1), the estimator O* simplifies as follows:

•,

1 +x(ri) R~

s_l 2 x ( r ~ ) rl ,1

The latter expression can be found in the Ph.D. thesis of Houchens (1984, p. 26) (see also Arnold et al. 1998, p. 122). 8.3. Estimation o f the scale p a r a m e t e r in the Weibull case

In contrast to the representation of Weibull distribution functions stated in Table 6, we now consider sequential order statistics based on three-parameter Weibull distributions defined by F(t)=l-exp

I(~~ ~} -

,

t_>/~, a , v a > 0 .

This distribution is denoted subsequently by Weibull(#, Va, a). We aim at estimating the parameter va supposing a and # to be known. TH~OREM 8.5. The M L E Theorem 4.1)

o f va is given by (cf. Cramer and K a m p s

1998b,

@i=1@/j=l (ri)q_1)(X~')_ #)a] 1la

va, = LRI-2"~2''~['mij[]

with R = ~-~~i~=1ri, and mÜ), 1 <_ j <_ ri, 1 < i < s, defined by m(ri) ij = (ni - j + l )Œij - (ni - j)c~i,j+l - 1 , m i,ri (ri) = (nj - ri + 1)c~i«i - 1~

1 <_ j <_ ri - 1 ,

1 -< ri -< ni "

The case of ordinary order statistics is considered in Harter (1965). In the case of an unknown location parameter p, explicit expressions of the MLEs of p and z9 are not available. The MLEs ~ and O are connected via 0 a = S(ft), where S(kt)= + 1)(X~) _ /~)a. Moreover, ~t has to satisfy ft _< minl<_i<_,X}] ). For given data (xij)l
l~i=lS ~~/1/~mij(r~)

Sequential k-out-of-n systems

$

r,

1

a s

ri

=

j=l

343

/a ~/~/~/ZZxi, ~~~/"'l;~~+~//~« ~/~-1 i=l j=l

-/

t~

given the constraint/~ _< minl_
=

ml;'l+ 1 Ri=I

~,~;I

"=

»

where (V,(/))i,j are sequential order statistics based on exponential distributions, it follows (cf. Cramer and Kamps, 1998b) that 0 *a has a gamma distribution with parameters R and Va~/R. Therefore, the distribution of va* is given by a so-called generalized gamma distribution with location parameter zero. This distribution was introduced by Stacy (1962). For a discussion and related references we refer to Johnson et al. (1994, Section 17.8.7). Moreover, it is easy to see that va* is a sufficient estimator of 0. We subsume these results in the following theorem (cf. Cramer and Kamps, 1998b, Theorem 4.4) by analogy with Theorem 8.2. THEOREM 8.6. Let va*= Va*(R) be the M L E of va in case of a known location parameter. 1. va*" is a gamma distributed random variable with parameters R and vaa/R. Hence, the moments o f order k are given by E(Va*")~ -- (R+k-1)! (R 1)! (%)k, k E N; in particular, EVa*~ = vaa and Var(va *~) ,ô2a Therefore, va*~ is an unbiased estimator o f vaa. The moments EVa*s, s > O, can be written in terms of the confluent hypergeometric function (cf. Johnson et al. 1994, p. 389). 2. va* is sufficient for va. 3. (Va*(R))R is strongly consistent for va, i.e., Va*(R) ----+ va a.e. w.r.t. R ---+oc. 4. (Va*(R))R is asymptotically normal, i.e., xffB(va*(R)/va-1) ~ JU(0, 1) w.r.t. R---+ oc. From a transformation result for MLEs it follows that va*a is the M L E of va" (see, e.g., Schervish, 1995, p. 308). This leads to the following theorem. THEOREM 8.7. The M L E O*a of the transformed parameter Oa attains the CramérRao lower bound. Since va*a is unbiased, it coincides with the U M V U E of vaa. We now focus on linear estimation of va. Linear estimates of # and va can be obtained by analogy. Suppose that (ni-j+l)c~ijTL(ni-k+l)o:ik

for

1 <j
l
.

Kamps and Cramer (1999) have calculated the moments of order v of sequential order statistics based on two-parameter Weibull(0, va~/c~i,a)-distributions pro-

344

E. Cramer and U. Kamps

vided that ( n - j 4 - 1)~j ¢; ( n - k 4 , 1)«k for j e k, 1 <_j,k <_ n. The moment of order v, v > - a , can be easily deduced from (6.5):

E(X(f)) ~ = cr_lOVV(v/a 4- 1) ~-~ aj[(n - j 4, 1)c~j] (u/a+l) = O~Sv,,-, say . j=l

(8.2) This expression can be applied to the three-parameter case by taking the following result into account (cf. Cramer and Kamps, 1998b, Lemma 3.1): Let the random variables y,(1)..., y,(~) be sequential order statistics based on two-parameter Weibull(0, Oa/o~i, a)-distributions, 1 < i < n. Then the random variables X(, i) = y,(i) + #, 1 < i < n, with ]A E IR are sequential order statistics based on three-parameter Weibull(]A, Oa/o~i, a)-distributions, 1 < i < n. Using this result and Expression (8.2), we obtain the first and second moments of the rth sequential order statistic based on three-parameter Weibull distributions:

Ex(r) = ]A + OSl,r , E(X(r)) 2 : ]2z 4" 2]AOSI,r q- O2S2,r , which yield VarX,(~) = ~92(S2,r -- Sl,r) 2 • Now we note the product moment o f X (i) and X~ ) to deterrnine the covariance of these sequential order statistics, 1 _< i < j _< n. For covariances of ordinary order statistics from Weibull distributions we refer to Lieblein (1955).

j

i a~/)(J') Z al(i) k=i+l I-1

E(x(i)x2")) = cj-la202 Z

× ~b((n - k + 1)~k, (n - l + 1)~l) ,

w h e r e a ~ O ( J ) = ~ jl=i+l,1¢k~~((n- l + 1)el - ( n - k+l):~k) -1, i + l < k < j , Ó(u,v)=

/oJo'

(xy)aexp{-uya-(v-u)xa}dxdy,

u,v>O

and .

Introducing the notation q = 1 + ä, Ó can be written in terms of hypergeometric functions (cf. (11.3)): ¢(u,v)

__ tl-ql) q

a(a+l)F(2q)F

(

q,l-q;q+l;1-

tl)

7

.

For v > u > 0, Lieblein (1955) obtained a simple representation of ¢ in terms of the incomplete beta function. Moreover, ~b can be expressed using Legendre functions. For details we refer to Cramer and Kamps (1998b). The above expressions for moments and covariances can be utilized to calculate the BLUE of O. Due to the independence of the samples X,(]),.. ,X~r*),

Sequential k-out-@n systems

345

1 < i < s, the covariance matrix of )~ : (X,(1),.. ,X}~~),X}ä),. . . . . . ,X(,'~s))' can be written as a block diagonal matrix 2; = diag(2;1,.., 27s), where 2;i E IRri×ri is the covariance matrix of (Ä~,(]),.. ,x,(~i))', 1 < i < s. It is seen that Zi = 02Ai, where Ai does not depend on the parameter 0. Moreover, the matrix Ai is positive definite, since the joint distribution of the corresponding sequential order statistics is absolutely continuous w.r.t, the r~-dimensional Lebesgue measure, 1 < i < s. Introducing the notations A = d i a g ( A l , . . . , As) and ~ = EX/O, the BLUE is given by the Gauß-Markov theorem %qBLUE =

(~'A-I~)-I~'A IX.

For previous results in the framework of ordinary order statistics we refer to Johnson et al. (1994, p. 645) and the references therein.

9. Joint estimation of location and scale parameters

In the model of sequential order statistics based on two-parameter exponential distributions, MLEs, UMVUEs and BLUEs of both location and scale parameter are found in Cramer and Kamps (2001). The results are summarized in Section 9.2. In the first subsection, we extend the results on maximum likelihood estimation to the Family ~ . Throughout this section we assume that the model parameters O:il,~i2, .. , 1 < i < s, are known.

9.1. Joint estimation of location and scale parameters in the family Analogous to the derivations in Cramer and Kamps (2001), the MLEs of # and 0 are stated based on s possibly differently structured (ni - ri + 1)-out-of-nj systems with a distribution function F E Y THEOREM 9.1. The simultaneous MLEs of # and O are given by = min g(X(,~)) 1
and

s

respectively, with X(°). = 9 l(~t), 1 < i < s, R = ~i=1 ri. In the next theorem, some properties of the estimators ft and Ô are shown. Noticing Remark 6.2 the proof is carried out as in Cramer and Kamps (2001). THEOREM 9.2.

1. The MLEs O and ~ are stochastically independent. s O~il)). 2. O ~ F ( R - 1,O/R), ft ~ Exp(#,O/~i=l(ni 3. (fr, O) is a complete sufficient statistic for (#, v~).

E. Cramer and U. Kamps

346

Basu and Singh (1998) report the independence result in the situation of ordinary order statistics from exponential distributions and one sample. REMARK 9.3. Suppose that X 1 , . . ,Xs form an iid sample from a normal population with parameters # and a 2. It is well known that the MLEs of # and «2, i.e., s ~ ){ ~ ~~=~ i and s 2 ~ ~~=1 (X~ - )?)2, are independent (cf. Bickel and Doksum, 1977, p. 20). Moreover, their distributions are given by ~ ( # , « 2 / s ) and 2 2 (s/«)Z~,-1. Considering statements 1 and 2 of Theorem 9.2, we observe an analogy in our situation, i.e., the estimators ft and Õ are independent with distributions Exp(#, O/ ~-~4-1 ~' nic~il) and (R/O)Z2(R_I). 2 For the case of one type II censored sample see Epstein and Sobel (1954) and David (1981, p. 153). If we consider # in both situations as location parameter and 0 and a 2 as scale parameter, respectively, the analogy is striking. From Theorem 9.2 we find the following results similar to those given in Theorem 8.2 for a known location parameter. COROLLARY 9.4.

1. E(O)k

--

(R+k-2)! (~)k, k E N; in particular, EO = g-~O and VarO = ~ß02. (R-2)!

Hence, Ô = ~ O is an unbiased estimator of O. 2. (õ(R)) R is strongly consistent for O, i.e., O(R) ----+ O a.e. w.r.t. R ---+oo. 3. (O(R)) R is asymptotically normal, i.e., ¥ # R ( O ( R ) / O - 1 ) ~ J V ' ( 0 , 1) w.r.t. R----+ oo,

The MLE ft of the location parameter possesses the following properties, which are directly obtained from Theorems 9.1 and 9.2. COROLLARY 9.5.

fl.

E(~)k = ~

~ i = l -niO~il #k-j;

in particular, 0 E~t = # + ~i21 niO~il

and

Var~=(

~9 ) 2 ~i21 ni°~il

Henee, the mean squared error is given by

~~~(~1= ~(\ ~ i =~1~ni~il )~

" oo

2. (~(R)) R is asymptotically unbiased provided 2i=1 ni~il = oc, i.e., e~(e) ~ # [ oo oo w.r.t. R ~ oc. ~ ~ i = 1 n~Œil < oc, the asymptotic blas is given by O/ ~i= 1 nic~il. 3. (~(R)) R is stron«ly consistent iiff 2i=l ni~il = cx~.

Sequential k-out-of-n systems

347

As in the one-parameter set-up, the preceding results can be applied to construct confidence sets for the parameters kz and 0. Proceeding similarly to the case of a known location parameter we obtain that, for a given level ~ c (0, 1),

~~=l~- ,~ ~,~~~~~1 ~

~ 1,1 ~, ~]

and

Y# =

2RÕ 2RO .] z2~R 1/(1 - ~/2)' Z2(R-1) 2 (~/2)

are confidence intervals for kz and O, respectively. Fp,q(1 - 00 denotes the 1 - 0~ quantile of the F-distribution with degrees of freedom p and q. Since in out situation p = 2, we can apply the relation F2,2q(1 - 0~) = q(o:-1/q - 1) (see, e.g., Epstein and Sobel, 1954). For results in the case of one sample and ordinary order statistics from exponential distributions we refer to Engelhardt (1995). In terms of progressive type II censored order statistics analogous results are obtained by Balakrishnan et al. (1999).

9.2. Joint estimation of location and scale parameters in the two-parameter exponential distribution In the situation hefe, being a particular case contained in the preceding section with 9(t) = t, all the results in Section 9.1 can be utilized. In contrast to the case of a known location parameter as considered in Section 8.2, the MLE, the U M V U E and the B L U E of the scale parameter v9 turn out to be different. The M L E s ~ and Õ are taken from Theorem 9.1 1

P = min ~
Õ :

s

Fi

ZZ(ni--j+

1)o:ijIX,Ui) -- X,~i"-1)]

(9.1)

ijl] ' = "=

with X (°),, = ~, 1 2, and both estimators are biased (cf. Corollaries 9.4 and 9.5). Since Ô = R-LrÕis an unbiased estimator of O, we obtain an unbiased estimator /~ of the location parameter kz via a bias correction, i.e.,

B :

ü--

(~)' niO~il

\i=1

O

.

/

The following theorem (cf. Cramer and Kamps, 2001, Theorem 3.11) gives the variances of/~ and Ô as well as their covariance. It extends a result quoted in Cohen (1995) for s = 1 and ordinary order statistics.

E. Cramerand U. Kamps

348 THEOREM 9.6.

1. The U M V U E of O is given by Ô = ~

Õ. The variance of Ô is

VarO- 02 R-1

2. The U M V U E of l~ is given by ~ = ~ - (~~_~ nicql)-lô. The variance of ) is Var) =

3. Cov(/, 0) =

R (R - 1)(2i=1 s

ni<) 2

02 .

1 02. (R 1/2~:,",~~1

Asymptotic properties of the U M V U E s can be derived directly from Corollaries 9.4 and 9.5. Since the UMVUEs/~ and 0 are nonlinear, we state the BLUEs for completion. The derivation is based on the joint covariance matrix Ig of the sequential order statistics X}I), • - - ,~~.1:¢(r')'X~~),.. ,X(~"). In the case of an underlying exponential distribution (F _= Exp(/~, zg)), the sequential order statistics X(1).i, . . . ,ù..~((n') follow the distribution of order statistics from a Weinman multivariate exponential distribution WMEù,(tz, 0, ~i), ~i = (cql,..., cq~i), 1 < i < s (cf. (6.1)). We utilize this fact to calculate the covariance matrix £ of the sequential order statistics X(} ) X(9 ) 1 < i < s . As mentioned in Section (8.3), the covariance matrix Ig is a block diagonal matrix £ = d i a g ( X 1 , . . , 2 ~ ) , where Zi E IW'×~~ is the covariance matrix of l
(x#l,,x}?l',

J

Cov(Xy),X~/k)) = Var(X.0i )) = 02 Z [ ( n i - v

+ 1)cq~] 2 = O2alj) '

say .

v=l

X(9))t is given by Hence, the covariance matrix of \(X*(1) i ~ " " " ~ *l ]

all/

a}2) a}3)

.. "" ..

a! 2) a! 3)

..:

a(r,I

la} 1) a} 1) a} 1)

la} 1) a} 2) a} 2) Zi=O21a}l)

/ La} l)

with a} 1) < a}2) < ' " < u_(ri) i . Choosing

a}2) a}3)

~92Ai,

say ,

i

o~ij -- 1 for all 1 _< j <_ ri, Y~i reduces to the covariance matrix in the case of a sample of ordinary order statistics from an exponential distribution (cf. Sarhan, 1954; Balakrishnan and Cohen, 1991). The inverse of Ai is given by

349

Sequential k-out-of-n systems

Ai 1 =

-bl 1) -}- bl 2) - bl 2) 0 - bl 2) b}2) -}- b}3) -bl 3) 0 -b} 3) bl 3) -- b}4)

''" 0 -b} 4)

"'"

0

0 -bl ~d

with b ~ ) = [ ( n i - j + 1)c~jl2, 1 <_j _< ri, (cf. Roy and Sarhan, 1956; Graybill, 1983, p. 187). The BLUEs of/~ and 0 are deduced from the GauB-Markov theorem, which yields the following matrix representation: OBLUE~ = ((fl, {R),A-I(fi, {s)) l(fl, {R),A-1j~,

(9.2)

//BLUE /

. . wherefi . . (ill,

, f l s ) , a n d f l i = E ( X } } ) ' .. . ,~-., , j - = ([nic~ill , , ~ / =2l [ ( n i - j + y(?))/~9

~i/]-a,... , ~ = l [ ( n i - j +

1)

1)ei/l-1), 1 < i < s. {R denotes the vector (1,..., 1)' of

dimension R, A = diag(A],. ' ' ,A,), and 2 ~ ~g(1) g(r,) ~ v(1) ,X.0)),. \~*1 ~'''~*1 . 2 ~''" The covariance matrix of the BLUEs is given by

COv(0BLUE~ = ((fl,~R)t~-l(fl,'~R)) 102 k [2BLUE / Evaluating these matrix expressions we obtain the following theorem (cf. Cramer and Kamps 2001, Theorem 3.12)• i

THEOREM 9.7. L e t X *l(°)=0, 1 0

and

The B L U E s o f # and 0 are given by

0BLUE = C

(nicxil) 2

(hi--J+

l)~iJ(Y(.Ji ) -X((.z 1)],

i=1 j=l

1 /ABLUE z C [ R ~f'~( ni°~il )22.(1)i 1_ i=1

2z¢. i=1

.z

i=1 j=l

REMARK 9.8• 1. The condition c 1 > 0 in Theorem 9.7 is fulfilled but for the case of record values with only one observation in each sample, i.e., ~il = 1/ni and ri = 1,

350

E. Cramer and U. Kamps

i = l , . . . , s . Applying the Cauchy-Schwarz inequality this is directly seen from the inequality S

c -1 >_ (R - s) ~ n i ~ i l

_> 0

i=l with equality in the first inequality iff (~il : 1/ni, i : 1 . . . ,S. If we consider this particular model, we have R = s and the matrix (fi, ~e)'A -1 (fl, ~R) given in (9.2) is

A = s({

I)" Since the matrix A is singular, a linear unbiased estimator of (#, #) /

does not exist. It is only possible to deal with so-called estimable linear combinations of/~ and #. Here, the BLUE of/~ + # is given h~,y s1~-,~ 2_~~=1X(1) ,~ 2. From Eq. (9.2) we derive the alternative representation 1

OBLUE= ~

s

/ ri-1

i~.. 1 ~j~. i "=

( ( n i - - j + l)o:ij -- (ni -j)o~iß+l)(~~2i " (J) --~tBLUE)

+ ( * - ~~ + 1 ) « i , r , ( x 2 ) -

~ùLU~)) I

In the particular case of an ordinary k-out-of-n system and s = 1 this yields the result of Epstein (1957). 3. In the case s = 1 the estimators given in Theorem 9.7 simplify considerably. We obtain the representations: rl

1

Z(nl_j

+ 1)Ctlj(X~)_X ~ ,)),

OBLUE--rl- l j=2

~~~~~ = x}l ) - ~~~u~

•

4. Given that « = 1 and that ~lj = ( k + ( n - j ) ( m + 1 ) ) / ( n - j + 1), 1 _<j_< n, for some k > 0, m > - k / ( n - 1) - 1, Ahsanullah (1996) calculated the BLUEs as well. For completeness we give the covariance matrix of the BLUEs in an explicit form, i.e.,

Cov (OBLUE/ \ #BLUE// = 02

_1 R

s

~i=1 (ni~il) 2

s

2

~i=1 (hirn)

- ~i=1 nicql

s

R

(~i=117iO~il) k -- Ei=I ni~il

"

REMARK 9.9. As in the model of a known location parameter, particular cases are found in the literature. For references see Remark 8.4. Considering the model of record values, we have a similar simplification of the proposed estimators (for the BLUEsletR>s+I and f o r s = l l e t r l _ > 2 ) :

Sequential k-out-of-n systems

351

MLE:

» : minX(} ) l <_i<_s *~ s

s=l: X!~) '

1 rX(•i)

UMVUE:

( B=

s; 1 @R~~I

s=l

lrl

__

Ô

1 (x~rl) X}~))l

s N s=l:

) min X}] )

s

s V , ;((~i) R - 1 z..ù'-*,

l
i=l

(FlX(1)-X!;1)), 1

_

R

~7~ x ( r i )

S

lZ-~--*~

R

i=1

--

-

s=l

E ,

BLUE:

1

(~~)

1 # -- rl - 1

)

1 -

s

]

i=1

.]

1 V~y(!) V~x(Fi) #BLUE -- (R - s)s R L..~~~*z - s ~ *z I i=1

s--1

1 ( F 1 - 1 FlZl/(1) -- X ( ; 1 ) ) '

1

~(X(?)X(~))«:1

0BLUE - - R -- s i=1 ~ *z

rl

_1

1

(X2r,) 1

X(1),~ *1 ]

The expressions for the BLUEs (s = 1) are given in Ahsanullah (1995, p. 45) (see also Arnold et al. 1998, p. 127). It turns out that for s 2 2 all estimators are different whereas in the case s = 1 the BLUEs and the U M V U E s coincide. The preceding results can be utilized to construct statistical tests and confidence bounds w.r.t, the parameters # and O. For instance, we consider the location parameter #. We want to decide whether the location parameter is given by #0 = 0. The corresponding decision problem reads H0:#--0

vsA:#¢0

.

Considering the MLEs of # and 0, we make use of the ratio of the MLEs, i.e., T = )/Õ. By Theorem 9.2, ~ and ~ are independent and the distribution of T does not depend on the parameter 0 (cf. Remark 9.3). Hence, the ratio ( R / [ ( R - 1) E i 2 1 ni~il])T follows an F-distribution with 2 and 2(R - 1) degrees of freedom. For a similar result in the uncensored case we refer to David (1981, p. 153). 10. E s t i m a t i o n o f a c o m m o n l o c a t i o n p a r a m e t e r and o f s c a l e p a r a m e t e r s

In the preceding sections, we were concerned with the estimation of distribution parameters based on s independent observations of possibly differently structured sequential (ni - ri + 1)-out-of-ni systems, 1 < i < s.

E. Cramer and U. Kamps

352

F o r a m o m e n t , suppose to have two systems, namely a sequential (n0) - r0) + 1)-out-of-n (1) system based on F (1) E o~ with a function g and parameters/~ and 01 and a sequential (n (2) - - r (2) 4-1)-out-of-n (2) system based on F (2) E ~- with the same function g and parameters # and 02, and to be interested in estimating the c o m m o n location p a r a m e t e r #, which m a y be interpreted as a m i n i m u m life or as a guarantee time of the components. On the other hand, we are interested in estimating the scale p a r a m e t e r s v~i and 02 for b o t h assuming # to be k n o w n and unknown. In what follows, results are shown in a very general sampling situation. Ler sequential (nlk) - rl k) 4- 1)-out-of-nl k) systems be observed with known model parameters ~~), 1 < j _< r}k), 1 < i < s (~), and which are based on F (k) E .~, k = 1,2. T h a t is, we suppose to have s (1) and s (2) independent, possibly differently structured sequential systems based on F (1) and F (2), respectively. The respective failure times are modeled by sequential order statistics X,O/),

1 < j < r} 1), 1 < i < S(1) ,

i1,0.)

1 < j < r/(2), 1 < i < S(2) ,

and

where all the r a n d o m vectors ( X ~ ) ) j , . . . , (X~l~,)j , ( Y ~ ) ) j , . . . , (YOi)))j are jointly independent. The corresponding observations are denoted by

(xij)id

and

(Y,Y)id"

We use the following notations. F o r k = 1,2 let Z=min{

min

l
g(X}~)] \ /

s(k)

~(~)

min g ( Y ( ] ) ) } l_
s(k)

V " n(k)~(k)

R (k) = ~ r}k)

i-1

i=1

r! 1)

j + 1"~,~<1/[g/xo»~

g@° 9 = 0,, « < ~/l/ =

(2 (2) n i O~il j=

n

-

j+ ,',) U ~,~> rg,>,>', - g(«-'9] Lt.,~/

9(Y(,°))

= 0, 1 < i < S (2),

353

Sequential k-out-of-n systems

s(k)

V(k) = / ~V ' n(h) ~(h) ~(~) _ p(h)Z i il i=1

The likelihood function is given by L(#,O1,O2;xij, 1 < i < s

(1), 1 < j < r}l); Yij, 1 < i < s (2), 1 < j < r } 2))

h=, ~=,

¢1of2

/ x exp

1

S(I)

I ~

(1)(1)

(1)

- Z- 2_., ni ~i~ wi

" S(2) I K-~

'Ul i=1

where

W i(1)~ W i(2)

(2)(2)

(2)

- Z- 2-_., ni ~il wi '[f2 i=1

and z are the respective realizations of W~(1),

W/(2)and Z.

THEOREM 10.1. 1. Let the common location parameter # be known. The M L E o f Oh is given by 1 vs(k) , - , (k) (h) .... (h) 0k=~2._ ni eil tvvi - # ) ,

k=1,2

.

i=1

2. Simultaneous estimation o f # , 01 and 0 2 leads to the M L E s

1 V(k) ' ~=z, Oh=~

k=l

2 .

For exponential distributions, i.e., 9 ( t ) = t, this result is shown in Cramer (2001). In the situation of part 1 of Theorem 10.1, the likelihood function obviously factorizes w.r.t. 01 and 02. Hence, the MLEs 01 and 0~ are already stated in Theorem 8.1 in a different notation. Properties are presented in Theorem 8.2. In the second part, there is no such factorization, such that a proof of the assertion is along the lines of the proof of Theorem 9.1 (cf. Cramer and Kamps, 2001). The hext theorem contains the independence of (V(1), V(2)) and Z as well as the distributions of these random variables, and hence the distributions of the above MLEs (cf. Cramer, 2001). THEOREM 10.2.

l. (V (1), V(2)) and 2. Z ~ E x p

Z are independent random variables.

#,~W+o2j

•

E. Cramerand U. Kamps

354

3. The joint distribution function of V (1) and V (2) is given by

P(V(I) ~/)1, V(2) ~ 1)2) B~(1) )~(23"~-I = ~~-1 @~-2)

["Y/(1) - 0ffvl(R(1) 1

1, 01)Fv2 (R(2), 02 )

9(2)

+-~2 F~~(R('), O1)F~~_(R (2) - 1,02)

}

,

where Fz(r, 0) = Of(r-l)! 1 fo« tr le t/Odt' z > O. The joint distribution in part 3 of Theorem 10.2 is a mixture of productgamma distributions, and V(1) and V(2) are seen to be asymptotically independent. More details can be found in Cramer (2001), e.g., the asymptotic normality

of (O~,õ2). THEOREM 10.3. Let R (1) lim

R(I),R(?)~ecR(1) -}- R(2)

-a~

(0,1),

and let limRu/R/2/~oo 7(1)/7 (2) exist and be finite. Then

~l)«:l»)

----+ ig"

((°0), ( 7

0

02/(1 -- b)

))

"

Finally we consider underlying exponential distributions, i.e., we assume g(t) = t, and we present the U M V U E of the common location parameter #. In this particular case, Theorem 10.2 includes results of Ghosh and Razmpour (1984) and Chiou and Cohen (1984) who were concerned with possibly censored samples from two exponential distributions Exp(#, 01) and Exp(#, 02). By analogy with their result it is possible to calculate the UMVUE/~ of #. The proof is based on the complete sufficiency of Z, V(1) and V(2) which is established similarly to Ghosh and Razmpour (1984) and Chiou and Cohen (1984). We arrive at the representation

(7i7V(1) - Z ) ( ~ ß ( 2 ) - « ) )=Z-

(R (1)- 1) (7@Z)V(2) -- / ) --(R(2) - 1)(~@I)V(1) - 2 )

In the case of uncensored samples (which correspond to parallel systems) and s (1) = s (2) = 1 from exponential distributions the expression reduces to the formulas of Ghosh and Razmpour (1984) and Chiou and Cohen (1984).

355

Sequential k-out-of-n systems

P(X <

11. Estimation of

I¢) with data from sequential k-out-of-n systems

Estimation of P = P(X < Y), where X and Y are independent random variables, has often been investigated in the literature. The results can be applied to socalled stress-strength models in reliability theory, where the strength Y of some component is subjected to a stress X. The component fails if and only if the applied stress is greater than its strength, and P is chosen as a measure of performance (see Beg, 1980; Constantine et al., 1986; Johnson, 1988 for further applications such as comparing two treatments). Several concepts of estimation are discussed in the literature such as maximum likelihood estimation, uniformly minimum variance unbiased estimation, Bayes estimation, parametric and nonparametric approaches. In this section, we consider the M L E and we focus on the U M V U E of the probability P, where the distributions of X and Y are elements of the class «~ of distributions introduced in Section 6. Noticing that for independent random variables

X ~ F (1) withF(1)(t):l - e x p f

g(t-)07-~}

t>g-l(/z)

and Y~

F (2)

with F (2) (t) = 1 - exp { -

the transformed random variables namely, g(X) ~

Exp(p, 01),

and

g(t)-! , 02 ~} , t>g-l(12) _

g(X) and g(Y) are g ( Y ) ~., Exp(p,

exponentially distributed,

02),

we find

?)2

P = P ( x < r ) = P ( g ( x ) < g ( r ) ) - o, + 02

(11.1)

Tong (1974) and Bartoszewicz (1977) obtained the U M V U E of P based on independent one-parameter exponential random variables in the non-censored and censored cases, respectively. Beg (1980) considered the U M V U E of P based on distributions from exponential families with unknown scale and truncation parameters generalizing the results of Tong (1977) for the case of two unknown parameters. Constantine et al. (1986) derived a simpler expression of the UMVUE of P compared with the one in Tong (1975) based on independent random samples from gamma distributions with known, integer-valued shape parameter and unknown scale parameters. Bai and Hong (1992) considered the U M V U E of P in the exponential case with possibly different sample sizes and an unknown common location parameter (see also Cramer and Kamps, 1997b). Maximum likelihood estimation is considered in Basu (1981), Chiou and Cohen (1984) and Johnson et al. (1994). Enis and Geisser (1971) and Ghosh and

E. Cramer and U. Kamps

356

Sun (1997) deal with Bayesian estimation procedures. Jana (1997) compares the MLE and the U M V U E of P in the one-parameter exponential case. For further details on the estimation of P we refer to Constantine et al. (1986) and Johnson et al. (1994, p. 530/2). In what follows we are concerned with the estimation of P based on failure data from sequential k-out-of-n systems in the sampling situation of Section 10: Sequential (nlk)- r}k)+ 1)-out-of-n}k) systems are observed with known model parameters e}~), 1 _<j _
Oä

1 ~¢kl , v - , (k) (k) .... (k) vgk =~£T2_..,n i Ctil kvvi - - # ) ,

with

#

k=l,2

,

i=1

with v~~ and ~9~ as in Theorem 10.1. From Theorem 8.2 we deduce that 0 k~F

R (k),

,

k=l,2

.

Since the estimators are independent by the model assumption, the distribution of the ratio P~ -- zg~/(zg~ + 0~) is given by a so-called generalized three parameter beta distribution G3B(fil, f12; tl) with density function t//~~t/h 1(1 _ t)/~2-1 h(t;/~1,/~2, ~) =

B(fil,

fi2)[1

-

(1

-

tl)tl [Jl+fi2'

tE(0,1),

fll,fl2, t / > 0

(11.2)

Sequential k-out-of-n systems

357

(cf. Johnson et al. 1995, p. 251). In our setting, we have fll = R(2),

R(2)#l q -- R(1)va2

f12 = R(1),

Properties, moments, extensions and further applications of this distribution can be found in Libby and Novick (1982), Chen and Novick (1984) and Pham-Gia and Duong (1989). We now turn to minimum variance unbiased estimation of P in the above general sampling situation. The results in the particular case s (a) = s (2) = 1 are stated in Cramer and Kamps (1997b). In the two-sample exponential model (i.e., ~Ill) " .. = ~(1) .(23 = 1), Tong (1974) calculates the U M V U E lnO) = ~v(2) ~11 = "'" mm C~ln(2) of P for parallel systems (non-censored case), and Bartoszewicz (1977) for k-outof-n systems (censored case) (see also Johnson et al., 1994, p. 531). Let the hypergeometric function F be defined by

v(:, 8; v;z) = ~

(11.3)

where (.)j with (x)j = x(x + 1)... (x + j symbol, (x)0 = 1.

1), j _> 1, denotes Pochhammer's

THEOREM 1 1.2. The U M V U E of P is given by \

/5 =

, ,

1-Fall-R(')

~(1) ~ V(2),

,~~2~),

I'R (2)'v~2/'~

V~(1)> V~(2)

where V} ~) = V (k) + 7 (k) (Z - ~) s(k)

k = 1,2 .

= Z ,~(k)~,(k)W(k) i=1 Let R(1),R (2) > 2 and O = 01/02. Then gar j3 z

O R(2)

1

(1 + O)R/13+Æ(2/ B(R(a), R(2)) × { R(z)_I ~ R(2)_l ~ ( (1 -R(2))i(1 -R(2))/ i=0 j=o

(

(R(1))i(R(1))J

x F R (1) + R (2), 1;R (U + i + j +

R(U + i + j

1)

1;l~ß

E. Cramerand U. Kamps

358

R0) 1R0)-I ( 1 - R ( 1 ) ) i ( 1 - R ( 1 ) ) j

q- Zi=1 ~ j=l xF

-(~~)j

(

1

R(2) q_iq_j

~)} (1)

R(1)+R(2),I;R(2)+i+j+I,~TÕ

-

~Õ

REMARK 1 1.3. 1. The hypergeometric functions in the above expressions o f / 3 may be replaced by their finite series representations. We have F ( - a , 1;c;z) = ~ (-a)jzJ j=o

for a C N U {0}

(cf. Gradshteyn and Ryzhik, 1994, p. 1065) 2. Putting s (1) = s (2) = 1 and c~11~ ) = . "" ~ ~ l.(1) ù(2) ~ " ' " z ~ù(2) = 1, we obn(1) = ~11 ln(2) rain the results of Tong (1974) and Bartoszewicz (1977) (see also Johnson et al., 1994, p. 531) for non-censored and censored samples from two-parameter exponential distributions with known common location parameter, respectively. The variance of/~u in this two-sample case can be found in Cramer and Kamps (1997b). In the particular case considered by Tong (1974), Kelley et al. (1976) and Bartoszewicz (1980) derive expressions for the variance.

11.2. Estimation of P(X < Y) for an unknown common location parameter In contrast to the preceding subsection, we suppose the common location parameter to be unknown. We establish the MLE of P and its distribution, we give the U M V U E and its variance, and we state that the MLE and the U M V U E are asymptotically equivalent. THEOREM 1 1.4. The M L E of P is given by p* z

R (I) V (2)

O2

R(2) V(1) + R(1) V(2)

Öl -}- O2

with Õ~ as in Theorem 10.1, k = 1,2. The distribution of P* is a mixture of generalized three-parameter beta distributions (of. (11.2)) with density function 1 1+

Œ ah(t;R(2),R(1) - 1,11)+~'-h(t;R(2)-x->o 1,R (1) 17) '

where R(2) ~1 17 = R(I~ " zg~

and

a . .7(2) . . 01 9 (I) 02

'

Sequential k-out-of-n systems

359

T h e o r e m 11.1 yields that/:~ is a ratio of two independent gamma variates, and thus its distribution is a generalized three-parameter beta distribution. F r o m T h e o r e m 11.4 we deduce the following similar result. The distribution of P* is a ratio of two (dependent) r a n d o m variables, which follow a mixture of g a m m a distributions. Therefore, the distribution of P* is a mixture of generalized threeparameter beta distributions. F o r more details we refer to Cramer (2001). The U M V U E /5 of P in our setting is shown in T h e o r e m 11.5, which, for s (1) = s (2) = 1, can be found in Cramer and K a m p s (1997b). THEOREM 11.5. The U M V U E of P is given by

{

V(2)+(R(2)-1)~( 2) V(1)

. ~l~l'~_(~,~211.»l,)vl~l~

( ~ - 7 ) ~ \ - - . . . . . . . . 751

B =

(RO)-l))~(1)V(2)+(R(2)-l)9 (2)VO)

1-

(R(2)-I)9(2)v(l)+(R(1)-l)9 (t) V(2) ~,1 ,fll)) F \

........ (R0)- 1)~(1)V(2)+ (R(2) 1)9(2)V(1)

VO) < V (2)

,

_

v~Tyj

V (1) »

>

V (2)

In the particular case of different parallel systems (i.e., nll)¢; nl 2)) with s ( 1 ) = s ( 2 ) = l a n d % j .(k) = 1 for k = 1,2 and all j, Bai and H o n g (1992) derived the U M V U E of P based on underlying Exp(#,Ol) and Exp(#,O2) distributions. A corrected version is stated in Cramer and Kamps (1997a). In the particular case 9 (1) = 9 (2), the estimator/5 simplifies considerably: (/su = ) Æ =

(R(1) - 1)~(1) V(2) (R(1) - 1)p(1)V(2) q- (R(2) - 1)~(2)V(1) (R (1) _ 1)R(2)9(1)02 (R(1) - 1)R(2)9(1)01 + (R(2) -

1)R(1)~(2)~2

Although the assumption 9 (1) = 9 (2) seems to be very restrictive, it includes some important examples. First o f all, if s (1) = s (2) = s and the estimation of P is based on the observation of record values of the underlying distributions, we have 0{(1) 1 and c~}2) = ~(J~~, 1 < i < s (see Table 2). Hence the assumption is fulfilled il mm n.~rY

and thé simplified estimator can be used. Moreover, if all observed values are upper record values, the statistics W/(1) and W,.(2) are simple. We find W/i(1) = 9~/x(r}l)) (r(2)) ), such that only the largest and the smallest ,i ,) and W/(2) = g(Y,i*

observed record values in each sample are necessary to estimate P ( X < Y). A n o t h e r interesting model leading to ~(1) = ~(2) is given by progressive type II censored samples with s 0) = s (2) = s and the sample sizes N 1 , . . ,Ns. F r o m Tabie 2 we conclude that c~l~) = Ni/n i(1) and %(2) = N i / n i(2), l < i < s. Thus, for minimum variance unbiased estimation of P ( X < Y) by means of data from progressive type I1 censored samples, the simplified estimator can be applied in this situation, although the censoring schemes remain arbitrary. Since the structure of/5_ is similar to that of the M L E P*, it can be shown that its distribution is a mixture of generalized three-parameter beta distributions, too (cf. Cramer, 2001). Moreover, this mixture turns out to be a Gauss hypergeometric distribution with density function

E. Cramer and U. Kamps

360

(R(2) _ 1) R(2) v~(:)

R 1) + R (2) -- 2

q~(t) = \

R (I) -- 1

- - -1)R~7-1 ~- +- 1

(R~

(1 - t) R(~-2tR(~-2

×

t C (0,1) ,

{R(2)-I o~_ 1)t) R(1)+R(2)-I ' 1 + \R(~)-I where

~9=L91/0

( R (2) -

1 ) 0 1 = ( R (1) -

2

and R(I),R (2) _>2. It reduces to a beta distribution

if

1)02.

Gauss hypergeometric distributions have been introduced by Armero and Bayarri (1994, (4.1) and (4.2)) (see also Johnson et al., 1995, p. 253) as prior distributions for the traffic intensity in an M / M / 1 queue. The density function of a Gauss hypergeometric distribution with parameters ~ > 0, fi > 0, 7 and z is given by

c

X ~-1 (1 -- X)/3-1 (l+zx) ~

for0<x
,

where c is a normalizing constant. Further properties are shown in Cramer and K a m p s (1997b). The kth moments of t5= are given by Efik_ = B ( k + R (2) - 1,R (1) - 1) B(R(1) - 1,R(2) - 1)

F(R(1)+R(2)- j,k+R(2)- I ;k+R(I)+R(2)-2;-z) ×

1,

z E (-1,0), z~O~

F (R (1) +R (2) - 1~R(1) - 1;k+R 0) +R (2) - 2 ; ~ r )

z>0

,

1 O where z = R(a)R(~-rr~_ 1 • N - 1.

This yields the variance R(2)-IK-~ (R (I) -

1

1)j

(1 +

Var/5_

+

1

-

( - 1)e(') (1 + z)R(2)Z1-R(~)-R(2)

1 +0 ×

Z)j

B(R(1) - 1,R(2)) log(1 + z ) +

(1)2 1~0

~

(-1) j

=O(R(1)' R(2)'O)'

say, i f z E ( - 1 , 0 )

,

361

Sequential k-out-of-n systems

Var/5_ =

(R0) - 1)(R (2) - 1) (R(I) +R(2) - 2)2(R(~) +R(2) - 1)'

ifz = 0 ,

and Var/5= = ~ (R(2), R(1), 1 )

ifz>0.

In the special case of two parallel systems with s (1) = s (2) = 1, nl 1) = nl 2) and e(k) lj = 1 for k = 1,2 and all j, t5= is given in Chiou and Cohen (1984). Assuming ~(1) = ~(2) and s0) = s (2) = 1, the variances of the U M V U E s t5 and /5 corresponding to an unknown and a known common location parameter are compared via relative efficiency in Cramer and Kamps (1997b). Since the structure of the U M V U E / 5 of P is complicated in the general situation, it is near at hand to look for asymptotic results w.r.t, an increasing number of samples. Let R = R (1) + R(2). By analogy with Lemma 1 of of Bai and Hong (1992) it is shown in Cramer (2001) that (V(k)/R(k))R«~ I is strongly consistent for 0k > 0, k = 1,2. Hence, the sequence of MLEs (P*)R is strongly consistent for arbitrary (R (1), R (2)) with min{R (1) ,R (2)} » oc. Moreover, an application of the dominated convergence theorem yields that the MLE P* is an asymptotically unbiased estimator o f P (cf. Serfling 1980, p. 11). Utilizing Theorem 11.4, i.e., the distribution of P*, we obtain the asymptotic normality of the MLE. THEOREM 11.6. Let liml~l~~,Æi2~_+o~R(1)/(R(1) T R (2)) ---- ~ E (0 71) and let the limit limR/l/,R/2/_~~ ~(1)/~(2) exist and be Jinite. Then the following assertions hold:

B((

V(1)/R (1)

01

0

and

/ p2(1_»/2.)

(11.4)

Considering the asymptotic variance in (11.4) we conclude that it is preferable to choose the sequences (R (1)) and (R (2)) such that (R (1))_~(R(2)).P)~ This choice leads to the smallest asymptotic variance of P*, i.e., 4PZ(1 By analogy with Bhattacharyya and Johnson (1974) we obtain a similar asymptotic result for the U M V U E of P.

THEOREM 11.7. L e t lime~lR:~o~R(1)/(R(2) + R (2)) = c] E (07 1) and let the limit limR/l/R/2/~~ ~(1)/~(2) exist and be finite. The U M V U E /5 and the M L E P* are asymptotically equivalent in the sense that B ( / 5 - P*) ~

0

a.e. w.r.t. R ---+ oo .

Moreover, the asymptotic distribution o f x/R(/5 - P) is that given in (11.4).

362

E. Cramer and U. Kamps

12. Reliability properties of sequential order statistics As pointed out in the second section, the models of order statistics and record values are contained in the model of sequential order statistics as particular cases. There is a variety of important results for order statistics and record values in connection with aging properties, such as the IFR or D F R property, and with partial orderings, such as hazard rate ordering or dispersive ordering. An excellent review on stochastic orders is given in Shaked and Shanthikumar (1994). In reliability theory, classes of distributions are considered which describe the lifelength of components or systems. There are numerous articles concerning the analysis of such families of distributions (see Barlow and Proschan, 1975, 1981; Patel, 1983; Basu, 1988). Since an ordinary order statistic Xr,n from an iid sample of random variables X 1 , . . , X ù with distribution function F represents the lifetime of an (n - r + 1)out-of-n system, one is interested in aging properties ofXr,n (more precisely, of the distribution of Xr,~), and in the transmission of aging properties. We may want to know whether the I F R property o f F implies the I F R property of the lifetime Xr,~ of the system as well as whether au aging property is transmitted from one order statistic to another. Throughout this section we assume F -1 (0+) _> 0, since the distribution function F is interpreted as lifetime distribution. If F is supposed to be absolutely continuous with density function f , then the failure rate (hazard rate) 2(.) is defined by

)~(t)- 1 f~~)(t)'__ 0 ~ F-I(o --) <~ t < F-l(1) leading to the examination of classes of distributions having an increasing failure rate (IFR) or a decreasing failure rate (DFR). For ordinary order statistics, the IFR property of some distribution function F implies the I F R property of an arbitrary order statistic based on F (see Barlow and Proschan, 1965, p. 36; Barlow and Proschan, 1975, p. 108). This transmission result is taken up by Takahasi (1988) and generalized to the transmission of the IFR property from the rth order statistic to the ( r + 1)th order statistic. The D F R property of a distribution does not necessarily imply the D F R property of some corresponding order statistic (cf. Barlow and Proschan 1965; Patel, 1983). Takahasi (1988) notes that the D F R property of the rth order statistic implies the D F R property of the ( r - 1)th order statistic. However, there is no distribution such that all corresponding order statistics possess the D F R property. Aside from the I F R and D F R property, other aging notions have been considered in the literature such as I F R A (increasing failure rate average) and D F R A (decreasing failure rate average) via the monotonicity of - 7 l log(l - F ( t ) ) as well as N B U (new better than used) and N W U (new worse than used) corresponding to 1 - F(x +y) < (>)(1 - F(x))(1 - F(y)) for all x,y. Concerning interpretations and examples, we refer to the literature on reliability theory (cf. Barlow and Proschan, 1975, 1981, Basu, 1988). For instance, it is well known that the following implications hold:

Sequential k-out-of-n systems

'IFR ~

IFRA ~

NBU'

and

'DFR ~

363

DFRA ~

NWU'

For the distributions out of the family Y (see Section 6), the I F R or D F R property is noted in Table 6. Nagaraja (1990) generalizes the above-mentioned results w.r.t, these criteria and to other neighbouring order statistics. If the rth order statistic in a sample of size n possesses the I F R (IFRA, NBU) property, so do Xr+l,n, Xr,n--~, Xr+l,n+l. Under the restriction r _< @ , the assertions remain valid for Xr+l,n+» Analogously, the D F R (DFRA, NWU) property of Xt,ù implies the corresponding property of the order statistics Xt-ic» Xr,n+l, X~-I,ù-1. Restricting to r < ~ ! , the assertions remain valid for X~-1,~-2. The above results are described and discussed in detail by Nagaraja (1990). The results may be illustrated in the triangular scheme of the order statistics (see Figures 8 and 9: If Xt0,ù0 possesses the IFR (DFR) property, then the same property holds for all order statistics belonging to the shaded area). Record values are used in models of reliability theory as well. Kochar (1990) points out that records are closely connected with occurrence times of nonhomogeneous Poisson processes and refers to surveys of Ascher and Feingold (1984) and Gupta and Kirmani (1988) (cf. Section 2.2). By analogy with transmission of the I F R property in the case of ordinary order statistics it is shown in the latter article that the I F R property of the rth record is ensured by the I F R property of the underlying distribution. Kochar (1990) generalizes this result in the sense of Takahasi (1988) using the same argument. Thus, the I F R property of some record is transmitted to the following one and the D F R property of some record is transmitted to the previous one. Moreover, if we consider a strictly increasing distribution function, it is not possible that all records possess the D F R property. Gupta and Kirmani (1988) and Kochar (1990) also consider ordering of records and record differences and present some results by analogy with results in the case of ordinary order statistics (e.g. Barlow and Proschan, 1975, p. 108). For somewhat different transmission properties of kth record values we refer to the paper of Raqab and Amin (1997).

T

nO

nO

Fig. 8. Transmission of I F R property.

364

E. Cramer and U. Kamps

~~° nO

nO

Fig. 9. Transmission of DFR property. For sequential order statistics only a few results without any restrictions imposed on the parameters are available up to now. In K a m p s (1994, 1995a), Pfeifer's record values and generalized order statistics are considered. The results are presented below. However, assumptions on the model parameters are made of the form m l = m2 = . rar-1 w.r.t. X2(r) , where mi = (n - 1. + 1)~i - (n - 1)~i+1 - 1, l 0, which here are denoted by X,(I'~),.. ,X,(,k'k) in order to indicate the sample size. 1. L e t r <_n - 1,ml . . . . .

m r = m, say. Then

Xff'~)IFR » X,(r+L~)IFR (in p a r t i c u l a r : F I F R ~ X,U'")IFR, 1 < j _< n), and X(r+I'")DFR ~ x ( r ' " ) D F R . 2. L e t r <_ n,ml . . . . .

mn = m,say. Then,

i f m > -1,X(~'"+I)IFR ~ X,(~'")IFR, and Xff'~)DFR ~ X,ff'~+I)DFR, if m

< -1,x(,r'n+I)DFR ~ x~r'n)DFR, and X,(r'n)IFR ~ X,ff'n+I)IFR .

Sequential k-out-of-n systems

365

3. Ler r <_ n, mj . . . . .

m, = m, say, and ~~ >_ m + 1, i f r = n. Then X,(r")IFR ~ X,(r+l ~ 'n+l) IFR, and Xt,(r+l '~ + 1D) F R

X~r,,)DFR .

As mentioned in Theorem 12.1, part 1, the IFR property is preserved under the ordering operation, i.e., the sequential order statistics are IFR, too. This is well known in the setting of ordinary order statistics (corresponding to ordinary k-outof-n systems) (cf. Barlow and Proschan, 1975, p. 108; Takahasi, 1988), and of record values (Gupta and Kirmani, 1988; Kochar, 1990). The results also extend to the I F R A and N B U property as mentioned above. This particular property holds true for sequential order statistics without requiring additional assumptions as in Theorem 12.1. TI-IEOREM 12.2. Let X,(1),.. ,X,(~) be sequential order statistics based on an absolutely continuous distribution function F having the IFR (IFRA, NBU) property. (1) (n) Then all the sequential order statistics X2 , . . ,X~ are IFR (IFRA, NBU). Without any restrictions imposed on the model parameters, further results along the lines of Theorem 12.1 have still to be developed. From Theorem 12.1 we have that ifX, (k'k) is DFR, then Xff '') is D F R for all 1 < r < n < k. There are further results on the transmission of the D F R property. It is shown by Takahasi (1988) and Kochar (1990) that there is no distribution such that all order statistics or record values have D F R distributions. Moreover, we find (cf. Kamps, 1995a, Theorem V.l.14): Let ml . . . . . m,-1 = m, say. For every m >_ - 1 and for every c~, > 1 there exist r = r(F, m, Œ,) and n = n(F, m, «~), r _< n, such that Xff '~) does not possess the D F R property. However, the D F R property of the underlying distribution function F is transmitted to the distribution function of the spacings, which is shown by Gupta and Kirmani (1988) for record values and in Kamps (1995a, Theorem V. 1.15) for ml . . . . . m~-i. It should be noted that there is no restriction on the parameters needed in the proof. THEOREM 12.3. Let X . ( 1 ) , . . ,X(, n) be sequential order statistics based on an absolutely continuous distribution function F which possesses the D F R property. Then X ( r ) - X (~-1) isDFR,

2
.

Another important notion in reliability theory is partial ordering. With respect to partial ordering of ordinary order statistics and record values we refer to the monograph of Shaked and Shanthikumar (1994), and, in particular, to the articles of Arnold and Villasenor (1998) and Boland et al. (1998) in the Handbook of Statistics 16 devoted to order statistics, as well as to the many references cited in these works. Ler, throughout, X and Y be random variables with absolutely continuous distribution functions F and G, density functions f and g, respectively, satisfying F -1 (0-t-), G -1 (0-~-) ~ 0 ,

E. Cramer and U. Kamps

366

i.e., their supports are contained in the positive real line. We now introduce some partial orderings between random variables (or distributions). Stochastic ordering, failure rate ordering and likelihood ratio ordering are well known and can be found in Ross (1996, Chapter 9), e.g., st.

Stochastic ordering: X is said to be stochastically smaller than Y, written X _< Y, iff 1 - F(x) <_ 1 - G(x) for all x _> 0. f.r. Failure rate ordering: X has a smaller failure rate than Y, written X _< Y, iff f(x) < 9(x) for all x > 0. 1-F(x) -- ~

Likelihood ratio ordering: X is said to be smaller than Y in the sense of likelihood i.r.

ratio, written X < Y, iff f(x) > f(y) for all 0 < x < y. -

~(x)

-

We observe that likelihood ratio ordering implies failure rate ordering 1.r.

(X_< Y ~

f.r.

X_> Y, e.g., Ross, 1996), which itself implies stochastic ordering

f.r.

st.

(X _> Y and G(0) _< F(0) ~ X _< Y; cf. Gupta and Kirmani, 1987, Corollary 1). In Doksum (1969) the tail-ordering of distributions is introduced (cf. Deshpande and Kochar, 1983). Bagai and Kochar (1986) point out the relations between tail-ordering and failure rate ordering depending on an increasing or decreasing failure rate. Namely, if F o r G is IFR (DFR), then tail-ordering is stronger (weaker) than failure rate ordering. t

Tail-ordering: X is said to be tail-ordered with respect to Y, written X _< Y, iff G - I ( F ( x ) ) - x is non-decreasing in x, 0 _< x < F - l ( 1 ) .

Much work has still to be done in the field of partial ordering for sequential order statistics without any restriction imposed on the parameters. Theorem 12.4 states a result for likelihood ratio ordering of sequential order statistics in the particular case mi . . . . . mn-1, whereas Theorem 12.5 is valid in the general situation. THEOREM 12.4. Let the appearing sequential order statistics be based on an absoIutely continuous distribution function F. As in Theorem 12.1 we denote the rth sequential order statistic by ~X(.r'n) to indicate the sample size n. Moreover, let m l = m2 . . . . . m, say, and A = O~nl - (~?l 2 -}- (nl -- n2 -- (rl -- r2))(m + 1) and B = ~nl -- 0c/,/2 -t- (t71 -- n 2 ) ( m

+ 1).

I f the parameters satisfy the conditions

rl _> r2

and

min(A,B) _< 0 ,

(12.1)

we have

x(.rl,.~) '¢_x~:2, 2) It is known from the literature that ordinary order statistics as well as record values are partially ordered in the sense of likelihood ratio. The assertions are contained in Theorem 12.4. In the case of ordinary order statistics (m = 0,

Sequential k-out-of-n systems

367

en~ = c % = 1 , cf. Table 2), condition (12.1) simplifies to rl _>r2 and nl - n2 _< rl - r2 (cf. C h a n et al., 1991) for nl = n2, Shaked and Shanthikumar, 1994, 1.C.21, for rl = r2 = 1, n2 = nl + 1 and for rl = nl = r2 + 1 = n2 + 1!. In the case o f kth record values (m = - 1 , cf. Table 2), condition (12.1) reduces to rl >_ 1"2 and kl _< kz (cf. K o c h a r , 1990 for kl = kz = 1) . Obviously, in the situation of T h e o r e m 12.4, we obtain that the corresponding h a z a r d rates are ordered, since likelihood ratio ordering is stronger than failure rate ordering. However, for this case, we are able to show a result which generally holds true for sequential order statistics.

X,(1),..

THEOREM 12.5. Let , X (n) be sequential order statistics based on an absolutely continuous distribution function F. Then X(f-1) f ~ x (~)

2 < r < n

F o r the respective results in the particular cases o f ordinary order statistics and record values we refer to Boland et al. (1994), Shaked and S h a n t h i k u m a r (1994) and Boland et al. (1998), and to Baxter (1982). Finally, we deal with the c o m p a r i s o n o f different sequential k-out-of- n systems via the stochastic order o f contrasts, i.e., o f differences o f sequential order statistics f r o m different distributions. It is guaranteed by assuming tail ordering o f the respective underlying distributions. The theorem is stated in K a m p s (1995a, T h e o r e m V.2.7) with a restriction on the parameters. In the proof, this restriction is seen to be superfluous. As particular cases, the results o f Bartoszewicz (1985) for ordinary order statistics (ml = m2 . . . . . m, say, m = 0) and of K o c h a r (1990) for record values (m = - 1 ) are included.

X(1),..

THEOREM 12.6. Ler ,X(, n) and y,(1), .., y,(~) be sequential order statistics based on distributions functions F and G, respectively, let F be continuous and let t

X «_ Y f o r random variables X ~ F and Y ~ G. Then f o r 1 < r < s < n we have

x,(s~_ x,(r/~ y,(,/_ y?/ Acknowledgement The authors would like to express their kindest thanks to Mrs. Theresia Meyer for the excellent T E X i n g o f the manuscript.

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David, H. A. (1981). Order Statistics, 2nd edn. Wiley, New York. Deshpande, J. V. and S. C. Kochar (1983). Dispersive ordering is the same as tail ordering. Adv. Appl. Probab. 15, 686-687. Doksum, K. (1969). Starshaped transformations and the power of rank tests. Ann. Math. Statist. 40, 1167 1176. Dyer, D. D. and J. P. Keating (1980). On the determination of critical values for Bartlett's test. J. Amer. Statist. Assoc. 75, 313-319. Engelhardt, M. (1995). Reliability estimation and applications. In The Exponential Distribution, pp. 7191 (Eds. N. Balakrishnan and A. P. Basu). Gordon and Breach, Amsterdam. Enis, P. and S. Geisser (1971). Estimation of the probability that Y < X . J. Amer. Statist. Assoc. 66, 162-168. Epstein, B. (1957). Simple estimators of the parameters of exponential distributions when samples are censored. Arm. Inst. Statist. Math. 8, 1556. Epstein, B. and M. Sobel (1954). Some theorems relevant to life testing from an exponential distribution. Ann. Math. Statist. 25, 373 381. Freund, J. E. (1961). A bivariate extension of the exponential distribution. J. Amer. Statist. Assoc. 56, 971-977. Gajek, L. and U. Gather (1989). Characterizations of the exponential distribution by failure rate and moment properties of order statistics. In Extreme Value Theory, pp. 11~124 (Eds. J. Hüsier and R. D. Reiss). Springer, Berlin. Gather, U. (1988). On a characterization of the exponential distribution by properties of order statistics. Statist. Probab. Lett. 7, 93-96. Gather, U., U. Kamps and N. Schweitzer (1998). Characterizations of distributions via identically distributed functions of order statistics. In Handbook ofStatistics, Vol. 16, pp. 257-290 (Eds. N. Balakrishnan and C. R. Rao). Elsevier, Amsterdam. Ghosh, M. and A. Razmpour (1984). Estimation of the common location parameter of several exponentials. Sankhyä Ser. A 46, 383-394. Ghosh, M. and D. Sun (1997). Recent developments of Bayesian inference for stress-strength models. In Frontiers in Reliability. The Indian Association for Productivity Quality and Reliability (IAPQR). Glaser, R. E. (1976). The ratio of the geometric mean to the arithmetic mean for a random sample from a gamma distribution. J. Amer. Statist. Assoc. 71, 480-487. Glaser, R. E. (1982). Bartlett's test of homogeneity of variances. In Encyclopedia ofStatistical Sciences, Vol. 1, pp. 189-191 (Eds. S. Kotz and N. L. Johnson). Wiley, New York. Gradshteyn, I. S. and I. M. Ryzhik (1994). Table oflntegrals, Series, andProducts, 5th edn. Academic Press, Boston. Graybill, F. A. (1983). Matrices with Applications in Statistics, 2nd edn. Wadsworth, Belmont. Gumbel, E. J. and R. D. Keeney (1950). The extremal quotient. Arm. Math. Statist. 21, 523-538. Gumbei, E. J. and J. Pickands III (1967). Probability tables for the extremal quotient. Ann. Math. Statist. 38, 1541 1551. Gupta, R. C. and S. N. U. A. Kirmani (1987). On order relations between reIiability measures. Comm. Statist. Stochastic Models 3, 149-156. Gupta, R. C. and S. N. U. A. Kirmani (1988). Closure and monotonicity properties of nonhomogeneous Poisson processes and record values. Probab. Engrg. Inform. Sei. 2, 475-484. Harsaae, E. (1969). On the computation and use of a table of percentage points of Bartlett's M. Biometrika 56, 273-281. Harter, H. L. (1965). Point and interval estimators, based on m order statistics, for the scale parameter of a Weibull population with known shape parameter. Teehnometries 7, 405-422. Harter, H. L. (1988). Weibull, log-Weibull and gamma order statistics. In Quality Control and Reliability, Handbook ofStatistics, Vol. 7, pp. 433-466 (Eds. P. R. Krishnaiah and C. R. Rao). NorthHolland, Amsterdam. Harter, H. L. and A. H. Moore (1965). Maximum-likelihood estimation of the parameters of gamma and Weibull populations from complete and from censored samples. Teehnometries 7, 639-643; Erratum (1967), Technometrics 9, 195.

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Nevzorov, V. B. (1987). Records. Theory Probab. Appl. 32, 201-228. Nevzorov, V. B. and N. Balakrishnan (1998). A record of records. In Order Statisties." Theory and Methods, Handbook of Statisties, Vol. 16, pp. 515 570 (Eds. N. Balakrishnan and C. R. Rao). Elsevier, Amsterdam. Patel, J. K. (1983). Hazard rate and other classifications of distributions. In Encyclopedia of Statistical Sciences, Vol. 3, pp. 590-594 (Eds., S. Kotz and N. L. Johnson). Wiley, New York. Pfeifer, D. (1982a). Characterizations of exponential distributions by independent non-stationary record increments. J. Appl. Probab. 19, 127-135. Correction: 19, 906. Pfeiler, D. (1982b). The structure of elementary pure birth processes. J. Appl. Probab. 19, 664~667. Pham-Gia, T. and Q. P. Duong (1989). The generalized beta and F distributions in statistical modelling. Math. Comput. Modelling 12, 1613-1625. Pike, M. C. (1966). A method of analysis of a certain class of experiments in carcinogenesis. Biometrics 22, 142-161. Raqab, M. Z. and W. A. Amin (1997). A note on reliability properties of k-record statistics. Metrika 46, 245351. Rényi, A. (1953). On the theory of order statistics. Acta Math., Acad. Sci. Hungar. 4, 191 231. Rockette, H., C. Antle and L. A. Klimko (1974). Maximum likelihood estimation with the Weibull model. J. Amer. Statist. Assoc. 69, 246 249. Rosenbusch, K. (1997). Varianzhomogenitätstests und ihre Anwendungen. Master's thesis, Aachen University of Technology. Ross, S. M. (1996). Stochastic Proeesses, 2nd edn. Wiley, New York. Roy, S. N. and A. E. Sarhan (1956). On inverting a class of patterned matrices. Biometrika 43, 227-231. Sarhan, A. E. (1954). Estimation of the mean and standard deviation by order statistics. Arm. Math. Statist. 25, 317-328. Sarkar, T. K. (1971). An exact lower confidence bound for the reliability of series system where each component has an exponential time to failure distribution. Technometrics 13, 535-546. Schervish, M. J. (1995). Theory of Statistics. Springer, New York. Scheuer, E. M. (1988). Reliability of an m-out-of-n system when component failure induces higher failure rates in survivors. IEEE Trans. Reliab. R 37, 73 74. Sen, P. K. (1986). Progressive censoring schemes. In Encyclopedia ofStatistical Sciences, Vol. 7, pp. 296 299 (Eds. S. Kotz and N. L. Johnson). Wiley, New York. Serfling, R. J. (1980). Approximation Theorems ofMathematical &atistics. Wiley, New York. Shaked, M. and J. G. Shanthikumar (1994). Stochastic Orders and Their Applications. Academic Press, Boston. Shanthikumar, J. G. and L. A. Baxter (1985). Closure properties of the relevation transform. Naval Res. Logist. Quart. 32, 185-189. Stacy, E. W. (1962). A generalization of the gamma distribution. Ann. Math. Statist. 33, 1 I87-1192. Sukhatme, P. V. (1937). Test of significance for samples of the 72-population with two degrees of freedom. Ann. Eugenics 8, 52 56. Takahasi, K. (1988). A note on hazard rates of order statistics. Comm. Statist. Theory Methods 17, 41334136. Tong, H. (1974). A note on the estimation of Pr(Y < X) in the exponential case. Technometries 16, 625; Correction (1975), Teehnometries 17, 395. Tong, H. (1975). Letter to the editor. Teehnometrics 17, 393. Tong, H. (1977). On the estimation of Pr{Y _<X} for exponential families. IEEE Trans. Reliab. R 26, 54-56. Varde, S. D. (1970). Estimation of reliability of a two exponential component series system. Technometries 12, 862875. Viveros, R. and N. Balakrishnan (1994). Interval estimation of parameters of life from progressively censored data. Technometries 36, 84-91. Weinman, D. G. (1966). A multivariate extension of the exponential distribution. Ph.D. thesis, Arizona State University.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

] "~

I ù.y

Progressive Censoring: A Review Rita Aggarwala

1. Introduction

There are many scenarios in life-testing and reliability experiments in which units are lost or removed from experimentation before failure. The loss may occur unintentionally, or it may be designed into the study. Unintentional loss may occur, for example, in the case of accidental breakage of an experimental unit, or if an individual under study drops out, or if the experimentation itself taust cease due to some unforeseen circumstances such as depletion of funds, unavailability of testing facilities, etc. Pre-planned and intentional removal of units may be considered in order to free-up testing facilities for other experimentation, to save time and cost, to physically disassemble live units at various points during experimentation, or to exploit the straightforward analysis that often results. In some cases, when there are live units on test, intentional removal of items or termination of the experiment may be due to ethical considerations. Although the idea of progressive censoring dates back almost half a century (Herd, 1956), the subject of progressive censoring has received considerable attention in the past few years, due in part to the availability of high speed computing resources, which make it both a feasible topic for simulation studies for researchers, and a feasible method of gathering lifetime data for practitioners. Progressive censoring allows for the removal of items at various points during experimentation, and has also been called multi-censoring or multi-stage censoring. Conventional censoring schemes form a subset of the possibilities available to the experimenter under progressive censoring. This summary is intended to be an overview of rauch of the work which has accumulated in the area of progressive censoring. For a thorough overview of the subject, the reader is referred to Balakrishnan and Aggarwala (2000). Section 2 will explore the current literature and introduce notation. Section 3 will discuss obtaining moments of progressively Type-II right censored-order statistics, a subject for which a large amount of literature has been written. Section 4 will address inference under progressive censoring. Finally, Section 5 will explore related topics, including simulation algorithms and optimal censoring schemes.

373

374

R. Aggarwala

The topics included are those which are directly applicable to the analysis of progressively censored samples. Mathematical justification has been included for many of the results given. A number of additional mathematical results have been obtained for progressively censored samples, and are not included here. The reader is referred again to Balakrishnan and Aggarwala (2000) for a more extensive account of mathematical properties of progressively censored order statistics.

2. Review and notation

Consider the following Type-I censoring scheme: m censoring times, Tl,. •., Tm are fixed such that at these times, R1,..,Rm surviving units are to be randomly removed (censored) from the test, respectively. Here, the Ris are fixed, with the provision that there a r e R i surviving items at time Ti, i = 1 , 2 , . . , m - 1, i.e. the number of items removed at time T/is R °bs = min(Ri, number of items remaining at T/). Furthermore, the experiment terminates at time Tm with R,cbs being the number of surviving units at that time. This type of censoring is referred to as progressive Type-I right censoring, and is a generalization of conventional Type-I one-stage right censoring, where Ri = 0, i = 1 , 2 , . . ,m - 1 so that no items are removed from experimentation until some pre-determined time Tm. Progressive Type-I right censoring has been discussed by Cohen (1963, 1966, 1975, 1976), Ringer and Sprinkle (1972), Wingo (1973), Cohen and Norgaard (1977), Bain (1978), Sherif and Tan (1978), Lawless (1982), Nelson (1982), Gibbons and Vance (1983), Cohen and Whitten (1988), Balakrishnan and Cohen (1991), Bain and Engelhardt (1991), Wingo (1993), and Wong (1993) for various lifetime distribution and scenarios. Gajjar and Khatri (1969) have considered a case of progressive Type-I right censoring in which at each time of removal T~, the population parameters change (for example, due to adjustment of temperature settings). These references have mainly developed likelihood techniques for estimation of multiple population parameters. A similar Type-II censoring scheme can be constructed, which we can naturally call progressive Type-H right censoring. Under this censoring scheme, n units are placed on life-test at time zero. Immediately following the first failure, R1 surviving units are to be removed from the test at random. Then, immediately following the second observed failure, R2 surviving units are to be removed from the test at random. This process continues until, at the time of the mth observed failure, the remaining R m = n - R 1 - R 2 . . . . . Rm_ 1 - m units are all to be removed from the experiment. In this censoring scheme, R1,R2,..,Rm (and therefore m) are all pre-fixed. The resulting m ordered values which are obtained as a consequence of this type of censoring are appropriately referred to as progressively Type-H right censored order statistics. Note that if R1 = R2 . . . . . Rm-1 = 0 so that Rm = n - m, this scheme reduces to conventional Type-II one-stage right censoring, where just the first m usual order statistics are observed (see Arnold et al., 1992 for a comprehensive review of usual

Progressive censoring: A review

375

order statistics). Also note that if R1 ~-R2 . . . . . Rm = 0 so that m = n, the progressive Type-II right censoring scheme reduces to the case of no censoring (complete sample case), where all n usual order statistics are observed. Both the Type-I and Type-Il progressive censoring schemes described above can be generalized in order to accommodate censoring on the left as well. In the case of progressive Type-Il censoring, we may assume that the observation of failures begins at the time of the (r + 1)th failure, at which time Rr+l surviving units are removed from the sample. The exact failure times of the r units known to have failed before this starting time are unknown. Similarly, progressive Type-I censoring can be generalized by starting the observation of failures at a pre-fixed time TL > 0. Mann (1969a, 1971), Thomas and Wilson (1972), Cacciari and Montanari (1987), Viveros and Balakrishnan (1994), Balasooriya and Saw (1998) and Montanari et al. (1998) have all discussed inference under progressive Type-II right censoring when the lifetime distributions are Weibull or exponential. Mann (1971) and Viveros and Balakrishnan (1994), among some others, have used linear inference based on progressive Type-II censoring as an approximation for progressively Type-I censored data. An interesting real application of progressive Type-II right censoring has been illustrated by Montanari and Cacciari (1988). These authors have studied the wear on insulated cables by assuming a Weibull lifetime distribution. Balakrishnan and Sandhu (1995) have discussed efficient simulation of progressively Type-II censored-order statistics, and Balakrishnan and Sandhu (1996), Aggarwala (1996), Aggarwala and Balakrishnan (1996, 19982001), Balakrishnan and Rao (1997a, b) and Aggarwala and Childs (2000) have explored inference and moment determination under progressive Type-II right censoring when samples are from truncated exponential, Pareto, power function, Laplace and uniform distributions. These authors have also established some mathematical properties which hold for arbitrary continuous distributions and general symmetric distributions. Balasooriya and Saw (1999) and Tse and Yuen (1998) have carried out computational studies for calculating moments of progressively Type-II right censored order statistics. Balakrishnan and Aggarwala (2000) have included a discussion of optimal censoring schemes for progressively Type-Il right censored samples in their work. Since the number of observed failures is known in advance in progressive Type-II censoring, linear inference and other mathematical properties are tractable in many cases. In addition, right censoring schemes are generally more common than left censoring schemes in industrial applications. Therefore, we will focus mainly on progressive Type-H right censoring in this review. For a more thorough discussion of a range topics in progressive censoring, the reader is referred to Balakrishnan and Aggarwala (2000). For the purposes of this discussion, we will adopt the following assumptions and notation: Suppose n independent units are placed on a life-test with the corresponding failure times X1,)22,.. ,X~, being identically distributed with cumulative distribution function F(x) and probability density function f(x). Suppose further that the pre-fixed number of failures to be observed is m and that the

R. A g g a r w a l a

370

progressive Type-II right censoring scheme is (R1,R2,.. ,Rm). Then, we shall . ~~(Rt ,R2 .,.Rm) denote the m completely observed failure times DyXi:m: n ' ' , i = 1 , 2 , . . , m. For simplicity in notation, when it is clear as to what the censoring scheine is, we will use the simplified notation Xi:m:n,i 1 , 2 , . . ,m, to denote these failure times bearing in mind that these still depend on the particular choice of (R1, R a , . . . , Rm) used. These completely observed failure times are the progressively T y p e - H right censored order statistics from F ( x ) arising from a sample of size n with censoring scheme (R1, R 2 , . . , Rm). It should be noted that X~:m:~is not the same as X~:n, t h e / t h usual order statistic from a sample of size n from F ( x ) for i _> 2. This is clearly evident from the fact that there is a possibility that the unit corresponding to the ith ordered failure time from the original sample of size n may be removed (or censored) before the observation of the /th progressively Type-II right censored order statistic. Therefore, we can say that ~ : m : n ~ X i : n . It is true that X - l : m : n : X I : n , since no censoring takes place before the first failure. It is not immediately evident what the probability density function for t h e / t h progressively Type-II right censored order statistic should look like. However, using simple probability arguments, we can write down the j o i n t probability density function of all m progressively Type-II right censored order statistics as =

m

fXZ:m:ù~V2..........)&,ù:n(Xl,X2,.. ,Xm) = « ]--[ f ( x i ) { 1 - F(xi) } R', i=1

Xl ~.X2

< "'" <X m

,

(1)

where C =

rl(n

-- R 1 -

1 ) . . . (n

-

R 1 -

R 2 .....

Rm_ 1 - m + 1)

.

Note that immediately preceding the first observed failure, n units are still on test; immediately preceding the second observed failure, n - R 1 - 1 units are still on test, and so on; finally, immediately preceding the mth observed failure, n - R1 - R2 . . . . . Rm-I - m + 1 are still on test. Having already motivated the study of progressively Type-II right censored order statistics from a life-testing point of view, we may now just regard them as a set of special order statistics, and allow the support of the parent distribution F ( x ) to range over the entire real line. In fact, even in reliability studies, data which take on only positive values are sometimes modeled by normal and extreme value distributions, with location and scale parameters suitably adjusted so that the probability of obtaining a negative value is very small; see Bain and Engelhardt (1991).

3. Moment determination

Much of the literature on progressive Type-II censoring addresses the issue of obtaining the moments (usually first and second order) LOI _« Ai:m: v(&,Æ2,-.,Rm) n . Moments

Progressive censoring: A review

377

can then be used to develop inference for parameters and functions of parameters in lifetime distributions, and are also often used in conducting goodness of fit studies and model selection. In this section, we will discuss obtaining moments of progressively Type-II right censored order statistics through explicit expression, recursion, and approximation. Bounds on moments of progressively censored samples have been recently established by Balakrishnan et al. (1999a), although they are not discussed here. An extensive discussion on moments of progressively censored order statistics from symmetric distributions, as weil as a few additional moment relationships, is given in Balakrishnan and Aggarwala (2000), as are many of the results presented here. 3.1. Explicit moments by distribution 3.1.1. The exponential distribution In developing explicit expressions of moments of progressively censored order statistics, we will begin with the familiar exponential distribution. Let us consider a progressively Type-II right censored experiment in which lifetimes of the units are assumed to have come from a (standard) exponential distribution with probability density function f(x)=e

x,

x>0

,

(2)

and with cumulative distribution function F(x)= 1-e-X,

x>0

.

(3)

Snkhatme (1937) established the following theorem for usual order statistics from the exponential distribution. THEOREM 1 (Sukhatme, 1937). Let Xl:n,X2:n,...,Xn:~ denote the usual order statistics from the standard exponential distribution in (2). Then, the "«pacings" defined by Z1 = nXI:~,Z2 = (n - 1)(X2:~ - X I : ~ ) , . . ,Z~ =X~:ù -Xù 1:~ are independent and identically distributed as standard exponential.

(4) []

This particular result has been used in a variety of different ways, including deriving explicit expressions for the single and product moments of usual order statistics from the exponential distribution. A generalization of this theorem has been established by Thomas and Wilson (1972), and is given as follows. THEOREM 2. Let XI:m:n,... ,Xm . . . . denote a progressively Type-H right censored sample from the standard exponential distribution, with censoring scheme (R17 . . . , R m ) . T h e " p r o g r e s s i v e l y Type-Il right censored spacings" Z1, Z2, • • •, Zm, as defined in (5), are all independent and identically distributed as standard exponential.

378

R. Aggarwala

PROOF. Let us consider the following transformation: Z 1 = nXl:m:n~

22 = ( n - R

1 -- 1)(X2 . . . . - X l : m : n ) ,

23 = (n - R 1

-R2

- 2)(Äz3 ..... - 2 2

Zm = (n - R ~ . . . . .

. . . . ),

1)(Xm.... - X m

Rm-1 - m +

1.... )

(5)

,

In order to derive the distribution ofZ~, Z 2 , . . , Zm defined in (5), we first consider the inverse transformation given by Z1

2 2 ..... - 2 3 :m:n --

x~.~.~_ •

n Z1

n

Z1 n

z~

~-]-

Z2

n-R1

Z2 n-R1

ZB - 1

z:

+

n

-1'

n-R1

n-R1

z~

+...~ -

1

-2'

-R2

-R1

n

.....

(6)

Rm_ 1 - m +

1

The Jacobian of this transformation is, therefore, 1, and consequently, we obtain the joint density function of Z1, Z 2 , . . , Zm as fZ~,Z2,..,Zm(Zl,Z2, . .,Zm) = C exp

--

zi

,

zi>O,

i= 1,2,.•,m

;

(7)

hence, the variables Z 1 , Z 2 , . . , Zm defined in (5) are all independent and identically distributed as standard exponential. [] Observe that if we set R1 = R2 . . . . . Rm = 0 (so that there is no censoring at all), the above transformation (5) simply reduces to that given in Theorem 1. Theorem 2 can now be used to derive exact explicit expressions for the single and the product moments of progressively Type-II right censored order statistics from the standard exponential distribution. We begin with the fact that X/:m:n = z i

j-IZJ -- (~k=0 Rk) -- j + 1

i = 1,..,m

(8)

,

where R0 = 0, and Z1, Z 2 , . . , Zm are all i.i.d, standard exponential random variables. Recognizing now that the progressively Type-II right censored order statistics have all been written as linear combinations of independent standard exponential random variables, we immediately have i E(Xi .... ) = ~

1 j-1

j---i" - (}-~~k=0Rk) -- j + 1'

i = 1, . . , m

,

(9)

Progressive censoring: A review

379

and Var(X/ .... ) =

1 , - (E~-ò

i= 1,..,m

.

(10)

R~) - j + 1

Furthermore, since we can write Xj:m:~ = X/:m:, + (terms which are independent of)(/ .... )

( j > i) ,

(11) we also readily have the property that Cov(Xi:m:n,Xj:m:n) = Var(Xi:m:n)

for j > i .

(12)

These results will enable us to develop exact inference for the parameters of both one- and two-parameter exponential distributions based on progressively Type-II right censored samples. This will be explored further in Section 4. 3.1.2. The uniform distribution

Let us move on now to the Uniform(0,1) distribution with probability density function f(u)=

1,

O
1

(13)

and with cumulative distribution function 0
F(u)=u,

1 .

(14)

Ler us denote here the usual order statistics obtained from a random sample of size n from the above uniform distribution by U1:, U 2 : , . . , U,:, One of the results that has been used quite extensively is the following result due to Malmquist (1950). THEOREM 3 (Malmquist, 1950). The random variables 1 __ Un:t/

l-U, ½-

1:ù'

1 - U.-l:, 1 -

U,,-2:,

'

(is) 1 _

Vn_ I

Vn

-

U21 n

-

1 -

UI:.'

1 - UI:,

are independent Beta random variables, distributed as follows:

380

R. Aggarwala

V l d B e t a (1, 1), V2 d Beta (2, 1),

ù.

(16)

V~_I d_ Beta (n - 1, 1), G d Beta (n, 1) .

This result has been exploited by Lurie and Hartley (1972) for developing an efficient computer generation algorithm for conventionally Type-II right censored samples from uniform and thus from any arbitrary continuous distribution. It can also be used to obtain moments ofusual order statistics from uniform distributions. Aggarwala and Balakrishnan (1998) have presented the generalization of this result for progressive Type-II censoring, along with various other mathematical properties of progressively censored order statistics. The generalization of Malmquist's result for the case of progressively Type-II right censored order statistics is presented in the following theorem. THEOREM 4. Let

Ui:m:n,i = 1 , 2 , . . , m , & n o t e a progressively Type-H right censored sample f r o m the Uniform (0, 1) distribution obtained f r o m a sample o f size n with the censoring scheme (R1,R2, ..,Rm). L e t the random variables Vi, i = 1 , 2 , . . , m, be as defined in (20). Then, Vi, i = 1 , 2 , . . , m a r e all statistically independent random variables with

V/=dBeta(i+j=m_i+l ~ Rj, I ) ,

i=l,2,..,m.

(17)

PROOF. From (1), (13) and (14), the joint density of Ui..... i = 1 , 2 , . . , m , is obtained as m

fu, ...... u2......... Um.. (tAl, b/2»' "", b/m) = C I " I ( 1 - - Ui) Ri,

(18)

i--1 O
<'''
< I ,

(19)

where, as before, c = n(n - R1 - 1)... (n - R 1 . . . . .

R m - 1 -- m -r- 1)

.

Let us now consider the following transformation: 1 -

Um

....

1 -- Um-l:m:n '

v2-

Vm

1 -

Um-1

.....

1 - Um-2:m:n ' 1 -

1--

U2

....

U1..... ' Vm = 1 - U1.... 1 -

(20)

Progressive censoring: A review

381

Upon inverting the transformation in (20), we find that Ui....

1-

12I ~, j--m-i+l

i=l,2,..,m

,

(21)

and the Jacobian of the transformation is [Iim_2 Vii-1. From (19), we then obtain the joint probability density function of the random variables 1~, 1 ~ , . . , Vm as

fvi,v2,..,V,ù(Vl,V2,---,Vm) = Using

the

factorization

m . m R i+1 j , C I I Vt--l-r-~J=m i i-1

theorem

on

(22),

0 < Vl, . . . , Um

it is clear

that

< 1

.

(22) V/ has a

Beta i + ~j=m-i+l Rj, 1 distribution, and that the variables V1, V2,.., Vm are all statistically independent. [] Theorem 4, like its counterpart for usual order statistics, will prove to be useful in facilitating the exact and explicit derivation of single and product moments of progressively Type-II right censored order statistics from uniform distributions and also in developing exact estimation methods as well as efficient simulation algorithms. For the purpose of deriving explicit moments of U,...... we first of all adopt the following notations: 1. ai = i +

~ Rj, i = 1, . . , m . j-m-i+ l ai 2. gi -i = 1 , . . ,m. 1 +ai' 1 3. f i i = ( a i ÷ 2 ) ( a i + l ) , i=l,...,m. 4. 7i=O~i--fli,

i= l,...,m

.

(23)

With these notations, note that E(Vj) = c~; and

Var(Vj) = c~}fij, j = 1 , . . , m

(24)

.

Now, beginning with E(Ui .... ) = 1 -

~ E(Vj), j=m-i+ 1

i= 1,..,m

(25)

and

Cov(Ui:m:n, Uj .... )

= C o v ( k . j = m I~Ii+l Vj,

k=m-j+llI Vk),

l
,

(26)

382

R. Aggarwala

we obtain, after some algebraic simplifications, the following expressions: E(Ui .... ) = 1 -

]YI

eJ' i = 1 , . . , m ,

j--m-i+l

Var(U,. . . . . ) ~ - C = m H i + l ( Z j ) C = m ~ i + l T J - - j = ~ _ i + l O ~ j ) , I

Cov(U/ ....... U j ..... , =

k

/

N

( k = m- _- ~ I i + l O ~. -k-) ( i = m-_- H i + , T j - - j = ~ + -l Œ- j ) ,

-

i<j

.

(27) These expressions have been used by develop exact estimation methods for parameter uniform distributions based samples. They will also be used later methods for obtaining moments.

Aggarwala and Balakrishnan (1998) to the parameters of both one- and twoon progressively Type-II right censored in this section to develop approximate

3.1.3. The Pareto distribution

Independent ratios similar to those derived for the Uniform(0,1) distribution can be established for the Pareto(v) distribution which has probability density function x > 1, v > 0

f ( x ) = vx " 1

(28)

and with cumulative distribution function F(x)= 1-x

v

x> 1 .

(29)

The result is given here without proof. (See Aggarwala and Balakrishnan, 2000) THEOREM 5. Let X1 ..... X2 . . . . . . . . ~Xm .... be a progressively Type-H right censored sample obtained by placing n independent and identical units on a liJè-test with the censoring s cheme (Rs, R2~..., Rm ) and wi th individual lifetime distribution as standard Pareto(v) with probability density function in (28). Further, let _

~:m'.n

Y1 =Xl:m:~, Y// X/ l:m:~' i = 2 , 3 , . . , m

.

(30)

Then, the random variables I71, Y2,-.., Y,, are all statistically independent and are distributed as follows: Y1 d Pareto (vn), Yi d Pareto (v(n - R1 - R2 . . . . .

Ri-1 - i + 1)),

i=2,3,..,m

.

(31)

383

P r o g r e s s i v e censoring: A r e v i e w

Using this result, we can derive exact and explicit expressions for the single and product moments of progressively Type-II right censored order statistics from the Pareto(v) distribution. For this purpose, let us use the notations 1. ~1 = v r /

2. 7 i = v ( n - R l - R 2 3. f i i -

.....

Ri-l-i+l),

~ icq _l ~ i=l,2,..,m,

i=2,3,..,m.

cq>l.

(32) 4.7i-5. (~i

i_2 ,

i=l,2,..,m,

~i -- 1 ~i--2'

~)i

Bi

~i»2.

i= 1,2,..,m

.

We then have the following expressions: i

E(X/:m:~) = H f i k , i

1 , 2 , . . ,m,

k-1 Cov(X/:m:n,Xj .... ) =

6k -k=l

fik =

fik,

1< i < j < m .

(33)

=

These values will enable one to derive explicit expressions for efficient linear estimators of the parameters of a Pareto distribution. 3.1.4. The FVeibull ( l o g - e x t r e m e value) and e x t r e m e value distributions

The Weibull distribution is perhaps one of the most popular distributions used for modeling lifetimes of experimental units. As such, a large proportion of the literature which has been published on progressive censoring is specific to the Weibull lifetime distribution. See, for example, Mann (1969a, b, 1971), Ringer and Sprinkle (1972), Thomas and Wilson (1972), Wingo (1973), Cohen (1975), Sherif and Tan (1978), Robinson (1983), Gibbons and Vance (1983), Cacciari and Montanari (1987), Montanari and Cacciari (1988), Wong (1993), Viveros and Balakrishnan (1994), Montanari et al. (1998), Tse and Yuen (1998). Because ofits intimate association with the extreme value distribution, a common approach to parameter estimation for the 2-parameter (scale/shape) Weibull distribution is to consider the log-lifetime, modeled as a 2-parameter (location/scale) extreme distribution. See, for example, Viveros and Balakrishnan (1994), Mann (1969a, b, 1971) and Thomas and Wilson (1972). Consideration of moments of these distributions is commonly associated with linear estimation. Mann (1971) and Thomas and Wilson (1972) have considered exact and approximate moments of progressively Type-II right censored order statistics from extreme value distributions. Mann (1971) has developed explicit expressions for the moments of (RI ,Rg....,Rm) X i . . . . -' » 1 < i < m for the 2-parameter extreme value distribution. Tse and Yuen (1998) have directly explored the first moment of Xr~(R~,R ......Rm) (expected . . . . -' experiment duration) from the Weibull distribution, and developed expressions for this moment when the number of items removed at each of the first m - 1

R. Aggarwala

384

failures is random. Aggarwala (1996) and Aggarwala and Balakrishnan (2000) have used the exact computational algorithm given in Thomas and Wilson (1972) to calculate moments of progressively Type-II right censored order statistics from the extreme value distribution, which were subsequently used in obtaining Best Linear Unbiased Estimates and optimal censoring schemes. We will present the general algorithm for obtaining moments of progressively Type-II right censored order statistics from arbitrary continuous distributions given in Thomas and Wilson in the next subsection. This can be used if moments of usual order statistics from the distribution of interest are known or more easily obtained. We will give the expressions derived by Mann (1971), and the approximate expressions given in Thomas and Wilson (1972) for the moments of progressively Type-II right censored order statistics from the extreme value distribution in this subsection. Suppose X l:m:n (R~'R2'''''R°')~ X Z:m:n "(Rl'R2'''''Rm), . ,~~m:m:n ~(~(RI'R2""'Rm) are progressively Type-II right censored order statistics from the Weibull distribution with cumulative distribution function

F(x)= 1-exp -

,

x>0

.

(34)

v(RI,R2,..,Rm) v(RI,R2,. Rm) (RI,R~,..Rm) Then the logarithms "l:m:n , "2:m:n , ' " , Y~:m:nof the progressively censored sample will form a progressively Type-Il right censored sample from the extreme value distribution with cumulative distribution function

G(y) = 1 - e x p - e x p / - - ~ - ) ] ,

(35)

where the location parameter of the extreme value distribution, c~= ln(a), and the scale parameter /~ corresponds to the shape parameter /? of the Weibull distribution. Thus, using extreme value theory on the log-lifetimes will allow us to effectively carry out inference on the Weibull population, keeping in mind that certain procedures, such as maximum likelihood estimation, will hold through the transformation, while others properties, such as estimator unbiasedness and distributional properties, may not. Mann (1971) gives the following expression for the joint density function of Y/ImRif2''Rm) and Yj(:Rml:22"'R')l<_i<j<_m, from a standard (c~=0, f l = 1) extreme value distribution i

j--i

9i~i(Yi,Yj) = b ~ ~-~~(-1)r+«Kr,s exp(yi + yj) r=l

s=l

× exp(-t/3«,seY~

-

t]4,«eYJ),

-0(3

where (

j-1

b=n(n-Rl-1).., n--~--~~(Rk+l) k=l

) ,

< Yi < Yj

< 0(3 ,

(36)

385

Progressive censoring." A review Kr,s = t/l(r, 0, i)r/2(r ,/)th(s , i , j ) ~ 2 ( s , j )

depends only upon the censoring scheme (R1,.. ,Rm), as do t/3..... and t/4,~, specifically,

i;~, ~r

]1

Z

rll(r'l'i)= L t=2

k=i-r

(Rk ÷ 1)

F~ i-~+,-i ~2(r'i) = [ ~ I

~

L,-2 k--i

,

t+2

]

)l (Rk+l

r=l,2,..,i-l-1,

i-l>2,

1

,

r= 2,3,..,i,i>_

2,

r+ 1

tli(i- l, l,i) = q2(1,i) = 1, i--s

(Rk + 1),

F]3,r,s z

r=

1,2,..,i,

s=

1,2,..,j-i,

k=i-r+l

q4,s =

L

(RÆ+I),

s=l,2,..,j--

i.

(37)

k-j-s+l

y(Rl ,R2, ..,Nm) i = 1 , 2 , . . ,m for the standard extreme The marginal density of "i'm'n value distribution is then given as i gi(Yi) = ~ Z ( - -

1) r+171 (r, 0, i)~2 (r, i) exp(yi - q5«eVl), - o c < Yi < oc,

r=l

(3s) where e=n(n

~5«=

-R1

- 1)...(n

~

(Rk+l),

-R1 .....

Ri 1 - i + 1 ) ,

r= 1,2,..,i.

k=i-r+ 1

Performing the required integration over these marginals in order to obtain E(Y/ .... ) and E(Y,':m:ùYy:m:n)gives i j--I

E( Yi...... Yj:m:n) =

b~

Z

(-- l )r+sKr,s~(1~3,r,s, g/4,s),

r ls--1 i

E(Y~.... ) =~; Z ( - I ) r + l t / 1 r=l

where

(r, 0, i)g/2(r, i)hl @/5,r) ,

(39)

386

R. Aggarwala

~[3(C1,«2)=2~~1C2 (c2--c1)h2(«1@c2)-}-«2[g1(Cl)]2-} -2L -

i «1)12~/ In

~

,

1 -}-~2

c1 < C%

1 {

(ca)

q~(Cl, c2) = 2clc~2 (c2 - cl)h=(cl + c2) + c~[91(c2)]2-2L 1 + -]-~-

4~(«i, «1) = ~

1

,

C1 > C2,

(7 + ln«~) 2 ,

(40)

(see Lieblein, 1953). Here, L(1 + x) is Spence's integral, which is equal to OO

Z ( - - 1 ) k + l x k / k 2 for --1 < x <

k-1

1,

hl(c) = - ( 7 -- ln«)/c, ha(c) = [7r2/6 q- (7 -c ln«)2]/c , are polygamma functions and y is Euler's constant. (See Abromowitz and Stegun, 1965.) The above moments for the extreme value distribution may be used, for example, for linear inference and moment estimation of « a n d / L They are cumbersome but explicit, and many standard mathematical software packages include such special functions in their libraries. As discussed earlier, the estimators of and/3 may then be used to obtain estimators of Weibull parameters. Thomas and Wilson (1972) used the fact that Z i : m : n = e r~.... (c~ = 0,fl = 1) form a progressively Type-II right censored sample from the standard exponential distribution. Then, using expressions (9), (10) and (12) for the means and covariances of progressively Type-II right censored order statistics from standard exponential distributions and a Taylor's expansion of ~.. . . . In(Z/ .... ) about the mean of ~:~:n, approximate means and covariances of ~-.... are obtained as =

E(Yi..... ) ~ ln[E(Zi.m.n)] •.

Var(Zi:m:n) 2[E(Z/:m:ù)]2'

Var(Zi:m:n) Cov(Y~:m:~, Yj:m:n) ~ [E(Z,".... )] [E(Zj:m:~)I' i <_j .

(41)

Thomas and Wilson (1972) compared the exact and approximate means and variances of the standard extreme value progressively censored order statistics for some selected censoring schemes, and found the above approximations for E(Yi:m:n) to be quite accurate, while the approximations for Cov(Y/:m:n, Yj:m:~) were consistently low for the cases considered.

3.2. Algoriihm for computing moments, arbitrary continuous distributions A method of obtaining the single and product moments of progressively Type-II right censored order statistics from a sample of size n from an arbitrary contin-

387

P r o g r e s s i v e censoring." A review

uous distribution, provided the mean vector and variance~covariance matrix of the usual order statistics from a sample of size n from that distribution are known, was given by Thomas and Wilson (1972). This method will be useful for computing moments of progressively censored order statistics from distributions such as the normal and log-normal, where explicit expressions for moments of progressively censored order statistics have not been developed. The method works as follows. Suppose we denote the usual order statistics (that is, order statistics from the complete sample where no progressive censoring has taken place) from a sample of size n from the distribution of interest by Zi:n,i = 1 , 2 , . . ,n. Furthermore, suppose we denote the m progressively Type-Il right censored order statistics from the sample of size n obtained with the censoring scheine (R1,R2,.., Rm) by Zj ..... j = 1 , 2 , . . , m. Then, each progressively Type-II right censored order statistic corresponds to some usual order statistic from the original n items on test, i.e., Zj:m:n= Z/(j:n where the rank of Zj ..... Kj, can take on the values Kj_I+I,Kj_I+2,..,j+RI+R2+...+Rj_I for j = 2 , 3 , . . , m and K I = I . T h o m a s and Wilson (1972) have shown that the joint probability mass function of the rank vector can be written as m

P ( K , , . . ,Km) = P(K1) I'IP(Ki I K 1 , . . ,Ke 1) with P(K1

=

1)

1

=

i=2

(42) where

e(~lK1,..,Ki

,)= {-- ~ - | v-~i-1 R

kx2..~j= 1

j t

,

. . . .

l -- Æi-1 --

1

i = 2,..,m

.

(43)

)

Thus, if all possible rank vectors (say there are M of them, M = (m_~)) n 1 can be listed for a particular censoring scheine (R1,R2, .. ,Rm), then for each rank vector, we can define an m x n indicator matrix Dl, l = 1 , 2 , . . ,M, whose (r,s)-th element is 1 i f s = Kr and 0 otherwise, so that Zps = D/Zu for some l, where Zps is the m x 1 vector of progressively Type-II right censored order statistics and Zu is the n x 1 vector of usual order statistics from the particular distribution of interest. Then, denoting E(Zù) by bu, we can obtain the means of the progressively Type-II right censored order statistics as

E(Zp~) = rt = EE(Zp, ID/) = E(Dz~u) =

D/p/ ,/=1

lt~

,

(44)

/

where p / i s the probability mass function of the rank vector corresponding to De, l = 1 , 2 , . . , M , given in (42). Similarly, denoting the variance-covariance matrix

R. Aggarwala

388

of Zu by 12u, we can obtain the variance-covariance matrix of the progressively Type-II right censored order statistics as Var(Zp~) = 1~

--~(Z~s~;) ù ù , : ES(D,ZuZ'»',I»,)

- ~~'

= s[»,(Xù

-

+ ùJù)»',l

~~'

M

= ~-~ Dl(12~ + g~~ü)D'zpz - ~~' .

(45)

i=1

Tables of means, variances and covariances of the usual order statistics are available for various sample sizes for numerous distributions. For example, the following is a complete listing of distributions for which the required moments of usual order statistics are available in Harter and Balakrishnan (1996). 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Normal distribution Exponential distribution Weibull (log-extreme value) distribution Gamma distribution Log-normal distribution Log-Weibull distribution Log-Gamma distribution Cauchy distribution Double exponential distribution Half normal distribution Logistic distribution Pareto distribution t distribution Half logistic distribution Generalized logistic distribution

Hence, using the formulas in (44) and (45), we will be able to compute the means, variances and covariances of progressively Type-II right censored order statistics fÆom these or any other continuous lifetime distribution for which we have means, variances and covariances of corresponding usual order statistics.

3.3. Recursive computation of moments Many situations arise in which moments of progressively censored order statistics can not be computed explicitly, however can be computed efficiently through recursion. This idea has been explored for usual order statistics for several distributions, see for example, Balakrishnan and Sultan (1998). For many distributions, although explicit expressions for moments of progressively censored order statistics from the un-truncated form of a distribution may be obtained, the

389

Progressive censoring: A review

same is not necessarily true for truncated forms. This is one common situation where recurrence relations can be useful. We will present in this section recurrence relations for first and second moments of progressively Type-II right censored order statistics from the doubly truncated exponential distribution. These were established by Aggarwala and Balakrishnan (1996). Similar relations have been established for the Pareto and power function distributions, and their truncated forms by Aggarwala and Balakrishnan (2000). These are also given in Balakrishnan and Aggarwala (2000). The proofs for these relations will be omitted here, in view of their lengthiness.

3.3.1. Recurrence relations for single moments The probability density function of the doubly truncated exponential distribution is e-X

f(X)-p_Q,

QI<x
•

(46)

Hefe, 1 - P is the proportion of right truncation on the standard exponential distribution and Q is the proportion of left truneation. Thus, P = 1 - e -P~ and Q = 1 - e -Qa. The eumulative distribution function of the doubly truncated exponential distribution is then given by

1

F(x) - p _ Q (e -QI - e -x),

Q1 < x < P1 •

(47)

Thus, the characterizing differential equation for this distribution is

f(X)-p1-Q_Q

F(x)=(I~QQ

1)+[l-F(x)]

.

(48)

This relation is then used in order to derive complete recurrence relations for the single and product moments of progressively Type-II right censored order statistics from the doubly truncated exponential distribution. In what follows,

A(p,q) = p ( p - R 1

- 1)(p-R1 -R2 - 2)...

× (p-R1-R2

.....

Rq-q)

.

(49)

THEOREM 6. For k > - 1 , B(O)(k+l) (k jl_ 1~ , (0) (/Q ['l-P'~pk+j 1:1:1 : *]/~1:1:1-- k p _ QB 1

/'1 - Q'~ rf~+l -- k p _

QBk31

.

(5o)

TUEOREM 7. For n >_ 2 and k > - 1 ,

#(n_l)(k+l/ i:i:o

k+l -

~

(n 1)««/ //1-P'~

(~_2)(k+1~ ( 1 - Q ) Q k + I

~~:i:~ - k v ~ ) ~ 1 : , . : ~ _ l

+

5 - 7 -0

(51)

390

R. Aggarwala

THEOREM 8. F o r 2 <_ m < n - 1, R1 >_ 1 a n d k > - 1, F(RI,...,Rm)(k+l)

1

I

[

D" (R'""R')(k)

1)/~(Rz+R2,R3,..,em)(k+l)

n ( n -- R1 -

x

-

~--1-

/'/

D

1- P

~:,,,-1:,,-1

, (RI

-~- tl -- ] 1tlffl'm:n-l"

l,R2,,Rm)(lc+l) 1 j - (n - R 1

(Rz+R2+LR3,..,R~)(k+~I

-

1)

1 -- Q

(52)

THEOREM 9. F o r 2 < m < n, R1 = 0 a n d k > - 1 , • (0'R2"'"R")(k) n ~ 1 - P ' ~ . (R2,R3,..,Rm)(k+l) BlO:'22'''''Rm)('~+') = (k T l)/.tl:m: n -- ~p~ßPtl:m l:n-1

_

(lq

__,\(R2+I.R3,.,Rm)(TM) (1p~QQ) )#l:m-l':n ~- fl

(53)

Q~I + 1

THEOREM 10. F o r 2 _ 1 a n d k > - 1 , (R 1 ....

Rm)(k+l)

B~:~:; '

1

~). (Rl,...,Rm)(k/

1 -- P

- R~ + 1 \ (k ×

[.4 A(n,i)

#(e,..,R~,R~+R~+,e~+2,..,e°,) ~+~1

( n ~ 1-5-- 1) /:,ù-1:o-1 A(n,i-

1)

A(n,i-

1)

+AÕ;--~ii-x 1)

p(R1,..,R~ 2,R~_I+aùR~+~,..,Rm)(k+~ R /k+~l

:~:~-

J

- (n - R1 . . . . .

i" (Rl'""Ri-l'Ri+Ri+l--l'Ri+2""'Rm)(k+l) R i -- )#i:m-lln

+ (n - R 1

Ri-1 - i+

X

.....

1)

#(R,...,Æ~ 2,R~_~+R~+I,R~+,. .,(~+~/} Ro,) i-l:m l:n

THEOREM 1 1. F o r 2 - 1,

(54)

Progressive censoring." A review

391

Bi:m:'f1(Rl ... .Æi. 1. 0. Ri+l . ... Rm) (k+l) = (k + _;t~i:m:n| ] , (R1,..,Ri-1,0,Ri+I,...,R~~) (k)

1-P

AÕ;-i[;;-2) [g(R~,..,R~_~,R~+,e~+~,..,R~)(~+~)_/~(R~,..,R,_~,R,_~,e~+~,..,Ro,)(~+I)] X [ i:m-l:n-1

i-l:m-l:n-1

J

-- (n -- R 1 . . . . .

"R :~ ,(RIù. ,Ri 1,Ri+l--l,Ri+2,.., m)~(~+1) R i - 1 - t)#i:m l:n

+ (n - R1 . . . . .

Ri-1 - i + l j i-i:m-l:n

"~[~(RI,...,Ri 2,Ri l+l,Ri+I,...,Rm) (k+l)

(55)

THEOREM 12. For 2 <_ m <_ n -- 1, Rm >_ 1 and k > - 1 , B(Rl ...Rm)(k+l) m:~£:n'

__

1 { Rm-+ l (k + l )u'(Rl"'"R")(kl +

[ A(n,m-- 1)

1- P

#(N,,..,N~ 2,XmI+R,ù)(~+')

X [A(n - 1,m - 2) m l:m-l:n-1 A ( n , m - 1) R ' (R~,..,R,, 1,R,ù-1)(~+~)] -A~~l--,m -1) mt~. . . . . -1 J +(n-R1

.....

R,n-1-m+l)

X ]~(RI,...,Rm 2,Rm-l÷Rm+l) (k+I) I m l:m-l:n

THEOREM 13. For 2 < rn < n,

R m =

(»6)

J" "

0 andk>

-1,

B(mR~;~»Rm1 '0)(k+') = (k + 1)#m:m:n,... (R~R,ù 1,0)(k) + (n - R 1 X

-R2

.....

Rm-I - m + 1)

B(Rl,...,Rm 2,Rm-l+l) (k+l) m-l:m-l:n

1-P A ~ ~ - i~, ~n -- 2)

rp~+l X L i

. (~~, ,~,ù-~s,,-l;~+l~l --]Xm

l:m l : n - 1

J

"

(57)

The recurrence relations presented in this section are complete in the sense that they will enable one to compute all the single moments of all progressively Type-II right censored order statistics from doubly, right (Q--+ 0), and left (P--+ 1) truncated exponential distributions for all sample sizes and all censoring schemes. Also, if we let P --+ 1 and Q ---+0, these results reduce to the recurrence relations for single moments of progressively Type-II right censoredorder statistics from the standard exponential distribution (see Aggarwala and Balakrishnan, 1996).

R. Aggarwala

392

3.3.2. Recurrence relations f o r p r o d u c t m o m e n t s

Again exploiting the characterizing differential equation for the doubly truncated exponential distribution, we arrive at the following recurrence relations for product moment of progressively Type-II right censored order statistics. THEOREM 14. For 1 _ 1, B!Rl,...,Rm) ',, ....

1 :Rj~-

f (RI,...,Rù,) (l-P) i ~ ~i:":° - ~-0

[ A(n,;)_ x '[A(n- 1,j-

1)

bt(R,...& ,R:+R,+I,R,+2,..:Ro,) ij:m-l:n-1

A(n,j1) #(R~,..,R:_2,R:<+R»Rj+~,..,Rm) -A(nTT,j--2) i,j-l:m-l:~-i A(n,j-

+ A[dTT,7-- (n -- R1 .....

1) R . ' (R~,..,R, 1,Rj--I,Rj+I,...,I~m)] 1) Jt*i,j:m:n-1 ] j) # (RI ,.. ,Rj-I ,Æ]q-Rj+l+I,Rj+2,..,Rm) Rj -

+ (n - < . . . . .

] i,j:m-l:n

Rj_l - j + 1)

X ]Ai,j_l:m_l: n(RI'''''Rj-2'Rj-I+Rj+I'Rj+I'''''R") }

(58)

THEOREM 15. For 1 < i < j < rn - l, m < n and R j = O,

B(RI,...,Rj_I,O,Rj+I,...,Rm) #(R1,..,R]_I,O,Rj+I,...,Rm) i,j:m:n

= i....

__

(l-P)A(n,j~Z~

A (n--- l - j - - 2 )

i,j l:m-l:n-1

J

1 ..... Rj_ 1 - j ) • (RI,...,Rj-I,Rj+I+I,Rj+2,..,Rm) X tAij:m_l:n

-- ( n - R

..... Rj 1 - j + l ) (RI,...,Rj-2,R:-~+I,Rj+I,...,Rm)

+(n-R1

)< ~/i,j l:m-l:n

THEOREM 16. For 1 1,

(59)

Progressive censoring: A review

Rm+l "i:m:~

i,m:~:n

-

E A(n,m-~ X -- A ~ 7 i ; m

393

V2- d

.(R1,..,Rm2,Rm-l+em) 2)#i,m_l:m_l:n_ 1

A ( n , m - 1) (Rl ...eù, 1Rm-1)] B A~-_--iT, m_--l)~'~#i,m:'m:;;-1 ' J q- (n -- R 1 . . . . .

(el,. eo, l:'n 2em-~+eù,+l/} . (60) Rm-1 - m + 1")#i,m-1;m

THEOREM 17. For 1 < i < m - 1, m < n and R,~ = O, (RI,...R~_,0) Bi,m:m:n =

(R1 ...Rm_l.0) _ ( 1 -- P ~ A ( n , m - ? ' I p _ Q j A(n - 1,m

2)

#km:n'

~r~ . (RI,...,Rm 1) (RI ...Rm-2,Rù, )< l't-l~i:m-l:n 1 --#i,m"-l':m-l:n-1

+ (n - R1 . . . . .

.,

1)']

3 (Rl,....Rm-2,Rm

Rm-1 -- m @ l)#i,m_l':m

l:n

(61)

1+1)

Using these recurrence relations, we can obtain all the product moments of progressively Type-II right censored order statistics from the doubly truncated exponential distribution for all sample sizes and all censoring schemes ( R 1 , . . , R m ) . The recursive algorithm is described in the following subsection. It is interesting to note that for the case of the doubly truncated exponential distribution, explicit expressions for the single and product moments of progressively Type-II right censored order statistics are not readily obtained in an explicit form. Therefore, the recurrence relations presented in this section will be useful to efficiently calculate the required moments in a simple recursive manner.

3.3.3. Recursive algorithm Using the recurrence relations just presented, the means, variances and covariances of all progressively Type-II right censored order statistics from doubly truncated exponential distributions can be readily computed as follows: Setting k = 0, (50) will give us the value #(0) 1:1:1 , which in turn, again using (50) with k 1, will give us the value /~1:1:1 • (°)(2/• From these values, we can recursively compute #I~~1) and /q:l:n • (n a)(2~ for n = 2, 3 , . . using (51). Thus, all the first and second moments with m = 1 for all sample sizes n can be obtained. Next, using (53), we can determine all moments of the form #(0,n-2) 1:2:n , n = 2, 3 , . . which can in turn be used, with (53), to determine all moments of the ~'orm B(0,n 2) (2) l:2:n n = 2, 3, Eq. (52) can then be used to obtain #(R1,R2) for R1 = 1,2, and n >_ 3, and these values can be used to obtain all moments of the form (n-2,0) and # (Rl'R2)/2/l:2:"using (52) again. Now, (57) can be used again to obtain ~2:2:~ '

"

"

l : 2 : n

"

•

•

•

# 2:2:n (~-2'°)/2/ for all n and (56) can be used next to obtain, for R1,R2

~

1~ 2,

"' '

and

n _> 3, all moments of the form #~R~ff2) and # (R~'R2)(2/2:2:n. This process can be con-

R. Aggarwala

394

tinued until all the desired first- and second-order moments (and therefore all variances) are obtained. From (61), all moments of the form #(<,..,Rm_l,0) m--l,m:m:~ , m = 2, 3 , . . , n , can be determined, since only the single moments, which have already been computed, are needed to calculate them. Eq. (60) can then be used to obtain #(R~,..,ëù,) m 1,m:m:n for

• (RI,'",Rj-I,O,Rj+I,'",Rm)

Rm = 1 , 2 , . . .

Then, using (59), all moments of the form Vj-l,j:m:~ (R~,...R°,) . (j < m) can be obtained, and using (58), all moments of the form #)-l,j:m:~, J < m , Rj > 1, can be calcnlated. From this point, using (61) and (60), we can obtain all • (R,,..,Rm) moments of the form #m--2,m:m:n and, subsequently, using (59) and (58), all mo(R,,..,R,n) ments of the form #)-2j:m:n, J < m, can be determined. Continuing this way, all the desired product moments (and therefore all covariances) can be obtained. As mentioned earlier, efficient recurrence relations have also been established by Aggarwala and Balakrishnan (2000) for progressively censored samples from the Pareto and power function distributions and their truncated forms.

3.4• First-order approximations to moments Using the explicit results and notation we obtained in Section 3.1.2, for progressively Type-II right censored order statistics U/I~I:~''Rm) from the Uniform (0, 1) distribution, we can obtain approximations to means, variances and covariances of progressively Type-II right censored order statistics from an arbitrary continuous distribution F(.) using the inverse probability integral transformation, d -1((R1,..,Rm)) -E(Rl""'g~),:m:n= F Ui:m: n ,

(62)

where F -1 (.) is the inverse cumulative distribution function of the lifetime distribution from which the progressively censored sample has come, and E (el ' ' 'Rm) - t:m:n are the progressively censored order statistics from that distribution. Expanding the function on the right-hand side of (62) in a Taylor series around (R~I Rm) m E(Ui:m:n'" ) = 1 -- I~j=m-i+l eJ = FIi and then taking expectations and retaining up to the first term, we obtain the approximation

E(I<(e~'"R»')~ k:,:m:n / --~ F -1 (Il/),

i= 1,2,..,m

,

Proceeding similarly, we obtain the approximations ~,/v(<,..,e,o)~ 2 (R~ R,ù) V«,~, i:ù,:~ j - ~ {F ~("(II/)} Var(Ui:m:'/," )

= {F-l(l)(rIi)}2C=m~ii+l)~J-j=~_i+l°~J )

(63)

395

Progressive censoring: A review

and CA. [/v(Rl, ...,Rm) y(Rlp",Rm)~ uvkli ..... ~j:m:n J

_~ F -1°) (n,)F 10)(i~ff)Cov (u}Rt:n..,R,~), ~j:m:n(:(R*"'"Rm)'~ff =F

l(l)(I~i)F-lO)(l~j)C=m_~ii+ 1

j=m-i+l

(65) k=m -j+ 1

(d/du) F -l(u).

using notation (23) from Section 3.1.2. Here, F -I<~>(u) = This type of Taylor series approximation for the usual order statistics was given by David and Johnson (1954) and if we set R1 . . . . Rm = 0, the above approximations reduce to the first terms in the corresponding formulas given by David and Johnson (1954). If more precision is needed in the approximations, then more terms could be retained in the Taylor series expansion. Balasooriya and Saw (1999) compared first- and second-order approximations of moments of progressively Type-II right censored order statistics from the extreme value and normal distributions. For the first moment of these order statistics from the extreme value distribution, they also compared the inverse transformation method presented in this section with the approximate method using the relationship between extreme value and exponential random variables presented earlier in Section 3.1.4 and found a reasonable correspondence.

4. Inference under progressive censoring We will present in this section results for inference when observed samples are progressively censored. As the Weibull distribution is such a commonly used distribution for modeling lifetimes, a large body of work has been written regarding inference for the parameters of Weibull and extreme value distributions when observed samples are pro gressively censored. Mann (1971) presented tables of coefficients of progressively Type-II right censored order statistics for Best Linear Invariant Estimators BLIEs for sample sizes up to n = 6 from the extreme value distribution. Thomas and Wilson (1972) compared Best Linear Unbiased Estimators (BLUEs), BLIEs approximations to these, a linearized Maximum Likelihood estimator (MLE), and unweighted regression estimators for the extreme value parameters. Cohen (1975) considered maximum likelihood estimation of the Weibull parameters. Viveros and Balakrishnan (1994) discussed conditional inference, including quantile estimation, interval estimation and prediction, for parameters of the extreme value distribution under progressive censoring. Gibbons and Vance (1983) compared maximum likelihood and least squares median rank estimators

R. Aggarwala

396

for 2-parameter Weibull distributions through a simulation. Montanari and Cacciari (1988) compared a number of methods for extreme value parameter and quantile point estimation, including maximum likelihood estimation, best linear invariant estimation, least squares median rank estimation, their "equalization method" of estimation, a method proposed by Bain and Antle, and an Efron estimator. They also compared various methods of interval estimation, including Bootstrap, Jackknife, Monte Carlo, the Greenwood approximation and a modified Bain and Englehart method. Montanari et al. (1998) went on to compare estimators of Weibull parameters using maximum likelihood, least squares rank and White estimation. Aggarwala (1996) and Balakrishnan and Aggarwala (2000) have provided tables of coefficients of progressively Type-II right censored order statistics for BLUEs of the parameters of extreme value distributions under "optimal" progressive censoring schemes, for sample sizes up to n = 30. Other distributions for which inference under progressive censoring has been addressed include the exponential, gamma, uniform, normal, burr, Pareto, lognormal and laplace or double-exponential. As these cannot all be addressed here, interested readers are referred to Balakrishnan and Aggarwala (2000) for a more complete overview.

4.1. Best Linear Unbiased Estimation 4.1.1. One-parameter models Consider an arbitrary "standard" continuous distribution F(x). Then, if the single and product moments of the progressively Type-II censored order statistics from F(.) are known, the BLUEs for a scale parameter may be obtained following the steps in, for example, Arnold et al. (1992, pp. 171-173). We will briefly outline the results here. If we are to consider the linear transformation Y = 0X, where the vector X represents a vector of progressively Type-II censored order statistics from the standard distribution F(x), then the best linear unbiased estimator of 0 is obtained by minimizing with respect to 0 the generalized variance Q(O)= (Y - 0la)'~:-l(Y - 0la), where la is the mean vector of X and 12 is the variancecovariance matrix of X. The minimum occurs when B'Z1 0* - - Y .

la'12-~la

(66)

This is the BLUE of 0. The variance of 0* is then easily obtained as 02 Var(0*) -- la,£_~la . Alternate explicit expressions for the BLUE and its variance are as follows:

(67)

397

Progressive censoring. A review

y~ m ~ k = l C k'i, ' tXk i:m:n "m

(68)

O* = ~ i = 1m

and 02

Var(0*) = ~ j =ml ~ km =l

(69)

ck'J#k]2j

Here, c iJ are the elements of the inverted variance-covariance matrix of the progressively censored order statistics from the standard (0 = 1) distribution. These can offen be attained explicitly, for example, in the case when the elements of Z can be written as ai,j = &tj. 4.1.1.1. The exponential distribution For a progressively Type-II right censored sample from the standard exponential distribution, let us denote the corresponding progressively Type-II right censored order statistics by X1..... X2...... ..., Xm:m:n. NOW, let us represent the progressively Type-II right censored order statistics with the same censoring scheme from the Exponential(O) distribution, that is, with probability density function

f(y;o)=le-y/°,

y>_0, 0 > 0

,

(70)

by Y~:m:n, i = 1 , 2 , . . , m . Using the expressions for the means, variances, and covariances of Xl:m:n, X2..... ...,Xm:m:~ obtained in Section 1, we see that the m x m variance-covariance matrix is of the special form aij = sitj (where tj = 1 for all j) which can be inverted explicitly as given in Graybill (1983, p. 198). As discussed in Arnold et al. (1992, pp. 17~175), the general form for the inverse of a non-singular k × k symmetric matrix (cij), where Cij = s i t j (i < j), is given by (cij) where

I -(si+lti- siti+l) 1 • C"J =

j = i + l,

si+~t,_l s i i t i + ~

i=j=2,

(siti l--Si lti)(Si+lti--siti+l) ~ S2[Sa(S2t 1

-- slt2)] -1,

tÆ-1I&(sktk 1--Sk-ltk)] 0,

1

i=j=

1,

i=l,...,k-1 .k-1

""

(71)

i=j=k j>i+l

We can therefore obtain the exact B L U E of 0 and its variance, using (66) and (67). Explicit expressions of the BLUE and its variance are as follows: 1 m

02

O* = m~-~'(Ri~=l @ 1)~i. . . . It can be shown that freedom.

2mO*/O is

and

Var(0*) = --m

(72)

distributed as chi-square with 2m degrees of

398

R. Aggarwala

REMARK 18. It is of interest to observe that the precision of the BLUE of in (72) depends only on m and n, and not on the progressive censoring scheme

(R1,..

,Rm).

REMARK 19. Balakrishnan et al. (1999b) have considered estimation of the scale parameter 0 based on k progressively censored samples of varying sizes and censoring schemes. 4.1.1.2. The Pareto Distribution For a progressively Type-II right censored sample from the standard Pareto(v) distribution, let us denote the corresponding progressively Type-Il right censored order statistics by Xl:m:~, X 2 . . . . . ' ' ' , X m : m m . Now, let us represent the progressively Type-II right censored order statistics with the same censoring scheine from the Pareto (v, 0) distribution, that is, with probability density function f(y;v,O)=õ

,

y>0,

v>0,

0>0

(73)

by Y/:m:~, i = 1 , 2 , . . , m . Using the expressions for the means, variances, and covariances of X1..... X2:m:~,.. ,Xm .... obtained in Section 3.1.3, we see that the m x m variance-covariance matrix is again of the special form «ij = sitj which can be inverted explicitly using (71). The elements of the symmetrie tri-diagonal inverted variance-covariance matrix of X, (ci,J), are given by ~)iTi+l __ f12i f12i+~

ci,i =

i = 1,2,..,

m -- 1,

(I~~=i ~k)(])i- f12)(~)i÷1- fl2÷i) ' 1

C m~m

fii+l

ci,i--1 = i

i = 1 2,..,

m -- 1,

f12

.

c ''J = 0 otherwise ,

(74)

where cq, fii, and Yi, i = 1 , 2 , . . , m, are as given in notation (32) in Section 3.1.3. 4.1.1.3. First-order approximation to the BLUE Let us again assume that the progressively Type-II right censored sample y(R1,..,Rù,) l:m:n ~ v(RI,...,I~~) *2:m:n

V(<,..,Rm) » • • • ~ ~m:m:n

has arisen from a lifetime distribution belonging

to a scale-parameter family, that is, with density function f ( y ; O) = ~ f @ . In cases where the inverse of the variance-covariance matrix !2 is difficult to obtain, we may consider using a first-order approximation to the BLUE of 0 in (66) by making use of the approximate expressions of la and 12 presented earlier in Section 3.4. To this end, we first note from (64) and (65) that the first-order approximation to the variance-covariance matrix 12 = ((cij)) has the special form

399

Progressive censoring: A review ai4 = sitj,

(75)

1 < i < j <_ rn ,

where

si=FI(I)(IIi)C

~I

=m--i+1

~J-- II

j=m -i+ 1

Recall that IIi = 1 - IIjmm i+l ~J and F kl=

II

7j-

j=m-i+l

II

j=m i+l

o9

O~j),

(d/du) F -1 (lA). Letting

1(1)(ü) =

and

bi=

II

ei ,

i=m-j+l

is a symmetric tri-diagonal matrix given by / k2 i =j= {F 1(~/(ii1)}2kl (k2bl-klb2)

£ - 1 = ((ci,J))

ki+lbi-] ki-lbi+l

•

CI'J

=

{F-t0)(II~)} 2 (kibi_1 ki ibi)(ki+lbi-kibi+l)' (Æ t(l)(I~m)}2

bml bm(kmbm 1-km lbm),

-1 F-l(1)([Ig)F-I(I)(IIi+l)(ki+lbi-kibi+i) ~ 01

1,

i = j = 2,..

,m -- 1,

i=j=m,

i=j-l=l,.., otherwise

m-l,

(76) Upon using the expressions in (63) and (76), we obtain the first-order approximation to the BLUE of 0 in (66) as (Balakrishnan and Rao, 1997a) 0*

I [{ F-l(II1)k2 =Ä {F 10)(I~1))2 kl(k2b 1 -klb2) F -1 ([12) --

+

), V(RI,...,Rm)

F -1(~~ (II1)F -1~1~(1-I2) (k2b~ - klb2/

m-1V.~f B'~f'_

-f ~ l:m:n

_F-1 (i~i_l) l(1)(~Ii-1)F-l(I)([Ii)(kibi l - ki_lbi)

F-1 ([Ii+l) F-I(~/(IIi)F

1('1(IIi+l)(ki+lbi - kibi+l)

F-l(IIi)(ki+lbi-1 - ki-lbi+l)

+{F_l(1)(~

+

Y

~

7 kii_~_kibi+l)

] y(R~,..,R,,) f .i:m:n

--F-1 (Rm_l)

].F-11~l(IIm-1)F-l(ll(II,ù)(kmb,ù-I

- km 1bin)

F-1 (II~,)bm 1

B {F-l(1)(IIm)} 2 b~(kmbm

1 -km-lbm)J

y(e~,..,e~) -m . . . . .

(77)

400

R. Aggarwala

where A=

{F l(lI1)} 2 k2 {F -1(~) (]~1)} 2 kl (k2bl - klb2) {F-1 (ili)} 2 (k_i+lbi_l=ki_lbi+l) i=2 {

([Ig)} (kibi-1 - k i - l b i ) ( k i + l b i - kibi+l) {F l(IIm)} 2 bm_ 1

+

{F-l(1)(l-[m)} 2 bm(kmbm_l - km_ibm) m-1

-

2 7F"~

1(~)

F-l(IIi)F-l(l-Ii+l) I l . " k i+lbi - kibi+i) ( z ) F - l ( l l (,I I i+l)(

=

(78)

F u r t h e r m o r e , the first-order a p p r o x i m a t i o n to the variance of the B L U E of 0 in (67) is given by Var(0*) = 0 2 / A .

(79)

EXAMPLE 1. Nelson (1982, p. 228, Table 6.1) reported data on times to breakd o w n of an insulating fluid in an accelerated test conducted at various test voltages. F o r illustrating the m e t h o d of estimation developed in this section, let us consider the following progressively T y p e - I I right censored sample of size m = 8 generated f r o m the n - : 19 observations recorded at 34 kilovolts in Nelson's (1982) Table 1, as given by Viveros and Balakrishnan (1994): In this case, let us assume a scale-parameter exponential distribution with density function f ( y ; O) = ö1 e y/0,

Y -> 0 ,

0>0

as the t i m e - t o - b r e a k d o w n distribution. Using the exact results from (72), we find that the B L U E of 0 is 1

m

-- ~-~,(Rg + 1)yg. . . . . m ~.

=

9.110

with a standard error of 0 * / v ~ = 3.221.

Table 1 Progressively censored sample generated from the times to breakdown data for insulating fluid tested at 34 KV by Nelson (1982) i

1

2

3

4

5

6

yi:8:19

0.19

0.78

Ri

0

0

7

8

0.96

1.31

2.78

3

0

3

4.85

6.50

7.35

0

0

5

Progressivecensoring."A review

401

Using the first-order a p p r o x i m a t i o n to the B L U E of 0, we note that the standardized variable X = Y/O has a standard exponential distribution in which case

F(x)=l-e

x, F - l ( u ) = _ l n ( l _ u )

and

F l/'/(u)=l/(1-u)

.

F r o m (77), we determine the B L U E of 0 to be a p p r o x i m a t e l y 0* = (0.12305 x 0.19) + (0.12305 × 0.78) + (0.47255 x 0.96) + (0.12458 × 1.31) -4- (0.47153 × 2.78) -4- (0.12798 x 4.85) + (0.12808 × 6.50) + (0.82641 × 7.35) = 9.57 . F r o m (79) we have Var(0*) = 0.11713 02 so that we obtain the standard error of the estimate 0* to be

SE(O*) = 0"(0.11713) 1/2 = 3.28 . These exact and a p p r o x i m a t e values are very close, with n = 19 and m = 8.

4.1.2. Two-parameter mode& Consider again an arbitrary "'standard" continuous distribution F(x). Suppose n o w that we believe our progressively censored observations to be represented by the linear t r a n s f o r m a t i o n Y =/~1 + eX, where the vector X represents a vector of m progressively censored order statistics f r o m the standard distribution F(x); then the best linear unbiased estimators of/~ and « will be obtained by minimizing the generalized variance Q(0) = ( Y - A0)'12 -1 ( Y - A0) with respect to 0 where B = (#, «)', A is the m x 2 matrix (1, la), 1 is the m × 1 vector with c o m p o n e n t s all l's, and la is the m e a n vector of X and 12 is the v a r i a n c e - c o v a r i a n c e matrix of X. The m i n i m u m occurs when #*=-laTY

and

cr*=lTY

,

(80)

where F = 12-1(1la/- lal')12 1lA and A : (1'12 11) (laQ2-1la) -- (lt•-lla) 2. F r o m these expressions, variances and the covariance of the estimators are readily obtained as

a21a'12-11a/A, Var(«*) = 0-21112-il/A, Var(#*) =

Cov(~*, «*) = _«21a,~-i 1/~ .

(81)

Alternate explicit expressions for the B L U E s and their variances and covariance are as follows: m

j=l

m

i--1

m

m

l--1 k - 1

R. Aggarwala

402

(7*

«'i k (#1 - #~)cldYj:m:. ,

=

(83)

j = l i=l l=l k=l

where

(

/(,~~#,#jcz'j)(- ~i~=ip,c")'

A= ~-~.~ci'i i=1 j=l

"

,/ \ i = 1 j = l

(84)

i=1

and c'« are the elements of the inverted variance~ovariance matrix of X. Furthermore, V a r ( Æ * ) = r 7 2 ( ~ "i=1 ~ ~ ~1=1 --~~#ißjci'J)/A'\

(85)

Var('*)='=('~~=1~_ici'Y)/A

(86)

and (87)

4.1.2.1. The exponential distribution Consider the two-parameter exponential distribution with probability density function

f(y; ~, «) =-le-(y-")/«,

y > ~,

o-

« > 0.

(8a)

Let us denote the progressively Type-II right censored sample from this distribution by Y~:m:~, i = 1 , 2 , . . ,m. Then, again using the moments for the standard exponential distribution obtained in Section 3.1.1 and the formulas in (80), we derive, after rauch algebraic simplification, the BLUEs of g and a to be (see Viveros and Balakrishnan, 1994; Balakrishnan and Aggarwala, 2000) 1

P* = Y1....

m

n(m-- 1) +E i( :R2z

1)(~:m:n

--

Yl:m:n)

(89)

and

«,

m

_1

~( 1 = Ri @

1)(Y[ .... - Yl:m:n) ,

(90)

respectively. Further, upon using (81), the variances and covariance of these estimators are given by

Progressive censoring." A review

403

mo- 2

Var(#*) - n 2 ( m

- 1)

(91)

O-2

Var(a*)

-

(92)

m--1

and O-2

n(m-

Cov(#*,o-*) -

1)

(92)

We can show that 2(m - 1)o-*/o- has a chi-square distribution with 2(m - 1) degrees of freedom. This fact can then be used to develop confidence intervals or tests of hypothesis about o-. Using the fact that 2(Yl:m:n - #)/o-* has a chi-square distribution with 2 degrees of freedom and that the random variables 2 ( m - 1)o-*/o- and 2(Y1.... - - # ) / o - * are statistically independent, we find that u ) u t- ~ has an F-distribution with (2,2m - 2) degrees of freedom. This pivotal quantity can be used to perform tests of inference concerning #; see Viveros and Balakrishnan (1994). REMARK 20. It is important to note here that the precision of the BLUEs of # and o-, for the exponential distribution depend only on m, and n, and not on the progressive censoring scheine (R1,..,Rm). REMARK 21. A number of results have also been established for the more general case of left and right progressive censoring, or general progressive TypeII censoring. They are presented, for example, in Balakrishnan and Aggarwala (2000). 4.1.1.2. The Pareto distribution Let us denote the progressively Type-II right censored order statistics from the location-scale shifted Pareto (v) distribution, that is, with density function

f(y;v,#,a)

= - v (~~ff_) (7

~ ~, y > # + o - ,

v>0,

-oo<#
a>0 (94)

by Y~:m:n,Y2:m:n,.--, Ym.... . In other words, Y/.... = # + «X,.:m:~ where X/:m:~, i = 1 , 2 , . . . , m, are progressively Type-II right censored order statistics from the standard Pareto(v) distribution. From the expression of the inverted variancecovariance matrix for the standard Pareto (v) distribution given in (74), and the expression of the mean vector given in Section 3.1.3, we can obtain the exact best linear unbiased estimators of the location and scale parameters, # and a, respectively, using (80) (see in Aggarwala and Balakrishnan, 2000). We will illustrate this with an example.

R. Aggarwala

404

EXAMPLE 2. Consider a sample of size n = 15, and the following progressive Type-II right censoring scheme: R1 = 5,R2 = 0,R3 z 2,R4 = 0,Rs = 3 (so that m = 5). Using the inverse transformation resulting from (30), specifically, i

X/:m:,~ = H { i n d e p P a r e t o [ v ( n - R 1 - R2 . . . . .

Rj ~ - j + 1)l } , (95)

j=l

a progressively Type-II right censored sample from the Pareto (3) distribution with location parameter # = 0 and scale parameter a = 5 was generated. The generated sample was: 5.11073, 5.34932, 5.36434, 5.70137, 5.90067. F r o m (82), we are able to compute the coefficients of the progressively Type-Il right censored order statistics corresponding to the B L U E for #. These are: 8.00439, - 1.18584, -2.71754, -0.94208, -2.15893. Using (83), the coefficients corresponding to the B L U E for « are: -6.84874, 1.15948, 2.65715, 0.92115, 2.11096. Therefore, the estimates of the location and scale parameters based on this sample are #* = 1.87680,

a* = 3.16207 ,

and their standard errors and estimated covariance are SE(#*) = 1.83036,

SE(c*) = 1.79113,

Cov(#*,«*)=-3.27578

.

It should be noted here that it has been assumed that the shape parameter (v = 3) is known. This may not always be the case. Aggarwala and Balakrishnan (2000) have conducted a sensitivity analysis with respect to v for this example in the event that this parameter is not known to the practitioner and is obtained through some form of estimation.

4.1.2.3. Extreme value distribution F r o m (35), the standard extreme value distribution has cumulative distribution function given by

F(y) = 1 - e x P l - exp(x)] . In Section 3.1.4, we discussed expressions for first and second moments of progressively censored order statistics from the extreme value distribution, as well as first-order approximations to these. In Section 3.2, we also discussed an alternative method for obtaining exact moments of progressively censored order statistics if moments of the usual order statistics are known. Using this alternative method and tables of means, variances and covariances of the usual order statistics for the standard extreme value distribution given in Balakrishnan and Chan (1992) for sample sizes up to n - - 3 0 , Aggarwala (1996) numerically determined coefficients of progressively censored order statistics for obtaining location and scale BLUEs based on progressively Type-II right censored samples from extreme value distributions for various sample sizes and censoring schemes.

405

Progressive censoring." A review

This was also performed for the normal and log-normal distributions. (see also Balakrishnan and Aggarwala 2000). EXAMPLE 3. Consider a progressively Type-II right censored sample from the extreme value distribution with censoring scheme (22, 0, 0), that is, all censoring is performed immediately following the first observed failure out of n = 25 items on test. In order to determine location and scale BLUEs, we must first compute the (single and product) moments of the progressively Type-Il right censored order statistics using the moments of the usual order statistics given in Balakrishnan and Chan (1992). We then use (80) to obtain the following coefficients for #* : 0.0169, 0.0804, 0.9027, and for «* : -0.2256, -0.0876, 0.3132. Furthermore, using (81), we have Var(#*) = 0.5613« 2 and Var(a*) = 0.13110-2. In comparison, for the conventional Type-Il right censoring scheme (0, 0, 22), we can compute (simply using the moments of the usual order statistics) Var(#*) = 2.9046a 2 and Var(a*) = 0.48470- 2. These values are much higher in the case of conventional censoring than for our selected scheme, and as it turns out (see Aggarwala, 1996; Balakrishnan and Aggarwala, 2000), we find that based on a sample size of n = 25 and a progressively Type-II right censored sample of size m = 3, our selected censoring scheine (22, 0, 0) is the m o s t efficient censoring scheme possible if our efficiency criterion is defined by Var(#*) + Var(«*). 4.1.2.4. First-order approximations to the BLUEs Let us assume that the progressively Type-II right censored sample v(RI,...,R,, ~1. . . . . ) y2(RI'''''R'°) , v(RI ,"qRm) ..... • " *m:m:n has come from a lifetime distribution belonging to the location-scale parameter family, that is, with density function f ( y ; #, a) = 1 f ( y _ # / a ) . As we did for one-parameter models, we can use the first-order approximations to moments obtained in Section 3.4 and the approximate inverse variance-covariance matrix obtained in Section 4.1.1. to obtain approximate BLUEs of # and a , as well as their variances and covariance. This would be done by substituting the approximate values of la and Z 1 into (80) with all of these approximate values (See Balakrishnan and Rao, 1997b, 1999). EXAMPLE 4. Consider the log-times to breakdown in the accelerated test considered in Example 1. These are presented in Table 2. Now, let us assume an extreme value distribution with density function

Table 2 Progressively censored sample generated from the log-times to breakdown data for insulating fluid tested at 34 KV by Nelson (1982) i

1

2

3

4

5

6

7

8

xi:8:~9 -1.6608-0.2485 -0.0409 0.2700 1.0224 1.5789 1.8718 1.9947 Ri

0

0

3

0

3

0

0

5

R. Aggarwala

406

f ( y ; #, er)

=

le(y-u)/« e eC~-~//«,

--Oo
< co

,

er

for the distribution of log-times to breakdown, so that the natural times to b r e a k d o w n would be modeled by a scale-shape Weibull distribution. In this case, for the standardized log-times variable X = ( Y - #)/er, F(x) = 1 - e -et,

= l n ( - l n ( 1 - u))

F-l(u)

and

-1 F-I(1)(ü)

=

(1 -

u)ln(1

- u)

'

F r o m (63), (76) and (80), we determine the a p p r o x i m a t e B L U E ' s of # and er to be #* = (-0.09888 x -1.6608) ÷ (-0.06737 × - 0 . 2 4 8 5 ) + (-0.00296 x -0.0409) + (-0.04081 × 0.2700) + (0.12238 x 1.0224) + (0.00760 x 1.5789) + (0.04516 x 1.8718) + (1.03488 x 1.9947) = 2.456 and er* = (-0.15392 x -1.6608) + (-0.11755 × - 0 . 2 4 8 5 ) + (-0.11670 x -0.0409) + (-0.10285 × 0.2700) + (-0.03942 x 1.0224) + (-0.07023 × 1.5789) ÷ (0.04037 x 1.8718) ÷ (0.64104 x 1.9947) = 1.314 . The a p p r o x i m a t e variances and covariance are given by Var(#*) = 0.16442« 2, Var(cr*) = 0.10125 «2, Cov(#*, «*) - 0.06920 a 2 so that we obtain the standard errors of the estimates #* and «* to be SE(#*) = er*(0.16442) 1/2 = 0.533 and SE(c*) = «*(0.10125) 1/2 = 0.418 . Viveros and Balakrishnan (1994) determined in this case the (biased) m a x i m u m likelihood estimates of # and a to be 2.222 and 1.026, respectively. 4.2. B e s t linear invariant e s t i m a t i o n

M a n n (1969b) has considered BLIEs for location-scale distributions. Here, instead of requiring linear estimators that have m i n i m u m variance in the class of all linear unbiased estimators, we find the linear estimator that has m i n i m u m m e a n square error. In the case of location-scale family of distributions, M a n n (1969b) showed that the BLIEs of # and er are given by

407

Progressive censoring: A review

#** = # , _

V3 «* (1--4@

and

«** -

~*

1+~

(96) '

respectively, where #* and a* are the BLUEs of # and Œ as given in (80), V2 = Var(«*)/~r 2 and //3 = Cov(#*, a*)/a 2 from (81). Therefore, we find Var(#**)=

V1

Cov(#**, a**) -

l+V2

l+V2

V3 l+V2

' (97)

where V1 -- Var(#*)/a 2. Using these expressions, one can determine BLIEs and first-order approximations to BLIEs given the moments of progressively censored order statistics from location-scale distributions. These estimates will be closer to the parameters they are estimating, on average. Mann (1971) gives tables of coefficients of progressively Type-Il right censored order statistics for BLIEs for sample sizes up to n = 6. These tables may be expanded in the light of the various methods we have presented for determining moments of progressively censored order statistics. Thomas and Wilson (1972) compared BLIEs and BLUEs for the extreme value distribution. 4.3. L e a s t squares median ranks estimator

Gibbons and Vance (1983) proposed the least squares median ranks estimator for estimating the parameters of a scale-shape Weibull distribution, whose cumulative distribution function is given by (34) F(x)=l-exp

-

,

x_>0

The procednre attempts to minimize the following function with respect to ~ and fi. l n [ - l n ( 1 - [ ' ( X / .... ) ) 1 -

ln(X/:m:n)-~lnc~

(98)

The function is obtained by simple manipulation of F ( x ) . Other distributions may be better suited to other manipulations of the cdf. A common non-parametric method to estimate F(.) based on complete samples is to use the step function/~(N:n) = (i/n + 1) where X~:n are the order statistics from a samp]e of size n from F(.). An alternative estimator for F(.) used in lifetime studies is F(X,.:~) = (i - 0 . 3 / n + 0.4) (see Johnson, 1964). This is again for complete samples. For progressively censored samples, Gibbons and Vance (1983) considered using /~(X/:m:n)= (ji - 0 . 3 / n + 0.4) where ji = ji-1 -- A, i = 1 , 2 , . . , m , J0 = 0 and A=

n + 1 -ji-~

2 - - R i 4- Ri+ 1 -}-... + Rm -}-m - i

408

R. Aggarwala

In their comparison of various Weibull parameter estimators, Montanari and Cacciari (1988) also considered Æ(Xi .... ) __ j i -- 0.5

n + 0.25 Alternative forms for F(X/:m:n) can also be considered, such as the product limit estimator (London 1988). Upon differentiating (98) with respect to ~ and 1//~, setting the expressions equal to zero and solving the resulting equations simultaneously, least squares median ranks estimators for ~ and 1//~ are obtained as _ m ~im=l blil)i -- ~ i m l ----

m

B

m ~~~i=~u2 - (~-~~i=1ui) 2

~=exp

(

(99)

bli ~'~~iL1 Vi m

'

m V

~~lui

:,(~)), ~i=li~

(100)

where ui=ln(X/

4.4.

Likelihood

....

)

and

vi=

ln[-ln(1-

f'(Xi:m:,))]

.

inference

For progressively Type-II right censored samples, the likelihood function to be maximized will be m

L(0)=cIIf(xi;0)[1-F(xi;0)] i--1

R~,

x~

<x2<'"<x,~

,

(101)

where c = n ( n - R1 -

1).-.

(n - R 1 . . . . .

Rm_ 1 - m +

1)

and 0 is the vector of parameters to be estimated. Progressively Type-I right censored samples may also be considered for parameter estimation using likelihood methods. Balakrishnan and Aggarwala (2000) provide an overview of various distributions and methods commonly considered. Following the description of progressively Type-I right censoring provided in Section 2, we can write down the likelihood function to be maximized when a progressively Type-I right censored sample is observed. If k items are observed to fail during experimentation, and the resulting observed failure times are x x , x 2 , . . , xk, the likelihood function to be maximized is given by k m L(O) = C Hf(xi; 0) H [ 1 - F(T,.; 0)] Æi , (102) i--1 i I

Progressive censoring." A review

409

where C is the normalizing constant, which we assume is independent of 0. Notice that this likelihood function is similar to the likelihood function given in (101) for progressively Type-Il right censored samples with k replacing m and T~ replacing x» Notice also that the times corresponding to failure are not the same as the times corresponding to censoring, hence for likelihood estimation for Type-I samples, both T/ and xi are present and distinct. Therefore, if we develop likelihood results for progressively Type-I right censored samples, they will be readily modified, and often simplified, to results for progressively Type-II right censored samples if, in addition to replacing k by m, T~. are replaced by xi.

4.4.1. The normal distribution We will begin our discussion of likelihood estimation with the first results on progressive censoring ever published in a widely read journal, the results given by Cohen (1963), and part of the eventual response given by Cohen (1966) to a query posed in Technometrics. Cohen first considered a progressively Type-I right censored sample from a Normal(g, a 2) distribution, i.e. the failure-time distribution of the n independent items on test is

f/Æ) = ~ e - ( Y crx/2n

~/2i/2"2/

- ~ < ~ < o~, - e c 0 . (103)

Denoting the observed (ordered) progressively censored sample by yi, i = 1 , . . , k, the log-likelihood function to be maximized for the location and scale parameters will be

lnL(p,a) = c o n s t a n t - n l n a - ~

l k~ß'~{Yi-- p'~2

i2__~_l~ , ~ )

~-~'~-

+2_.,t¢i

"=

ln(1-F/)

i=1

(104) where B-p (9"

and q~(.) is the cumulative distribution function of the standard normal (p = 0, « = 1) distribution. To maximize the log-likelihood, the score equations to be solved are

2ki-1Yi

k

2~='(Y'-Y/2 k

m Ri

= Y = P - « ~i=17 ; zi' «2 1 - ~ ~ ~ ~ z , - ~ i~l

zi

(10»)

i=l

where Zi = (p(~i)/1 - 4~(~i) and ~0(.) is the probability density function of the standard normal distribution. Cohen (1963) has discussed the numerical solutions

R. Aggarwala

410

of these equations in detail. Now, many mathematical computer packages contain algorithms to solve the above system of equations efficiently. Note that maximum likelihood estimation for progressively Type-II right censored samples will result in the same system of equations, with the censoring times, T~, replaced by the failure times, yi. REMARK 22. In determining the variances and covariance of the MLEs, one could examine (at least asymptotically) the asymptotic variance-covariance matrix,

(106)

4.4.2. The exponential distribution

We have already addressed best linear unbiased estimation for the one- and twoparameter exponential distributions. Let us denote the observed progressively Type-II right censored order statistics from the one-parameter exponential distribution by yi, i = 1 , 2 , . . . , m. The M L E of the scale parameter 0 is then given by =--

nyl +

/11

+ 1)(yj-Yl

=

j=l

+ 1)yj ,

(107)

"=

which becomes identical with the BLUE of 0 given in (72). Cohen (1963) derived the MLE for 0 when the observed sample is a progressively Type-I right censored sample. The estimator is k + K-,m R.T. = ~i=1Yi Total Accumulated Life ~i< "' = k k '

(108)

and it reduces to (107) when we adjust it for Type-II samples, that is T, are replaced by yi, and k, the number of observed failures times, is replaced by m. Let us now denote the progressively Type-II right censored order statistics from the two-parameter exponential distribution (location parameter /z, scale parameter a) by Yi, i = 1 , . . . ,m. The log-likelihood function based on this progressively Type-II right censored sample is given by m

lnL = ( C o n s t a n t ) - m l n a

-

Z(Ri+1)( i 1

y/:m:n- \

~) ,

(109)

O-

which is monotonically increasing in # so that the MLE of # is ~ = yl. In this case, the M L E of o- is given by 1

m

G = m~'(Ri~=2 q-

1)(y/-Yl)

•

(110)

Progressive censoring: A review

411

It may be noted that the MLEs ~ and ~ are both biased, and in fact, they are simply the BLUEs adjusted for their bias. Therefore, the pivotal quantities established in Section 4.1.2 for inference using BLUEs will also hold for MLEs. See Viveros and Balakrishnan (1994). EXAMPLE 5. Earlier in Example 1, we considered Nelson's data on times to breakdown, where a one-parameter exponential model seemed appropriate. The one-parameter B L U E in that case was found to be 0* = 9.110 with aA standard error of 3.221. Using (107), the M L E of 0 is found numerically to be 0 = 9.111. 4.4.3. The Weibull distribution In one of a series of papers on likelihood estimation under progressive censoring that appeared in Technometrics, Cohen (1975) considered both maximum likelihood and "modified maximum likelihood" estimation for the three-parameter [location(/z) scale(a) - shape(0)] Weibull distribution with cumulative distribution function F(y)=l-e

(y_~)o/«

y>/~,

-cxD<#0,

0>0.

(111)

The log-likelihood function for a progressively Type-I right censored sample from this distribution is given by k

in L = (Constant) + k in 0 - k in a + (0 - 1) Z

i n ( y / - #)

i=1

(Yi - #)o+ O-

Ri(Ti - #)o

,

(112)

i=1

where y~ is the observed (ordered) progressively censored sample. Again, the results which we will present for progressively Type-I right censored samples may be adapted to Type-II scenarios and simplified in the usual way. From inspection of the likelihood function, we see that we may consider the three cases 0 < 1, 0 = 1, and 0 > 1 separately. If it is known that Õ = 1, then we have a simple location-scale exponential distribution, whose estimators we have already derived in the previous subsection. If 0 < 1, then we see that the likelihood function becomes infinite as # ~ yl, and therefore ~ = yl [in fact, as Cohen (1975) has pointed out, ~ = yl - 5, where q is the unit ofprecision of measurements made.] The additional score equations to be solved for a and 0 in this case are given by

~17 -- 0 = ----Œ-~~

(Yi --

/2)0@

ZRi(Tii=l

- #)0

,

(113)

OlnL k k ~Õ- - 0 = ö + ~ ln(yi - ,u) i=1

-

(yi -

~)°ln(yi

-

#) +

/~,(r~

-

~,)°ln(r,

-

~)

(114)

R. Aggarwala

412

We can now eliminate « between these two equations, obtaining a single equation as Y'~~/k=l(Yi - #)°ln(» - #) + ~i%a Ri(Ti -/~)°ln(T/-/~) k ~ i = 1 (Yi -- ~)0[_ ~ m l R i ( T i _ #)0

1 ö

1 k lc/__~ll n ( y i - / ~ ) = 0 '

(115)

When # or its estimate is known, as in the present case and the two-parameter case discussed by Cohen (1966), one needs to solve only this equation for 0 and then obtain the estimate for a by substitution back into the original equation, lnL/aa = 0, given above. Finally, for the case when 0 > 1, one taust solve a system of three equations, consisting of the two equations ~lnL/êa = 0 and ôlnL/aO = 0 given above, along with the equation

~~n~ 0E~ -

o =

-

(y, _

~)o-1+

O"

~

Ri(r~

-

~~01

i=l k

- (0 - 1 ) Z ( y i - #)-1

(116)

i=1

Again, simplification by eliminating the parameter a is possible. Cohen (1975) has discussed the numerical solutions to these equations in more detail, using yl as an initial approximation for 3. Many available computer algorithms will have suitable methods for solving this system of equations directly. The reader is referred to Balakrishnan and Aggarwala (2000) for further discussion on maximum likelihood estimation. A number of interesting results have been obtained, including results for Laplace and Pareto parameters, as well as consideration of MLEs under left and right (or general) progressive censoring schemes. Other topics in inference, including linear prediction and conditional inference, are also presented there.

5. Related topics in progressive eensoring

5.1. Simulation Many of the studies carried out on progressive censoring are computation in nature. Balakrishnan and Sandhu (1995) presented the following simple algorithm based on the mathematical properties of progressively censored order statistics from uniform distributions, for simulation of progressively censored order statistics from arbitrary continuous distributions. 1. Generate m independent Uniform (0, 1) observations W1, W 2 , . . , 1/ i+~j=m i+lRJ 2. Set Vi = Wz. for i = 1 , 2 , . . , m.

Wm.

413

Progressive censoring: A review

3. Set Ui:m:n = 1 -

gingm 1"'" Vm i+1 for

i = 1,2,..»m. T h e n

U1 .....

U2:m:n,...,

Um.... is the required progressively Type-II right censored sample from the Uniform (0, 1) distribution. 4. Finally, we set X~:m:n = F-I(Ui .... ), for i = 1 , 2 , . . ,m, where F - l ( . ) is the inverse cumulative distribution function of the lifetime distribution under consideration. Then XI ...... X2:m:n,... ,Xm:m:, is the required progressively Type-II right censored sample from the distribution F(.). The above simulation algorithm requires exactly m pseudo-random uniform observations, and does not require any sorting. Similar algorithms can be established using the mathematical properties we have presented in Section 3 for exponential and Pareto distributions.

5.2. Optimal censoring schemes The question of choosing optimal values of R I , R 2 , . . ,Rm when considering a progressive Type-II right censoring scheine is an important one to consider from a practical point of view, and as it turns out, it also gives rise to a number of interesting problems, in the areas of optimization, numerical analysis, simulation and programming. The first question to consider is out definition of optimality. That is, what criterion will be used to determine whether one scheine is better than another? In mathematical terms, we taust specify an objective function to be optimized before we can determine the optimal progressive censoring scheine to be employed, as different censoring schemes may perform optimally under different objective functions. First, ler us consider a practitioner who is interested in designing a warranty. This individual will be interested in the estimation of the time after which, say 95 per cent of the product will survive, that is, the manufacturer of the product is willing to repair 5% of all items produced. In this case, we are interested in estimating the fifth percentile of the product's failure-time distribution. This being a single (univariate) estimator, we could define the best scheme as the scheine which minimizes the mean squared error of the estimator. That i~ the objective function (to be minimized in this case) would be MSE(~0.05)where 40.05 represents the estimator for the fifth percentile, the parameter of interest. This has been discussed by Aggarwala (1998). A similar objective function may be defined for any case where a single parameter is to be estimated. We have discussed two-parameter (for example, location-scale) estimation extensively in Section 4. In the case where more than one parameter is to be estimated, there is a variance-covariance structure of the estimators which one may wish to consider. In the case of two-parameter estimation, optimality may be defined in terms of the trace or determinant of the variance-covariance matrix of the estimators, which makes sense particularly if the estimators are unbiased. In the case of biased estimators, we may want to replace the variance terms in out objective functions by the corresponding mean squared errors. More complicated objective functions can also be considered. For example, suppose the cost of experimentation is of concern, and will be proportional to the

R. Aggarwala

414

time on test of the experiment. Then one may wish to add to the objective function terms containing

E(Y~m(RI....

'R2""'Rm)

) °r P(Y~:%'::2,'R~) >

TU)

,

(1 17)

where y(R~,e2,..,e~) is the last observed failure time in the experiment and Tv is a m:m:n time by which we would like to end experimentation with high probability. Similarly, one may want to include in the objective function terms involving the variance of ~vm(e~,e2,'',R'n) or may have an interest in recycling live items removed, :m:n » and therefore add a term such as m

Zconstanti ×

Ri ,

(118)

i--1

where, perhaps the constants decrease with i. It is clear that there is an infinite number of possibilities, and the practitioner is able to customize the objective function to be optimized according to his or her priorities. Once the objective function has been satisfactorily defined based on n units available to put on test, the next question to consider is the one of actually finding the optimal censoring scheine. Suppose that a practitioner would like to observe m complete failure times, and taust determine what the best scheine to use in this case is. In this finite sample situation, one may list all (m-a) ù-a possible choices of censoring schemes and corresponding values of objective functions, and determine the best value (i.e. the value which optimizes the objective function) or a region of satisfactory values from this list, and either pick the best scheine or one which gives a value very close to the best but which may be practically more convenient. Alternatively, there may be cases where a simple closed-form expression of the objective function is available, and optimization is somewhat more mathematically elegant. Notice that m and n are chosen in advance here. If one or both of these are also to be determined, one may proceed by deciding upon values of m and n which are feasible given an agreeable value of the objective function. Aggarwala (1996) and Balakrishnan and Aggarwala (2000) have considered both the trace and the determinant of the varianc~covariance matrix of BLUEs for two-parameter location/scale distributions as objective functions of interest. In Sections 3 and 4, we discussed various ways of calculating these values, and as such, the actual value of the objective function may be determined, either in closed form or through a computer algorithm, for any scheme and for a number of distributions. The feasibility of computation is an important part of defining an objective function. Balakrishnan and Aggarwala (2000) have presented tables of optimal censoring schemes for these two objective functions for the extreme value, negative extreme value, log-normal and normal distributions which were obtained by calculating the objective function based on all (m-l) n-1 censoring schemes and choosing the best and worst values. They also note that for the exponential distribution, we have already seen that the variance-covariance matrix of BLUEs is independent of the censoring scheine chosen, and therefore,

Progressive censoring."A review

415

for these objective functions, the censoring scheme may be chosen for the sake of practicality and convenience. Two objective functions may be compared to one another through their relative efficiency, which can be defined as Objective function (Scheme B) . Objective function (Scheme A) '

(119)

hence, if one is interested in determining a region of satisfactory schemes, she or he can define that region in terms of efficiencies. For example, an experimenter may be satisfied with choosing any scheme which is at least 98% as efficient as the optimal scheme. For the distributions and values of n and m considered in Balakrishnan and Aggarwala (2000), the conventional Type-II right censoring scheme is orten the least efficient in terms of the objective functions considered. This was seen to be true for the extreme value and normal distributions, whereas for the log-normal distribution, the conventional scheme was the most efficient. This highlights the importance of having some idea of the distribution of lifetime failures. Goodness of fit algorithms for progressively censored samples is an area which is currently being investigated. As an example, the results for trace and determinant optimal and least optimal (where the matrix of interest is the variance-covariance matrix of location-scale BLUEs) progressive Type-II right censoring schemes for the two parameter normal distribution are given in Tables 3-5 for selected values of m and n. The reader is referred to Balakrishnan and Aggarwala (2000) for a more indepth discussion of such tables. Note that since an optimal scheme in this case would minimize the objective function, large values for efficiencies indicate that Scheme A is more efficient than Scheme B (which is taken to be the conventional Type-II right censoring scheme in Tables 3 and 4). Table 3 gives efficiencies (line 1) and traces (line 2) of the optimal scheme where efficiencies are with respect to the conventional censoring scheme. Table 4 gives the same for determinants. Table 5 provides coefficients, variances and the covariance of BLUEs for selected schemes. In that table, the following notations are used beside censoring schemes: (Note: conventional censoring schemes are always included, regardless of whether or not they fall into one of the following categories): + = trace optimal, * = determinant optimal, - = least optimal (trace), / = least optimal (determinant).

5.3. Acceptance sampling plans Acceptance sampling plans are perhaps the oldest and most extensively used statistical tool in industry. Although there is currently a strong shift towards product improvement (versus product monitoring), acceptance sampling based on inspection of finished products will continue to be a component of most company-wide quality initiatives. One form of product inspection is through life

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testing experimentation (sometimes referred to as reliability or life-test [acceptance] sampling plans). Naturally, censored samples will be of interest in this context (sometimes described as failure-censored reliability [acceptance] sampling plans). This discussion will focus upon the construction of failure-censored reliability acceptance sampling plans based on progressively Type-Il right censored samples. It should be noted that these tools need not be applied only to finished products, but may be modified and implemented at earlier points in production. Owen (1964, 1969) discussed acceptance sampling plans for normal and nonnormal distributions, while Hosona and Kase (1981) considered sampling plans based on the double exponential distribution. Fertig and Mann (1980) developed reliability sampling plans for Weibull and extreme value distributions, Schneider (1989) discussed failure-censored reliability sampling plans for Weibull and lognormal distributions, and Balasooriya (1995) discussed failure-censored reliability sampling plans for the exponential distribution. Kocherlakota and Balakrishnan (1984, 1986) proposed robust acceptance sampling plans based on exponential distributions. With regard to progressively Type-Il right censored samples, Balasooriya and Saw (1998) have developed reliability sampling plans and operating characteristic curves for two-parameter exponential distributions, and Balasooriya and Balakrishnan (2000) have developed reliability sampling plans for the log-normal distribution based on progressively Type-II right censored samples. These are further addressed in Balakrishnan and Aggarwala (2000). Many of these references provide examples using MIL STD standards for producer and consumer risks. For the purpose of developing such acceptance sampling plans, we may either use exact (ML/BLU) estimators of the parameters and their exact distributional properties or we may use first-order approximate estimators of the parameters and their distributional properties. We will consider the exponential distribution for illustration using exact results. Balakrishnan and Aggarwala (2000) also address the log-normal lifetime distribution as an illustration using approximate results. Thus, we will consider the two-parameter exponential distribution with probability density function f(y; #, «) = l_ e_0~_~)/«, y>#,_ « > 0 cr

,

(120)

where # is the location parameter and cr is the scale parameter. In the present context, # is the guarantee period and a is the mean residual lifetime, the mean lifetime being # + a. 5.3.1. One-sided sampling plans

For one-sided acceptance sampling plans, based on the progressively Type-II (1 R~), right censored sample Yl:m:~"' ~m:m:nV(Rl'""Rm) (denoted from now on by 111...... . . . , Ym..... for convenience) observed from a lot of n units, the lot is accepted if either q51(Y1..... . . . , Ym.... ) _> L or ~b2(Yl:m:ù,.. , Ym.... ) _< U where the functions q51 and q52 are statistics based on the observed sample, determined so •

•

,

Progressive censoring." A review

421

that in each case certain probability statements are satisfied. Depending on whether a right-sided or left-sided acceptance sampling plan is desired, one of the above two inequalities will be used. Suppose that L and U (often referred to as the lower specification and upper specification limits, and often determined based on much larger collections of data over time) represent quantiles from the two-parameter exponential lifetime distribution. Let Kp = - l o g ( 1 - p ) and K~_p = - l o g p denote the lower and upper p-percentage points of the standard exponential distribution, respectively. Then, from the equations Pr(Y<_L)=p

and

Pr(Y>U)=p

,

(121)

we have L-#

Kp-

and

U-#

Kl-p---

er

(122)

er

The one-sided acceptance sampling plans we will discuss here will coincide with one-sided tolerance limits (see Engelhardt and Bain, 1978 for discussion based on complete samples), as well as one-sided eonfidence intervals for quantiles based on progressively Type-II right censored samples, diseussed, for example, by Aggarwala (1998) for extreme value lifetime distributions. 5.3.1.1. Case 1:/~ unknown and er known It may be the case that the location parameter # is highly variable, whereas the location parameter a is very stable, and therefore considered a known, constant quantity. In these cases, we will accept the lot of size n if either/z + kla >_L or /2 + k2er _< U, depending upon whether we are interested in a right-sided or leftsided acceptance rule. Here,/~ = Yl:m:nis the M L E of/z, and kl and k2 need to be determined so that

Pr(fi+k~er>_L)=l- 7 and

Pr(/)+k2er
7 .

(123)

Writing L in terms of Kp, the first probability statement in (123) is equivalent to pr{2n(fiä--#) < e n ( K p - k l ) } = 7 -

(124)

Since

2n(Yl:m:n- #) d 2

2n(fl - ~) (7

(7

which is equivalent to an exponential random variable with mean 2, (124) yields kl = K p + 1- log(1 - 7) = - l o g ( i - p ) + n

1- l o g ( l - 7 )

.

(125)

n

On the other hand, if we are interested in a left-sided acceptance sampling plan, the second probability statement in (123) is equivalent to

R. Aggarwala

422

pr{2n(/)-~-#) _< 2 n ( K l _ p - k 2 ) } = l - 7

(126)

which gives 1

1

/'/

n

k2 = Kl-p 4- - log 7 = - log p + - log ? .

(127)

5.3.1.2. Case 2: # known and a unknown In this case, we will accept the lot if either # + k~ä >_L or # + k~6 <_ U, depending upon the desired direction of the acceptance sampling rule, where ] m ä = m~i=l(Ri + 1)(Y/.... - #) is the MLE of a and k 1# and kZ# need to be determined so that Pr(#+k~ä_>L)=l-?

and

Pr(#+k~ô-
•

(128)

Following similar steps to Case 1, the first probability statement in (128) is equivalent to

P [2m& r~7

2mKp']

(129)

-< k~ J = Y "

Since

2m6

2w-,
(7

:m - #

d 2

i=1

(129) yields 2mKp k ~ - Z2 - -

2m log(1 - p )

(130)

X2 2m,?

2m,y

,

where Z22m,y is the 7-percentage point of the chi-square distribution with 2m degrees of freedom which can be obtained from tables or software, that is,

f(z~m) dZ2m = 7 •

(131)

f0X~,,,,

Similarly, for left-sided sampling plans, the second probability statement in (128) is equivalent to

( m^

Pr } a-<

2mK1 p'~ = 1 _ 7 kZ

]

,

(132)

which readily yields

2mK1 p _ k~-

)~2 2m, 1-'/

2mlogp Z2 2m,1 "y

(133)

Progressive censoring: A review

423

5.3.1.3. Case 3: # and a both unknown In this case, we will accept the lot if either ~ + gl Ô" ~ L or ~ + g2 ~ ~ U , again depending upon the one-sided sampling plan required, where

B = Yl:m:n and

ä = --1 ~ ( R i + 1)(Y/:m:ù- YI.... ) m

are the MLEs of # and «, respectively, and gl and f2 need to be determined so that Pr(/)+glä_>L)= 1-7

and

Pr o)+g2&__ 2nKp) = 1 - 7 , m

(135)

d 2 where Z1 =)~2 (an exponential random variable with mean 2) and Z2 mm « ;g2 2m--2 independently. Thus, (135) can be rewritten as

7=

B2mKp/gl ,10

(

2"

)

fz~,°_2(z) 1 - e -( ùKp-ve,z)/2 dz ,

(136)

where fz~°,_2(z) denotes the probability density function of the chi-square distribution with 2m - 2 degrees of freedom. Upon substituting for fz~,_2(z) and integrating the resulting expression, we obtain the equation 7=Fz~m-2\ gl J

(1

n gl)m-lFz~,2\

--m

(137)

gl

m

where Kp = - log(1 - p) and F.,2 (.) denotes the distribution function of •2m_ 2. • "~2m 2. Eq. (137) needs to be solved lteratlvely for gl. Similarly, the second probability statement in (134) is equivalent to

(+"m g2Z2 ~< 2nKl_p ) = 1 - 7 ,

Pr Z1

(138)

which can be rewritten as

1 - 7 = [ a0 2mKl p/e2fx~~ 2(z) ( 1 -- e (2nKtp-~ß2z)/2) dz ,

(139)

which, upon substitution and integration, yields

yl~~

1 - 7 = Fx~m2 \ - - 7 ~ 2 J

«~~

(1 --m-7-7~m--l~2)Ez~°'-2

(~

(1 -- nm g2)

),

(140) where Kl_p = - log p. Eq. (140) needs to be solved iteratively for g2.

R. A g g a r w a l a

424

5.3.2. Two-sided sampling plans Two-sided acceptance sampling plans call for acceptance of the lot of n units if L _< qS(Yl:m:n, • • •, Ym:m:n) <-- U, where the function q5 is determined so that a certain probability statement is satisfied. We will suppose L to be the lower Pl percentage point of the 2-parameter exponential probability distribution, and U to be the upper p2 percentage point of the lifetime distributions which the observed units are assumed to follow. Thus,

L-# Kp~ -- - -

--

log(I-p1)

and

U-# Kl_p2

-- - -

ff

-- Kl_p2

= - log p2

G

(141) following the notation used thus far. 5.3.2.1. Case 1: # u n k n o w n and a k n o w n We will accept the lot if L «_ [~ + k « <_ U, where fz Yl:m:n is the m a x i m u m likelihood estimator of # and k needs to be determined so that =

Pr(L
U)=l-7

(142)

•

Since 2n(fz

-

#)

_

2 n ( Y l : m : n - - 11) d )(2 2

(7

we can rewrite (142) as Pr{2n(Kp, - k) < Z 2 < 2n(Kl-p2 - k)} = 1 - 7 ,

(143)

which readily yields (144) or

k = llog(, 1----7- ,, n \e-n~pl - e-n*'-p2 J

(145)

5.3.2.2. Case 2: # k n o w n and cr u n k n o w n We will accept the lot if L _< # + k'g- < U, where 1

m

6 = mZ(Ri

+ 1)(Yi:m:n

- -

#)

i-"~ 1

is the m a x i m u m likelihood estimator of a and k* needs to be determined so that Pr(L_<#+k*a_< Since

U)=

1-7

•

(146)

,

Progressivecensoring."A review 2ma

__

425

2 PZ_,vm i R i + 1) ( Yi:~:~r- / { ) =d )~22m

O"

i=l

we can rewrite (146) as

(2mKpl

Pr\

k*

-<)~2m-<

2mKl-p2) k*

(147)

J =1-7

or

(2mKl_m)

Fz~m\

k*

(2mKpl "~

J -Fz~,ù\ k* J = 1 - y

.

(148)

Eq. (148) needs to be solved iteratively for k*. 5.3.2.3. Case 3: # and a both unknown Hefe, we will accept the lot if L _
B = 111....

and

m

6 = m~-~~(Ri + 1)(Y~..... - Yl:m:~~) i"~-2

are the maximum likelihood estimators of # and «, respectively, and g needs to be determined so that Pr(L_<~+«ô-_< U ) = I - ?

.

(149)

We can rewrite (149) as Pr (2flKp, -< Z1 _}_rll~lgg2 _<2nKl-p2) = 1 - 7 ,

(150)

where Z1

-

-

2 n ( £ - #) d 2 ~2 and -

-

(7

Z2 --

2m6 d o-

--)~2m_2

independently. Eq. (150)yields

( m2 y p l_
1-y=Pr +

_

/e e-n&l B 2mKpt a0

1

e-Z(I-~e)/2zm-2 dz

2m-lF(m - 1)

ri 2~KI ~2/g2m-lF(-m 1 e-41-~e)/2z~ 2 dz - 1)

e tIKI-p2 JO

(151)

Once we numerically determine the constant g for specified values ofpl,p2 and 1 - 7, we can obtain the required two-sided acceptance sampling plan which accepts the lot based on a progressively Type-II right censored sample if

426

R. Aggarwala

L <[t+g&<_ U ,

based on the specified confidence probability of 1 - 7, where L and U are chosen to control the lower Pl and upper p2 percentages of the product's 2-parameter lifetime distribution.

REMARK 23. In the case of the exponential distribution, the exact determination of the factors required in the acceptance sampling plans is made possible due to the explicit form of the estimators of the parameters based on progressively TypeII right censored samples and the convenient distributional properties that these estimators possess. For most other distributions, such an exact determination of the factors becomes impossible in which case approximate methods may be used. A discussion based on the log-normal distribution is given in Balasooriya and Balakrishnan (2000) and Balakrishnan and Aggarwala (2000).

References Abramowitz, M. and I. A. Stegun (Eds.) (1965). Handbook ofMathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York. Aggarwala, R. (1996). Advances in life-testing: progressive eensoring and generalized distributions. Ph.D. Thesis, McMaster University, Hamilton, Ontario, Canada. Aggarwala, R. (1998). Warranty design using progressive censoring schemes. Invited Talk presented at the International Conference on Combinatorics, Statistics, Pattern Reeognition and Related Areas. Mysore, India. Aggarwala, R. and Balakrishnan, N. (1996). Recurrence relations for single and product moments of progressive Type-II right censored order statistics from exponential and truncated exponential distributions. Ann. Inst. Stat. Math. 48, 757-771. Aggarwala, R. and N. Balakrishnan (1998). Some properties of progressive censored order statistics from arbitrary and uniform distributions with appiications to inference and simulation. J. Stat. Plan. Inference 70, 35~49. Aggarwala, R. and N. Balakrishnan (1999). Maximum likelihood estimation of the Laplace parameters based on progressive Type-II censored samples. In Advances in Methods and Applications of Probability and Statistics (Ed. N. Balakrishnan). Gordon and Breach Publishers, Newark, NJ. (to appear). Aggarwala, R. and N. Balakrishnan (2000). Resuhs for Progressive Censoring with Pareto Failure: Properties, Moments, Inferenee, Simulation and Recursive Computation. (submitted). Aggarwala, R. and N. Balakrishnan (2001). Unfolding Moments: Relationships for Single and Produet Moments of Progressively Type-H Right Censored Order Statistics for Populations Related Through Symmetry. (submitted). Aggarwala, R. and A. Childs (2000). Conditional inference for the parameters of Pareto distributions when observed samples are progressively censored, In Advanees in Stochastic Simulation Methods (Eds. N. Balakrishnan, V. B. Melas and S. M. Ermakov). Birkhäuser, Boston. Arnold, B. C., N. Balakrishnan and H. N. Nagaraja (1992). A First Course in Order Statistics, Wiley, New York. Bain, L. J. (1978). Statistical Analysis of Reliability and Life-Testing Models TheotW and Practice. Marcel Dekker, New York. Bain, L. J. and M. Engelhardt (1991). Statistical Analysis of Reliability and Life-Testing Models, 2nd edn. Marcel Dekker, New York.

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Balakrishnan and Aggarwala (2000). Progressive Censoring: Theory, Methods and Applications. Birkhauser Publishers, Boston. Balakrishnan, N. and P. S. Chan (1992). Order statistics from extreme value distributions, I: Tables of means, variances and covariances, Comm. Stat. - Simul. Comput. 21, 1199 1217. Balakrishnan, N. and A. C. Cohen (1991). Order Statistics and Inferenee: Estimation Methods. Academic Press, Sah Diego. Balakrishnan, N., E. Cramer and U. Kamps (1999a). Inequalities for means and variances of progressive Type II censored order statistics. SubmitIed. Balakrishnan, N., E. Cramer, U. Kamps and N. Schenk (1999b). Progressive Type II censored order statistics from exponential distributions. Submitted. Balakrishnan, N. and C. R. Rao (1997a). Large-sample approximations to the best linear unbiased estimation and best linear unbiased prediction based on progressively censored samples and some applications. In Advanees in Statistieal Decision Theory and Applications, pp. 431~444 (Eds. S. Panchapakesan and N. Balakrishnan). Birkhäuser, Boston. Balakrishnan, N. and C. R. Rao (1997b). A note on the best linear unbiased estimation based on order statistics. Am. Statistician 51, t81 185. Balakrishnan, N. and C. R. Rao (1999). On the efficiency properties of BLUEs. J. Star. Plan. Inf to appear. Balakrishnan, N. and R. A. Sandhu (1995). A simple simulational algorithm for generating progressive Type-II censored samples. Am. Statistician 49, 229-230. Balakrishnan, N. and R. A. Sandhu (1996). Best linear unbiased and maximum likelihood estimation for exponential distributions under general progressive Type-II censored samples. Sankhyä, Ser. B 58, 1-9. Balakrishnan, N. and K. S. Sultan (1998). Recurrence relations and identities for moments of order statistics. In Handbook o f Statistics, Vol. 16: Order Statistics: Theory and Methods, pp. 149~28 (Eds. N. Balakrishnan and C. R. Rao). North-Holland, Amsterdam, The Netherlands. Balasooriya, U. (I995). Failure-censored reliability sampling plans for the exponential distribution. 9". Star. Comput. Simul. 52, 337-349. Balasooriya, U. and N. Balakrishnan (2000). Reliability sampling plans for log-normal distribution based on progressively censored samples. IEEE Trans. Reliab. 49, 199-203. Balasooriya, U. and S. L. C. Saw (1998). Reliability sampling plans for the two-parameter exponential distribution under progressive censoring. J. Appl. Stat. 25, 707-714. Balasooriya, U. and S. L. C. Saw (1999). A note on approximate moments of progressively censored order statistics, Metron - Roma 57, 11~130. Cacciari, M. and Montanari, G. C. (1987). A method to estimate the Weibull parameters for progressively censored tests. IEEE Trans. Reliab. R-36, 87-93. Cohen, A. C. (1963). Progressively censored samples in life testing, Technometrics 5, 322329. Cohen, A. C. (1966). Life testing and early failure, Technometrics 8, 539-549. Cohen, A. C. (1975). Multi-censored sampling in the three parameter Weibull distribution. Technometrics 17, 347-351. Cohen, A. C. (1976). Progressively censored sampling in the three parameter log-normal distribution. Technometrics 18, 99-103. Cohen, A. C. and N. J. Norgaard (1977). Progressively censored sampling in the three parameter gamma distribution. Technometrics 19, 333 340. Cohen, A. C. and B. J. Whitten (1988). Parameter Estimation in Reliability and Life Span Models. Marcel Dekker, New York. David, F. N. and N. L. Johnson (1954). Statistical treatment of censored data. I. Fundamental formulae. Biometrika 41, 228-240. Engelhardt, M. and L. J. Bain (1978). Tolerance limits and confidence limits on reliability for the two-parameter exponential distribution Technometries 20, 37-39. Fertig, K. W. and N. R. Mann (1980). Life-tests sampling plans for two-parameter Weibull populations. Technometrics 22, 165-177.

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Gajjar, A. V. and C. G. Khatri (1969). Progressively censored samples from log-normal and logistic distributions. Technometrics 11, 793-803. Gibbons, D. I. and L. C. Vance (I983). Estimators for the 2-parameter Weibull distribution with progressively censored samples. IEEE Trans. Reliab. R-32, 95-99. Graybill, F. A. (1983). Matriees with Applications in Statistics, 2nd edn., Wadsworth, Belmont, California. Harter, H. L. and N. Balakrishnan (1996). CRC Handbook of Tables for the Use of Order Statistics in Estimation, 2nd edn., CRC Press, Boca Raton. Herd, R. G. (1956). Estimation of the parameters of a population from a multi-censored sample. Ph.D. Thesis, Iowa Stare College, Ames, Iowa. Hosona, H. O. Y. and S. Kase (1981). Design of single sampling plans for doubly exponential characteristics, In Frontiers in Quality Control, pp. 94-112 (Eds. H. J. Lenz, G. B. Wetherill and P. T. Wilrich). Physica-Verlag, Berlin, Germany. Johnson (1964). The statistical treatment of fatigue experiments. Elsevier, Amsterdam. Kocherlakota, S. and N. Balakrishnan (1984). Two sided acceptance sampling plans based on MML estimators. Comm. S t a t . - Theor. Meth. 13, 3123-3131. Kocherlakota, S. and N. Balakrishnan (1986). One- and two-sided sampling plans based on the exponential distribution. Naval Res. Logist. Quart. 33, 513-522. Lawless, J. F. (1982). Statistical Models & Methodsfor Lifetime Data. Wiley, New York. Lieblen (1953), On the exact evaluation of the variances and covariances of order statistics in samples from the extreme-value distribution. Ann. Math. Stat. 24, 282-287. London, D. (1988). Survival Models. Actex Publications, Connecticut. Lnrie, D. and H. O. Hartley (1972). Machine-generation of order statistics for Monte Carlo computations. Amer. Statistician 26, 26~7. Malmquist, S. (1950). On a property of order statistics from a rectangular distribution. Skandinavisk Aktuarietidskrift 33, 214-222. Mann, N. R. (1969a). Exact third-order-statistic confidence bouuds on reliable life for a Weibull model with progressive censoring. J. Am. Stat. Assoc., 64, 306 315. Mann, N. R. (1969b). Optimum estimators for linear functions of location and scale parameters. Ann. Math. Star. 40, 2149-2155. Mann, N. R. (1971). Best linear invariant estimation for Weibull parameters under progressive censoring. Technometrics 13, 521-533. Montanari, G. C. and M. Cacciari (1988). Progressively-censored aging tests on XLPE-insulated cable models. IEEE Trans. Elect. Insul. 23, 365 372. Montanari, G. C., G. Mazzanti, M. Cacciari and J. C. Fothergill (1998) Optimum estimators for the weibull distribution from censored test data, progressively-censored tests. IEEE Trans. Dielect. Electri. Insul. 5, 157-164. Nelson, W. (1982). Applied Life Data Analysis. Wiley, New York. Owen, D. B. (1964). Control of percentages in both tails of the normal distribution. Technometrics 6, 377-387. Owen, D. B. (1969). Variable acceptance sampling with non-normality. Technometrics 11, 631-637. Ringer, L. J. and E. E. Sprinkle (1972). Estimation of the parameters of the Weibull distribution from multicensored samples. IEEE Trans. Reliab. R-21, 46-51. Robinson, J. A. (1983). Bootstrap confidence intervals in location-scale models with progressive censoring. Teehnometrics 25, 179-187. Schneider, H. (1989). Failure-censored variables-sampling plans for log-normal and Weibu11 distributions. Technometrics 31, 199-206. Sherif, A. and Tau, P. (1978). On structural predictive distribution with Type-II progressively censored Weibull data. Statistische Hefte 19, 247-255. Sukhatme, P. V. (1937). Tests of significance for samples of the Z2 population with two degrees of freedom. Ann. Eugenies 8, 52 56.

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Thomas, D. R. and W. M. Wilson (1972). Linear order statistic estimation for the two-parameter Weibull and extreme value distributions from Type-II progressively censored samples. Technometrics 14, 679 691. Tse, S-K and H-K Yuen (1998). Expected experiment times for the Weibull distribution under progressive censoring with random removals. J. Appl. Stat. 25, 75-83. Viveros, R. and N. Balakrishnan (1994). Interval estimation of life characteristics from progressively censored data. Technometrics 36, 84-91. Wingo, D. R. (1973). Solution of the three-parameter Weibull equations by constrained modified quasilinearization (progressively censored samples). IEEE Trans. Reliab. R-22, 96-102. Wingo, D. R. (1993). Maximum likelihood methods for fitting the Burr Type XII distribution to multiply (progressively) censored life test data. Metrika 40, 203310. Wong, J. Y. (1993). Simultaneously estimating the three Weibull parameters from progressively censored samples. Microelect. Reliab. 33, 2217 2224. Yuen, H-K and S-K Tse (1996). Parameter estimation for Weibull distributed lifetimes under progressive censoring with random removals. J. Stat. Comput. Simul. 55, 57-71.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 ElsevierScienceB.V. All rights reserved.

| A 1 -T

Point and Interval Estimation for Parameters of the Logistic Distribution Based on Progressively Type-II Censored Samples

N. Balakrishnan and N. Kannan

The logistic distribution has been widely used as a growth model in many problems. The logistic model has often been selected as an alternative to the normal because of the similarity of the two distributions. In this article, we consider the estimation of the location and scale parameters of the logistic distribution based on progressively Type-II censored samples. The maximum likelihood method yields equations that do not provide explicit solutions under any censoring scheme. We use an approximation of the cumulative distribution function (cdf) that leads to simplified likelihood equations which yield explicit estimators. We examine numerically the bias and mean squared error (MSE) of the maximum likelihood estimators (MLEs) and the approximate estimators and show that the approximation provides estimators that are almost as efficient as the MLEs. The probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are shown to be unsatisfactory, especially when the effective sample size is small. We suggest the use of unconditional simulated percentage points for the construction of confidence intervals. We also develop estimators based on weighted least squares; these estimators may be used as effective starting values for the numerical algorithm to determine the MLEs, and are themselves quite efficient when the effective sample size is large. A wide range of sample sizes and progressive censoring schemes have been considered in this study. Finally, we present a numerical example to illustrate the methods of inference developed here.

1. Introduction

Censoring is a common phenomenon in life-testing and reliability studies. The experimenter may be unable to obtain complete information on failure times for all experimental units. For example, individuals in a clinical trial may withdraw from the study, or the study may have to be terminated for lack of funds. In an industrial experiment, units may break accidentally. In many situations, however, 431

432

N. B a l a k r ~ h n a n and N . Kannan

the removal of units prior to failure is preplanned in order to provide savings in terms of time and cost associated with testing. The two most common censoring schemes are termed Type-I and Type-II censoring. Let us consider n units placed on a life-test. In the conventional Type-I censoring scheme, the experiment continues upto a prespecified time T. Failures that occur after T are unobservable. The termination point T of the experiment is assumed to be independent of the failure times. By contrast, the conventional Type-II censoring scheme requires the experiment to continue until a prespecified number of units m < n fail. In this scenario, only the smallest lifetimes are observed. In Type-I censoring, the number of failures observed is random and the endpoint of the experiment is fixed, whereas in Type-II censoring the endpoint is random, while the number of failures observed is fixed. There is extensive research in the reliability and survival analysis literature dealing with inference under Type-I and Type-II censoring for different parametric families of distributions. For additional details and references, the interested reader may refer to the book by Cohen (1991), Balakrishnan and Cohen (1991) and Cohen and Whitten (1988). One of the drawbacks to the conventional Type-I and Type-II censoring schemes explained above is that they do not allow for removal of units at points other than the terminal point of the experiment. Cohen (1963, 1966) was one of the earliest to study a more general censoring scheme: fix m censoring times T l , . . . , Tm. At time T/, remove Ri of the remaining units randomly. The experiment terminates at time Tm with Rm units still surviving. This is referred to as Progressive Type-I right censoring. Cohen (1963) discussed the estimation of parameters under this scheme for the parameters of the normal distribution. This idea of progressive censoring can be adapted to the Type-Il censoring scheme as follows: consider an experiment in which n units are placed on a lifetest. At the time of the first failure, R1 units are randomly removed from the remaining n - 1 surviving units. At the second failure, R2 units from the remaining n - 2 - R I units are randomly removed. The test continues until the mth failure. At this time, all remaining Rm = n - m - R 1 R2 . . . . . Rm-1 units are removed. The Ri's are fixed prior to the study. If R1 = R2 . . . . . Rm = 0, we have n = m which corresponds to the complete sample. If R1 = R2 . . . . . Rm i = 0, then Rm = n - m which corresponds to the conventional Type-II right censoring scheme. This generalized censoring scheme is called Progressive T y p e - H censoring. Mann (1969, 1971) and Viveros and Balakrishnan (1994) have discussed inference for the Weibull and exponential distributions under Progressive Type-Il censoring. Viveros and Balakrishnan (1994) have derived explicit expressions for the best linear unbiased estimates (BLUEs) of the parameters of the 1- and 2parameter exponential distributions. Balakrishnan et al. (2001), in a recent article, have discussed MLEs and approximate estimators for the normal distribution based on progressively Type-II censored samples. For an exhaustive list of references and further details on progressive censoring, the reader may refer to the book by Balakrishnan and Aggarwala (2000).

433

Point and interval estimationfor parameters of the logistic distribution

In this article, we consider progressively Type-II censored data from a logistic distribution. In Section 2, we discuss the maximum likelihood estimation of the location and scale parameters of the logistic distribution. Section 3 provides explicit estimators by appropriately approximating the likelihood equations. In Section 4, we develop weighted least squares estimators of the parameters. The expressions for the observed and expected Fisher information are provided in Section 5. In Section 6, we provide results of a simulation study in order to evaluate the performance of the approximate estimators and the MLEs determined by numerical methods. In Section 7, we simulate the coverage probabilities for pivotal quantities based on asymptotic normality and show them to be unsatisfactory. For this reason, we provide unconditional simulated percentage points of these pivotal quantities. An illustrative example is presented in Section 8. Conclusions and a brief summary of the results are finally provided in Section 9. 2. Maximum likelihood estimators

Assume the failure time distribution to be logistic with probability density function (pdf)

f(x;/~, o-)= ax/3 {1

-}- e-r~(x-#)/«'fS} 2' -oc<x
-oo
a>0

(1)

and corresponding cumulative distribution function (cdf) 1 F(x; #, er) = 1 + e-~(*-")/«'/5

(2)

We use the notation X ~ L(/~, a 2) to indicate that the random variable X has the logistic distribution given above. Consider the random variable Y = (X - #)/a. Then Y has the standard logistic distribution with pdf and cdf given by

f ( y ) = V ~ { l _}_e_=y/,/5}2

(3)

and 1

F(y) -- 1 + e-~Y/"/5 '

(4)

respectively. The random variable Y has mean 0 and variance 1. For an encyclopedic treatment of the logistic distribution, the reader may refer to the book by Balakrishnan (1992).

N. Balakrishnan and N. Kannan

434

Let Xl:m:~,.. ,Xm:m:ù denote a progressively Type-II censored sample from (1) with ( R 1 , . . , Rm) being the progressive censoring scheme. The likelihood function based on the progressively Type-Il censored sample is given by m

L(#, «) : c H f(x~:~:~;#, « ) [ 1

-

F(xi:~:~;#, a)] R~ ,

(5)

i=1

where C

:

n(n

--

1 -- R1)(/7

2

-

-

R 1 -

R2).-.

(n

-

m + 1

-

R 1 . . . . .

Rm_l)

.

Using the relation f ( y ) = (7~/v/3)F(y){1 - F ( y ) } , the likelihood function may be rewritten as m

7~

L(#, «) = c H ~ F ( y i

..... )[1 - r ( y i : m : ~ ) l R'+I

,

where Yi:m:~ = (Xi:rn:n -- #)/~r. To simplify the notation, we will use xi and yi instead of Xi:m:n and yi ..... respectively. The log-likelihood function is then given by

lnL(#,a)=K-mlna+

i--1

lnF(y~)+~-~~(R~+l)ln{1-F(yi)},

i=l

(6)

where K is a constant. From Eq. (6), we derive the likelihood equations for # and cr as ~log 7 c + ä ß ~ ~ i ~1 (R~ + 2)F(y,) = 0 , 8# L - - mav/3

log L ~a

m a

TC

m

Tl[

(7)

m

~/~i~l y i ~ - ~ F ( ~ i @ 2)yiF(yi) : 0

.

(8)

Eqs. (7) and (8) do not yield explicit solutions for # and a and have to be solved numerically to obtain the MLEs of the two parameters. The book by Balakrishnan (1992) provides some numerical results on the bias and mean squared error of the MLEs for both complete and Type-II censored data. The asymptotic variances and covariances have also been derived for the MLEs. It should be mentioned that Gajjar and Khatri (1968) have considered maximum likelihood estimation for the logistic distribution under progressive Type-I right censoring.

3. Approximate estimators The likelihood equations are nonlinear in # and cx and do not admit explicit solutions because of the presence of the term F(y). Approximate solutions for

Point and interval estimation for parameters of the logistic distribution

435

MLEs have been discussed in the book by Tiku et al. (1986) for several specific distributions. Most of the approximate methods require linear approximation of the cdf or hazard functions. We approximate the logistic cdf F(yi) by expanding it in a Taylor series around E(Yi .... ) = Vi:m:n. From Balakrishnan and Sandhu (1995), it is known that F(Y~:m:n) £ U/:m:n , where ~ .... is the ith order statistic from a progressively Type-II censored sample from the uniform U(0, 1) distribution. We then have (9)

Yi:m:nd ]z:'-l(Ui:m:n)

and hence v,:m:ù = E ( ~ : m : ù )

- F-I(~,:~:ù)

,

(10)

where ai .... = E(Ui .... ) and is given by (see Balakrishnan and Aggarwala (2000)) = l -

~-[ j 4- Rm_j+l 4- "'" 4- Rm ::Z~-+lJ + 1-7 R~7~_j+~5~777 ~-Rm'

(11)

i=l,...,m.

For the logistic distribution, F 1(.) is easily noted to be F

l(u) = -

v / 3 1 n,(,1~- -u~ 7/

(12)

Expanding F(yi) around Vi:m:, we have (keeping only the first two terms), F(yi) ~ F(vi:m:n) + (,Yi - Vi:m:n)F'(Y)[y=v, ..... ,

(13)

= 7i + am ,

(14)

where ~i = F(vi:m:n) - Vi:m:nf(vi .... ), ~5i = f(Vi:m:n) >_ O,

i = 1, . . , m

i = 1, . . , m

(15)

,

(16)

.

Using the above expression, we approximate the likelihood Eqs. (7) and (8) by ô log L mzr z~ m ô# ~ - ~ + äß~-'(Ri,y..{_ log L

- - ~

~O"

m O"

7c

m

+ 2){7i + 6iYi} = 0 ,

7"C

Zyi+~~-~~(Ri+

0"~/3 /=1

(17)

m

oV3 i=1

2)yi{?i+6iYi}=O

.

(18)

N. Balakrishnan and N. Kannan

436

Eq. (17) m a y n o w be rewritten as m

m

--mq- Z ( R i 4 -

2)Yi q- ~-~~(Riq- 2 ) ~ i ( ~ )

i=1

(19)

=0

i=1

which yields the estimator of # as

B = ~i~_1 (Ri + 2)6i =K+La

Ri + 2)(Sixi + cr

Ri + 2)7i - m

,

(20)

where

K = ~i~=] (Ri + 2)aixi

(21)

m R i + 2)& 2,=1(

L = ~im] (Ri + 2)7i - m ~ i =m1 ( R i + 2)6i

(22)

Eq. (18) m a y n o w be rewritten as m

G

m

GX/3~=I \ 7~

O" ,I +

m

_

( X ~ )

Riq-2)Yi(T)

2

+~-~i~.l(Ri+2)6i_

=0

.

(23)

Replacing # n o w by K +LG as in (20), we have from (18) m

m

t-

Ri + 2)7i - 1}(xi - K) +

0-

Ri + 2)6i(x~ - K) 2

_

mL~

Lrc

2Lrc

~

m~

L2~ y_. m

a 7 ,__Ll/~'+ 2)~~+ ; 7 ~ ( ~ ,

«,~

+ 2)ô,

"'

Z(Rii-1

(24)

-~ 2)~)i(xi -- K) = 0 .

The last four terms can be shown to vanish, leaving us with the quadratic equation m

mG2 .

.

m

R i. + 2)yi.

1}(x/

K)

Ri + 2)6i(xi

-

K) 2 =

0

(25)

Point and interval estimation for parameters of the logistic distribution

437

or

ma 2 - A l a

-A2 = 0 ,

where 7E

m

A1 : ~ Z { ( R i

+ 2)7i - 1}(xi - K),

i=1

7~

m

A 2 : ~Z(Ri+

2)Õi(xi-K) 2 > 0 .

i=1

Eq. (25) is a quadratic equation in a, with the roots given by

ä=

A1 i vIA 2 + 4mA2

2m Since A2 > 0, only one root is admissible, and hence the approximate MLE of a is given by

A1 + ~/A~ + 4mA2 ä=

(26)

2m

The approximate MLEs are thus explicitly given by the expressions in (20) and (26). This procedure allows us to obtain estimators for # and a explicitly. We need to evaluate the performance of these approximate estimators by comparing their efficiency with those of the MLEs obtained by solving the likelihood equations numerically. These approximate solutions may provide us with an excellent starting value for the iterative solution of the likelihood Eqs. (7) and (8).

4. Weighted least squares estimation

Consider the cdf of the L(#, a 2) given by 1

F(x; #, a) = 1 + e-~(x ~)/o4g

(27)

Since F(Xi:m:n) ~ U/:m:n as mentioned earlier, we have

E(Xi:m:n) =F

v ~ 1 /Il C C~i:m:n) l(~i:m:n)=#--0"~n~- e~.... / '

where C~i:m:~is as given in (11). We can construct weighted least squares estimators of # and a using the observed data. The BLUE of the parameter vector 0T = (#, a) is given by

N. Balakr~hnan and N. Kannan

438

0L = ( w T ~ - l W )

-1WT~-IX

,

(28)

where X is the m x 1 vector containing the observed progressively Type-II censored data, W is the m x 2 matrix with the first column containing all ls and the second column containing the terms -(x/J/~)ln((1 -C~i..m:~)/C~i.... ), and ~2 is the variance-covariance matrix of the observed progressively censored order statistics which may once again be approximated using the moments of the uniform progressively censored order statistics. The first-order approximations for these quantities have been presented by Balakrishnan and Aggarwala (2000). The BLUEs are known to be asymptotically unbiased and efficient for large m. Furthermore, these estimators should also provide good estimates for large effective sample sizes, i.e. large m, and may therefore provide a good starting value to begin the numerical iterations required for the determination of the MLEs. It is of interest to mention here that upon using this approximate expression of Z and the resulting symmetric tri-diagonal matrix as its inverse, Balakfishnan and Rao (1997) have derived explicit expressions for the BLUEs of # and a in (28) and the variances and covariance of these estimators.

5. O b s e r v e d

and expected

Fisher information

In this section, we compute the observed and expected Fisher information based on the likelihood as well as the approximate likelihood equations. These will enable us to develop pivotal quantities based on the limiting normal distribution and then examine the probability coverages of these pivotal quantities through Monte Carlo simulations. We now derive the observed Fisher information for the likelihood Eqs. (7) and (8). We find 62 in L 6# 2 62 i n

L

Ô#6o-

~z m ~rg--/wZ(Ri-v»

rmr

--

+ 2)f(yi)

(29)

,

i=1

o'2 V/3

n

m

7"2

2-X/~ i~_l(Ri-2)F(yi) o" _

m

a B v / ~ i=1 Z ( R i + 2)yif(Yi) ,

(30) Õ2 in

L

_

m

2~

m

2~

m

m

From these expressions, we now compute the expected Fisher information. First of all, we have

Point and interval estimation for parameters of the logistic distribution

( ô21nq_ E

~1~2 J

439

~ m «SV~i~=l (Ri + 2)E[f(Yi)]

where E[f(Y~)] = ~ E [ F ( Y ~ ) { 1 - F(Y~-)}] = ~ 3 E[{1 - F ( Y i ) } ] - ~ 3 E [ { 1 - F(Y,-)} 2]

D

F r o m the joint density of Y1,. •, Ym, we readily find

E[{1 - F(y.)}] = A ( n , m

-

1) f--. f f(Yl){ 1 - F(y,)}<..

"f (Yi --1)

× {1 - F(y/_l)} ei 'f(yi){1 - F ( y i ) } R'+I ×/(Yi+l){ 1 - F(yi+1)} Æi+l''' × f(ym){l

-- F ( y ~ ) } R ° d y l

A ( n , m - 1) =A(n + 1 , m - 1 )

'

dym

'

where

A ( n , m - 1) = n ( n - 1 - R 1 ) ( n - 2 - R 1 - R2)... x (n - m + 1 - R 1 . . . . .

Rm-1)

and A(n + 1, m - 1) has the same form with n replaced by n + 1, and Ri replaced by Ri + 1. Proceeding similarly, we find El{1 - F(Y/)}2] = A ( n , m - 1) f . . - / f ( y a ) { 1

- F(y,)} "~.. "f(Yi-1)

× {1 - F(yi_l)} Ri 'f(yi){1 - F(yi)} Ri+2

x f(Yi+l){1 - F(yi+l)} Ri+' ''" × f(ym){ 1 -- F(ym)}R~dyl ... dym

A ( n , m - 1) =A(n + 2, m - 1 )

'

where A(n + 2, m - 1) has the same form as A(n, m - 1) above with n replaced by n + 2 and Ri replaced by Ri q- 2. By substituting for these expressions, we obtain (

E

821n L'~ ~2 ~ ~ A ( n , m - 1) A ( n , m - 1) 81~2 j = ~ ß ~ 2 ) _ q ( R ~ + 2 ) [ . A ( n + l , m _ X ) - A ( n + 2 , m_l) J . (32)

N. Balakrishnan and N. Kannan

440

Next, from Eq. (30), we have (

82 in L~ _

mrc

E - äßä-j

aT-x~~ + ~

~

m

Z ( R i + 2)ElF(Y/)] i=1

m

+

~--'(Ri + 2)E[Y~f(~-)]

.

Now~ A(n,

m -

1)

A(n+ l , m - 1 )

E[F(Y/)]=I-E[{1-F(Y/)}]=I and 7Z

E[Y/f (Yi)] = - ~ { E[Yi{ 1 - F(Y//)}]- E[Yi{ 1 - F(~-)}2] }

7r ( A ( n , m - 1 ) E E A(n,m-1) E Y. ) =-~~.A(n+l,m_l) (i:m:"+l) A(n-~-~,mZl) ( i:m:n+2)~ ' where Y~:m:,+l and Y/.... +2 denote the /th progressively censored order statistic from samples of size n + 1 and n + 2 with the corresponding censoring schemes ( R 1 , . . , R i 1,Ri+ 1,R¢+I,...,Rm) and (R1,..,Ri-I,Ri+ 2, Ri+I,...,Rm), respectively. Let Ui.... +1 and Ui:m:n+ 2 denote the corresponding progressively censored order statistics from the uniform U(0, 1) distribution, and ~i:m:ù+l and ~i:m:n+2 denote their expected values. We may approximate E(Y~:m:ù+I) by F-l(c¢i .... +1), and E(Y/ .... +2) by F-l(c¢i.... +2). Upon substituting for these expressions, we obtain

(~~ln~~_

E - e#aùJ

m~

~ ~

~7---v/3+~ Z(R/+2) i=l

~'~'m-~'l

El - A ( n + l , m

1)

7z2

+ ~a2 ~_.~(Ri + 2)A(n,m - 1) i=1

f F-l(c¢i:m:n+l)_

x I.A(n+l,m-1)

F-l(~i . . . . + 2 ) "~ A(n+2, m - 1 ) J "

(33)

Finally, from Eq. (31), we have E -

=-äS+a-~~.=

(Ri+I)E[Yi]

2~ m «~--~ ~ ( R ~ + 2)E[Z-{1 - F(Y,)} 1 + ~

7"C

m

~--~~(Ri~+ 2)E[y~2f(Y,.)] .

(34)

Point and interval estimationfor parameters of the logistic distribution

441

Now, E[y~2f(Y~)] = ~

{E[y~2{1 - F(~)}] - E[yi2{1 - F(Y//)}2]} A(n,m - 1)

~A~~-~~,;,-

11 r~Var(~ .... +11 + {E(~ .... +,/}2~

7~ A(n,m - 1) , ~ A ~ ~ 2 , r~--1) f].Var(Y/..... +2) + {E(Y/.... +2)}2j~. 1

We can approximate Var(Yi .... +1) by {F-l(1)(~i:m:n+l)}2Var(Ui .... +a), where F-l(1)(u) = dF -1 (u)/du, and Var(Yäm:~+2) by {F -~(1)(~äm:n+2)}2Var(~ .... +2). We may derive similar expressions for the approximate likelihood Eqs. (17) and (18): 521nL Ô}t2 62 In L

m

~/_~i_~1 R 0 .2 ( i + 2)8i ,

~

m7c

7c

(35)

m

7~

m

(36) ~21n L

m

2=

~

~0.2 ~ 7 -}- ~

m R O'~-VFji~_1( i-}- 2)yi(7i + cSiYi) 2=

i~_l yi

~

m

~ ~ ( R i @ 2)Ôiß .

(37)

(7

The expected Fisher information based on these expressions is then given by 7~ 521nL~ ~/22 j ~ ~

E

(~2~n~) E

E

~#S«j ~

(

Ô21nL'~ Ô0.2 ff ~

m~

(38)

_ (Ri+2)6i

~

m

0.2v/~+~Z(Ri+2){yi+aiF-l(oq:m:n)} vo i=1 m +~~= (Ri + 2)6,F '(~i . . . . . ) ,

(39)

m~v/.~m.~_lF-l(o:i:m:n ) (72 0.

+~

(e, + 2)(v~F-1(~~.... ) + ô~U(~.... )) Im=

q- ~ß'~~(Riq- 2)aiE(yi2m:n)

}

(40)

442

N. Balakrishnan and N. Kannan

with E(y~.2m:n) being approximated using the corresponding moment of the progressively censored uniform order statistic as before. Of course, from these expressions of expected information matrices, the variance-covariance matrices of the MLEs and the approximate estimators can be readily determined by inverting the 2 × 2 information matrices. Let iD2 In L ê,u2

V1 a2 '

~2 in L ~#~a

V2 a2 ,

and

~2 In L ~a 2

V3 0-2

The observed information matrix can be inverted to obtain the asymptotic variance-covariance matrix of the estimators as {aß~[V 2Fi V23J}-l:a2[V]~

W1222] ,

(41)

where V 11 _

V3

Vl~_ ~2,

VI2 _

V2

~v3_ ~2,

V22 _

V1

~~-~2

Similarly, we may obtain the terms V,11, V,12, V,22 from the observed Fisher information corresponding to the approximate likelihood equations. These are not presented here for conciseness.

6. Simulation results

In this section, we discuss results of a simulation study comparing the performance of the approximate estimators with the MLEs. Using the algorithm presented in Balakrishnan and Sandhu (1995), we generated progressively Type-II censored samples from the standard logistic distribution. We computed the approximate estimates from Eqs. (26) and (20). The MLEs of the parameters were obtained by solving the nonlinear Eqs. (7) and (8) using the IMSL nonlinear equation solver, in which the approximate estimates were used as starting values for the iterations. Tables 1 and 2 provide the average values of the estimates, their variances, and covariance. In addition, we also present the averaged values of variances and covariances determined from the observed Fisher information matrix for the purpose of comparison. All the averages were computed over 10,000 simulations. It may be noted that the MSEs for certain specific censoring schemes are comparable to those presented in Balakrishnan (1992) for conventional Type-II censored samples. From Tables 1 and 2, we observe that the approximate estimators and the MLEs are almost identical in terms of both bias and variance. The approximate estimators are almost as efficient as the MLEs for all sample sizes and censoring schemes. As the effective sample proportion m/n increases, the bias and variance

Point and interval estimation for parameters of the log&tic distribution

443

Table 1 Average, variances and covariance of the M L E s n 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 25 25 25 25 25 25 50 50 50 50 100 100 100 100

m 2

2 3 3 4 4 5 5 2 2 3 3 4 4 5 5 5 5 5 5 2 2 3 3 4 4 5 5 5 5 5 5 5 10 10 5 5 10 10 15 15 20 20 25 25 20 20 50 50

Scheme (0,8)

/)

-0.5519 -0.3151 (2*0,7) -0.2694 (7,2*0) -0.1473 (3*0,6) -0.1458 (6,3*0) -0.0902 (4*0,5) -0.0832 (5,4*0) -0.0559 (0,13) -0.6802 (13,0) ù0.3421 (2.0,12) -0.3598 (12,2.0) -0.1643 (3.0,11) -0.2138 (t 1,3.0) -0.1068 (4.0,10) -0.1389 (10,4.0) ù0.0722 (0,10,3"0) -0.0928 (2"0,10,2"0)-0.1083 (2,2,2,2,2) -0.1102 (4,4,2,2*0) -0.0920 (0,18) -0.7688 (18,0) -0.3544 (2"0,17) -0.4219 (17,2"0) -0.1712 (3"0,16) -0.2604 (16,3"0) -0.1141 (4"0,15) -0.1774 (15,4"0) -0.0796 (10,5,3"0) -0.0923 (5,5,5,2*0) -0.1099 (3,3,3,3,3) -0.1389 (0,15,3"0) -0.1034 (5,10,3"0) -0.0991 (9"0,10) -0.0392 (10,9"0) -0.0225 (4*0,20) -0.2065 (20,4*0) -0.0837 (9"0,15) -0.0543 (15,9"0) -0.0265 (14"0,i0) -0.0189 (10,14"0) -0.0118 (19"0,30) -0.0271 (30,19"0) -0.0105 (24*0,25) -0.0149 (25,24*0) -0.0045 (19"0,80) -0.0498 (80,19"0) -0.0136 (49*0,50) -0.0065 (50,49*0) 0.0000

(8,0)

6"

Var@)

Var(#)

Cov@, «) ~11

~22

~12

0.5175 0.7701 0.6935 0.8718 0.7883 0.9043 0.8399 0.9179 0.5094 0.7994 0.6825 0.8962 0.7762 0.9251 0.8265 0.9364 0.9120 0.8871 0.8648 0.9067 0.5054 0.8186 0.6772 0.9115 0.7705 0.9380 0.8202 0.9479 0.9304 0.9081 0.8608 0.9216 0.9243 0.9226 0.9629 0.8165 0.9560 0.9184 0.9684 0.9506 0.9713 0.9597 0.9847 0.9683 0.9850 0.9557 0.9920 0.9841 0.9923

0.4015 0.4554 0.2352 0.2963 0.1571 0.2170 0.1261 0.1798 0.5013 0.4963 0.2960 0.3110 0.1915 0.2218 0.1389 0.1821 0.1728 0.1636 0.1366 0.1631 0.5905 0.5282 0.3574 0.3232 0.2345 0.2270 0.1651 0.1853 0.i766 0.1695 0.1452 0.1812 0.1787 0.0597 0.0882 0.1939 0.1883 0.0622 0.0882 0.0408 0.0586 0.0306 0.0449 0.0235 0.0352 0.0502 0.0451 0.0118 0.0184

0.2461 0.2242 0.2047 0.1645 0.1745 0.1364 0.1391 0.1163 0.2466 0.2088 0.2084 0.1501 0.1810 0.1240 0.1457 0.1057 0.1072 0.1198 0.1220 0.1117 0.2469 0.1976 0.2103 0.1401 0.1846 0.1156 0.1495 0.0986 0.t002 0.1081 0.1207 0.0995 0.1000 0.0758 0.0596 0.1520 0.0933 0.0787 0.0566 0.0480 0.0410 0.0405 0.0292 0.0304 0.0243 0.0443 0.0259 0.0155 0.0127

0.2081 0.1594 0.1262 0.0818 0.0748 0.0427 0.0435 0.0307 0.2664 0.1842 0.1783 0.0963 0.1220 0.0544 0.0803 0.0405 0.0602 0.0725 0.0595 0.0576 0.3074 0.1979 0.2153 0.1037 0.1561 0.0603 0.1081 0.0454 0.0585 0.0721 0.0758 0.0686 0.0649 0.0226 0.0100 0.1299 0.0482 0.0340 0.0120 0.0095 0.0051 0.0169 0.0039 0.0092 0.0028 0.0372 0.0053 0.0044 0.0009

0.2297 0.1509 0.1952 0.1422 0.1619 0.1252 0.1310 0.1092 0.2350 0.1384 0.2024 0.1294 0.1708 0.1138 0.1398 0.0994 0.0980 0.1097 0.1117 0.1040 0.2377 0.1299 0.2063 0.1207 0.1756 0.1060 0.1447 0.0928 0.0938 0.1003 0.1107 0.0908 0.0922 0.0722 0.0578 0.1479 0.0878 0.0753 0.0551 0.0473 0.0407 0.0393 0.0289 0.0302 0.0243 0.0433 0.0258 0.0154 0.0126

0.1891 0.0967 0.1128 0.0636 0.0651 0.0379 0.0351 0.0228 0.2509 0.1098 0.1677 0.0762 0.1118 0.0486 0.0725 0.0318 0.0538 0.0660 0.0484 0.0511 0.2940 0.1169 0.2065 0.0830 0.1458 0.0544 0.1008 0.0366 0.0514 0.0648 0.0635 0.0613 0.0579 0.0198 0.0085 0.1230 0.0396 0.0309 0.0103 0.0081 0.0037 0.0162 0.0036 0.0084 0.0021 0.0361 0.0049 0.0043 0.0007

0.2521 0.3526 0.1609 0.2766 0.1194 0.2132 0.0983 0.1714 0.3562 0.3709 0.2230 0.2894 0.1521 0.2203 0.1094 0.1751 0.1592 0.1458 0.1096 0.1482 0.4483 0.3852 0.2859 0.2994 0.1939 0.2262 0.1351 0.1786 0.1645 0.1529 0.1149 0.1670 0.1652 0.0540 0.0880 0.1636 0.1815 0.0557 0.0885 0.0387 0.0590 0.0291 0.0447 0.0225 0.0358 0.0481 0.0451 0.0115 0.0180

444

N. Balakrishnan and N. Kannan

Table 2 Average, variances and covariance of the approximate estimators n 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 25 25 25 25 25 25 50 50 50 50 100 100 100 100

m

2 2 3 3 4 4 5 5 2

2 3 3 4 4 5 5 5 5 5 5 2 2 3 3 4 4 5 5 5 5 5 5 5 10 10 5 5 10 I0 15 15 20 20 25 25 20 20 50 50

Scheme

(0,8)

»

-0.5403 -0.3143 -0.2657 -0.1469 -0.1488 -0.0915 -0.0905 -0.0590 -0.6635 (13,0) -0.3401 (2"0,12) -0.3481 (12,2"0) -0.1627 (3"0,11) -0.2074 (11,3"0) -0.1059 (4"0,10) -0.1369 (10,4"0) -0.0738 (0,10,3'0) -0.0724 (2"0,10,2"0) -0.0793 (2,2,2,2,2) -0.1167 (4,4,2,2*0) -0.0769 (0,18) -0.7509 (18,0) -0.3518 (2"0,17) -0.4073 (17,2"0) -0.1688 (3"0,16) -0.2500 (16,3"0) -0.1115 (4"0,15) -0.1708 (15,4"0) -0.0797 (10,5,3"0) -0.0825 (5,5,5,2*0) -0.0869 (3,3,3,3,3) -0.1423 (0,15,3'0) -0.0808 (5,10,3'0) -0.0803 (9"0,10) -0.0427 (10,9"0) -0.0304 (4*0,20) -0.1977 (20,4*0) -0.0825 (9'0,15) -0.0548 (15,9'0) -0.0348 (14"0,10) -0.0229 (10,14"0) -0.0192 (19"0,30) -0.0273 (30,19"0) -0.0200 (24*0,25) -0.0163 (25,24*0) -0.0123 (19"0,80) -0.0476 (80,19"0) -0.0237 (49*0,50) -0.0072 (50,49*0) -0.0056 (8,0) (2*0,7) (7,2*0) (3*0,6) (6,3*0) (4*0,5) (5,4*0) (0,13)

6

Var())

Var(ä)

Cov(~, Õ-) ~11

722

712

0.5471 0.8029 0.7256 0.9242 0.8193 0.9666 0.8687 0.9823 0.5335 0.8286 0.7094 0.9444 0.8027 0.9843 0.8514 0.9995 0.9604 0.9375 0.9013 0.9717 0.5255 0.8451 0.6999 0.9563 0.7932 0.9943 0.8418 1.0088 0.9969 0.9721 0.8966 0.9661 0.9834 0.9396 1.0175 0.8354 1.0147 0.9336 1.0242 0.9633 1.0142 0.9673 1.0266 0.9752 1.0198 0.9603 1.0361 0.9873 1.0150

0.4109 0.4557 0.2366 0.3006 0.1566 0.2209 0.1254 0.1832 0.5190 0.4972 0.3024 0.3154 0.1937 0.2260 0.1394 0.1858 0.1779 0.1703 0.1357 0.1675 0.6125 0.5295 0.3670 0.3276 0.2389 0.2315 0.1671 0.1892 0.1807 0.1757 0.1449 0.1873 0.1839 0.0596 0.0892 0.1970 0.1924 0.0622 0.0892 0.0407 0.0590 0.0307 0.0453 0.0235 0.0354 0.0504 0.0455 0.0118 0.0185

0.2751 0.2437 0.2249 0.1850 0.1896 0.1572 0.1502 0.1354 0.2705 0.2243 0.2256 0.1668 0.1943 0.1415 0.1556 0.1225 0.1199 0.1352 0.1344 0.1305 0.2669 0.2105 0.2250 0.1543 0.1960 0.1308 0.1582 0.1135 0.1177 0.1261 0.1330 0.1103 0.1152 0.0791 0.0693 0.1596 0.1068 0.0817 0.0663 0.0496 0.0468 0.0412 0.0343 0.0309 0.0275 0.0448 0.0310 0.0156 0.0139

0.2258 0.1664 0.1335 0.0866 0.0774 0.0446 0.0438 0.0312 0.2875 0.1915 0.1894 0.1017 0.1280 0.0574 0.0834 0.0420 0.0671 0.0821 0.0613 0.0643 0.3287 0.2050 0.2275 0.1092 0.1635 0.0639 0.1125 0.0474 0.0636 0.0811 0.0792 0.0757 0.0720 0.0228 0.0090 0.1349 0.0506 0.0346 0.0110 0.0094 0.0042 0.0171 0.0029 0.0092 0.0022 0.0375 0.0042 0.0044 0.0006

0.2211 0.1341 0.1852 0.1200 0.1531 0.1061 0.1240 0.0945 0.2284 0.1218 0.t943 0.1076 0.1632 0.0946 0.1334 0.0841 0.0884 0.1000 0.1045 0.0923 0.2324 0.1137 0.1995 0.0995 0.1690 0.0872 0.1390 0.0774 0.0813 0.0889 0.1037 0.0812 0.0820 0.0695 0.0517 0.1428 0.0727 0.0728 0.0487 0.0459 0.0375 0.0385 0.0265 0.0296 0.0227 0.0427 0.0228 0.0152 0.0121

0.1931 0.1014 0.1172 0.0631 0.0702 0.0402 0.0401 0.0258 0.2515 0.1130 0.1680 0.0732 0.1130 0.0496 0.0744 0.0347 0.0467 0.0562 0.0526 0.0463 0.2932 0.1190 0.2051 0.0782 0.1451 0.0542 0.1009 0.0390 0.0477 0.0574 0.0674 0.0532 0.0516 0.0211 0.0105 0.1221 0.0415 0.0317 0.0126 0.0089 0.0047 0.0164 0.0046 0.0086 0.0027 0.0359 0.0061 0.0043 0.0009

0.2742 0.2808 0.1757 0.2253 0.1287 0.1860 0.1029 0.1569 0.3728 0.3060 0.2349 0.2371 0.1611 0.1918 0.1159 0.1597 0.1414 0.1325 0.1143 0.1403 0.4609 0.3245 0.2947 0.2461 0.2010 0.1967 0.1407 0.1627 0.1537 0.1436 0.1231 0.1467 0.1505 0.0553 0.0872 0.1680 0.1652 0.0574 0.0878 0.0389 0.0593 0.0295 0.0457 0.0227 0.0365 0.0482 0.0464 0.0115 0.0184

Point and interval estimation for parameters of the logistic distribution

445

Table 3 Variances and covariance of the M L E s from observed and expected information using the approximate covariance matrix n

m

Scheme

~11

~22

~12

zll

/22

/12

I0 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 25 25 25 25 25 25 50 50 50 50 100 100 100 100

2 2 3 3 4 4 5 5 2 2 3 3 4 4 5 5 5 5 5 5 2 2 3 3 4 4 5 5 5 5 5 5 5 10 10 5 5 10 10 15 15 20 20 25 25 20 20 50 50

(0,8) (8,0) (2*0,7) (7,2*0) (3*0,6) (6,3*0) (4*0,5) (5,4*0) (0,13) (13,0) (2"0,12) (12,2"0) (3"0,11) (11,3"0) (4"0,10) (10,4"0) (0,10,3"0) (2"0,10,2"0) (2,2,2,2,2) (4,4,2,2*0) (0,18) (18,0) (2"0,17) (17,2"0) (3"0,16) (16,3"0) (4"0,15) (15,4"0) (10,5,3"0) (5,5,5,2*0) (3,3,3,3,3) (0,15,3"0) (5,10,3"0) (9"0,10) (10,9"0) (4*0,20) (20,4*0) (9"0,15) (15,9"0) (14"0,10) (10,14"0) (19"0,30) (30,19"0) (24*0,25) (25,24*0) (19"0,80) (80,19"0) (49*0,50) (50,49*0)

0.25212 0.35264 0.16090 0.27660 0.11941 0.21322 0.09827 0.17140 0.35618 0.37095 0.22305 0.28935 0.15209 0.22029 0.10945 0.17511 0.15918 0.14582 0.10960 0.14822 0.44827 0.38516 0.28586 0.29936 0.19393 0.22621 0.13512 0.17861 0.16449 0.15290 0.11489 0.16704 0.16518 0.05396 0.08804 0.16356 0.18153 0.05568 0.08845 0.03867 0.05897 0.02907 0.04468 0.02254 0.03576 0.04807 0.04514 0.01145 0.01802

0.22969 0.15088 0.19515 0.14219 0.16189 0.12520 0.13104 0.10916 0.23499 0.13841 0.20244 0.12940 0.17076 0.11383 0.13976 0.09945 0.09804 0.10973 0.11172 0.10397 0.23768 0.12985 0.20627 0.12067 0.17557 0.10604 0.14473 0.09277 0.09383 0.10027 0.11066 0.09080 0.09224 0.07220 0.05775 0.14786 0.08777 0.07534 0.05510 0.04732 0.04069 0.03933 0.02891 0.03022 0.02428 0.04329 0.02578 0.01537 0.01259

0.18913 0.09672 0.11284 0.06365 0.06511 0.03786 0.03510 0.02282 0.25091 0.10982 0.16768 0.07624 0.11182 0.04865 0.07248 0.03179 0.05384 0.06604 0.04840 0.05107 0.29401 0.11692 0.20650 0.08298 0.14582 0.05444 0.10076 0.03660 0.05136 0.06480 0.06349 0.06133 0.05790 0.01979 0.00852 0.12299 0.03957 0.03089 0.01034 0.00810 0.00374 0.01621 0.00362 0.00837 0.00212 0.03614 0.00492 0.00427 0.00071

0.21987 0.32860 0.16008 0.24015 0.12381 0.18876 0.10201 0.15581 0.29373 0.38067 0.21052 0.26152 0.15223 0.19797 0.11232 0.15943 0.14869 0.14000 0.11234 0.13941 0.36363 0.42140 0.26467 0.27932 0.19058 0.20695 0.13668 0.16426 0.15355 0.14602 0.11920 0.15856 0.15541 0.05527 0.08417 0.16437 0.16873 0.05729 0.08440 0.03910 0.05744 0.02967 0.04368 0.02282 0.03517 0.04984 0.04415 0.01154 0.01789

0.28366 0.38553 0.25194 0.28868 0.21298 0.22520 0.17407 0.18272 0.27527 0.34619 0.24800 0.25634 0.21272 0.20049 0.17501 0.16364 0.15582 0.16703 0.15401 0.16139 0.27139 0.31856 0.24669 0.23389 0.21360 0.18329 0.17686 0.15028 0.14793 0.15131 0.14644 0.14183 0.14502 0.08707 0.08049 0.17835 0.14024 0.08936 0.07622 0.05536 0.05270 0.04392 0.03550 0.03341 0.02912 0.04793 0.03094 0.01632 0.01409

0.18199 0.20122 0.12186 0.10448 0.07506 0.05761 0.04259 0.03289 0.23508 0.23652 0.17418 0.12857 0.12205 0.07651 0.08137 0.04848 0.06957 0.08223 0.06105 0.06421 0.27342 0.25480 0.21213 0.14061 0.15681 0.08619 0.11091 0.05669 0.06778 0.08112 0.07714 0.08045 0.07556 0.02277 0.01165 0.13442 0.06158 0.03469 0.01457 0.00929 0.00467 0.01761 0.00472 0.00909 0.00262 0.03905 0.00671 0.00450 0.00081

446

N. Balakrishnan and N. Kannan

of the estimators reduce significantly. For a fixed n and m, we can determine the censoring scheine that is most efficient. For almost all choices, the censoring scheine RI = n - m , R 2 . . . . . Rm = 0 seems to provide the smallest bias and variance for the estimates. These results are similar to those obtained by Balakrishnan et al. (2001) in the case of the normal distribution. Table 3 provides a comparison of the variances and covariances of the estimators using the observed and expected information matrix. These values are computed from the likelihood equations. When n and m are small, the values are quite different. However, when the effective sample size m is 1arge, the values seem to be in agreement. The large differences may be attributed to the error involved in the approximation of the moments of Y~':m:n.To ascertain the veracity of this statement, we computed the moments of Yii:m:n using Monte Carlo simulations and used those values in the expression of the expected information. Table 4 provides a comparison of the variances and covariance based on the observed and expected information using the values of the simulated moments. It is clear that these values are indeed closer, and some of the significant discrepancies noted earlier in Table 3 no longer exist. So, there is a need for caution while using first order approximations to moments in case of small effective sample sizes. Table 5 provides a comparison of the weighted least squares estimators with the MLEs. The estimators are close both in terms of bias and variance. However, when the effective sample size is small, the BLUEs are not very reliable. This is not surprising because the BLUEs are only asymptotically efficient for large n and m. Another reason for the poor performance of the BLUEs may be due to the approximation of the variance-covariance matrix of Xi:m:nS. It is reasonable to assume that the performance of the BLUEs would improve considerably if the true variance-covariance matrix was available.

7. Coverage probabilities Since (~)

is asymptotically normally distributed, we have the asymptotic

distribution of Table 4 Variances and covariance of the MLEs from observed and expected information using the simulated covariance matrix n

in

Scheme

~11

~22

~12

111

i22

/-12

10 10 10 15 15 20 20 20

2 2 3 2 3 2 3 4

(0,8) (8,0) (7,2*0) (13,0) (12,2"0) (18,0) (17,2"0) (16,3'0)

0.25212 0.35264 0.27660 0.37095 0.28935 0.38516 0.29936 0.22621

0.22969 0.15088 0.14219 0.13841 0.12940 0.12985 0.12067 0.10604

0.18913 0.09672 0.06365 0.10982 0.07624 0.11692 0.08298 0.05444

0.11158 0.24160 0.20012 0.28581 0.22138 0.32059 0.23838 0.18390

0.09609 0.15392 0.13434 0.14932 0.12620 0.14479 0.11970 0.09849

0.07456 0.11053 0.06738 0.13424 0.08360 0.14861 0.09264 0.06021

Point and interval estimation for parameters of the logistic distribution

o

o o o o o ~ 0 0 0 0 ~ 0

d d d d d d

ù~

ô d d d d d d <~

3

;>

<

I

I

I

I

I

I

447

448

N. Balakrishnan and N. Kannan

Table 6 95% and 90% coverage probabilities for the pivotal quantities based on M L E s n

m

Scheine

10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 25 25 25 25 25 25 50 50 50 50 100 100 100 100

2 2 3 3 4 4 5 5 2 2 3 3 4 4 5 5 5 5 5 5 2 2 3 3 4 4 5 5 5 5 5 5 5 10 10 5 5 10 10 15 15 20 20 25 25 20 20 50 50

(0,8) (8,0) (2*0,7) (7,2*0) (3*0,6) (6,3*0) (4*0,5) (5,4*0) (0,13) (i3,0) (2"0,12) (12,2"0) (3"0,11) (11,3"0) (4"0,10) (10,4"0) (0,10,3"0) (2"0,10,2"0) (2,2,2,2,2) (4,4,2,2*0) (0,18) (18,0) (2"0,17) (17,2"0) (3"0,16) (16,3"0) (4"0,15) (15,4"0) (10,5,3"0) (5,5,5,2*0) (3,3,3,3,3) (0,15,3"0) (5,10,3"0) (9'0,10) (10,9"0) (4*0,20) (20,4*0) (9"0,15) (15,9"0) (14"0,10) (10,14"0) (19"0,30) (30,19"0) (24*0,25) (25,24*0) (19"0,80) (80,19"0) (49*0,50) (50,49*0)

P1 46.6 71.2 65.3 83.2 75.5 87.3 81.1 88.6 43.7 71.7 61.9 83.6 72.0 87.5 77.5 89.0 87.1 84.5 82.0 86.4 42.8 72.3 60.6 84.1 70.1 87.6 75.4 89.2 87.8 85.3 79.7 87.1 87.4 88.3 92.4 74.3 89.3 86.9 92.6 91.4 93.2 91.0 93.7 92.3 94.0 89.2 93.9 93.6 94.0

P2 43.1 66.7 60.7 78.3 69.8 81.6 75.1 83.3 40.3 67.3 57.9 78.9 67.4 82.0 72.4 83.7 81.9 79.3 76.3 81.1 39.1 67.6 56.3 79.1 65.7 82.4 70.5 84.0 82.6 80.3 74.5 81.6 82.0 83.0 87.0 69.4 84.1 82.1 87.3 86.2 88.2 86.1 88.5 87.2 89.0 84.5 88.7 88.5 89.0

90.8 91.8 91.4 94.1 92.9 94.7 93.1 94.5 92.9 90.4 92.2 93.8 92.9 94.5 92.7 94.3 94.0 93.7 92.8 93.8 94.2 89.6 93.0 93.4 93.2 94.3 93.i 94.2 93.7 93.6 92.3 93.8 93.8 94.5 95.0 93.5 94.0 94.0 95.0 94.7 95.0 94.6 94.9 94.5 95.2 94.6 94.8 94.7 94.5

P3 83.5 85.6 85.1 89.3 87.5 89.8 88.0 89.7 85.0 83.6 85.3 88.4 86.7 89.5 87.2 89.1 88.8 88.5 87.4 88.5 86.4 82.1 86.2 87.7 86.9 89.1 87.4 88.9 88.7 88.2 86.5 88.6 88.7 89.3 90.3 87.6 88.7 88.5 90.2 90.0 90.3 89.4 89.7 89.6 90.3 89.1 89.7 89.5 89.7

44.2 64.8 61.7 76.2 70.3 80.4 75.2 82.6 43.5 67.2 61.1 78.6 69.3 82.4 74.3 84.4 81.9 79.7 77.4 81.7 43.2 68.9 60.6 80.1 68.9 83.8 73.7 85.7 84.2 82.0 77.2 83.1 83.4 84.7 89.1 73.3 86.7 84.3 89.7 88.1 90.3 89.7 92.2 90.5 92.5 89.2 93.0 93.0 93.7

40.3 60.8 57.i 72.1 65.5 76.4 70.7 77.8 39.9 63.7 56.3 74.6 64.6 78.3 69.5 80.0 77.5 75.0 72.8 77.0 39.6 65.4 55.9 76.0 64.2 79.7 68.8 81.4 79.6 77.2 72.5 78.5 78.9 80.0 84.7 68.7 82.1 79.5 85.4 84.0 85.7 84.9 87.4 85.7 87.2 84.5 88.3 87.7 88.7

Point and interval estimation for parameters of the logistic distribution

449

Table 7 95% and 90% coverage probabilities for the pivotal quantities based on approximate estimators n

m

Scheine

10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 25 25 25 25 25 25 50 50 50 50 i00 100 100 100

2 2 3 3 4 4 5 5 2 2 3 3 4 4 5 5 5 5 5 5 2 2 3 3 4 4 5 5 5 5 5 5 5 10 I0 5 5 10 10 15 15 20 20 25 25 20 20 50 50

(0,8) (8,0) (2*0,7) (7,2*0) (3*0,6) (6,3*0) (4*0,5) (5,4*0) (0,13) (13,0) (2"0,12) (12,2"0) (3"0,11) (11,3"0) (4"0,10) (10,4'0) (0,10,3"0) (2"0,10,2"0) (2,2,2,2,2) (4,4,2,2*0) (0,18) (18,0) (2"0,17) (17,2'0) (3"0,16) (16,3'0) (4'0,15) (15,4"0) (10,5,3"0) (5,5,5,2*0) (3,3,3,3,3) (0,15,3"0) (5,10,3"0) (9"0,10) (10,9"0) (4*0,20) (20,4*0) (9"0,15) (15,9"0) (14"0,10) (10,14"0) (19"0,30) (30,19"0) (24*0,25) (25,24*0) (19"0,80) (80,19'0) (49*0,50) (50,49*0)

Ql 47.9 68.4 66.5 80.7 76.4 85.2 81.5 87.5 44.7 69.4 62.9 81.4 73.0 85.7 78.3 87.8 85.8 84.0 82.3 86.0 43.5 70.4 61.3 81.8 70.9 86.0 76.2 87.9 87.0 85.2 80.5 85.8 86.4 88.5 92.2 74.8 88.1 87.3 92.2 91.5 93.2 91.2 93.8 92.4 94.0 89.4 94.0 93.6 94.2

Q2 44.4 63.6 61.9 74.8 71.0 78.9 75.6 81.3 41.1 64.3 58.8 75.5 68.3 79.5 73.3 82.0 79.9 78.3 76.6 80.4 40.0 65.0 57.0 75.9 66.6 79.8 71.4 82.4 81.5 79.6 75.3 79.9 80.8 83.3 86.7 70.2 82.5 82.5 86.9 86.2 88.0 86.3 88.6 87.3 89.3 84.7 88.9 88.6 89.4

90.2 86.5 91.4 89.7 92.8 91.0 92.7 91.4 92.1 85.5 91.7 89.1 92.8 90.6 92.7 91.0 90.9 90.9 92.2 91.2 93.5 84.8 92.4 88.9 93.0 90.5 92.9 90.8 90.8 90.8 92.1 90.7 90.8 94.3 93.4 93.3 90.9 94.0 93.3 94.4 93.8 94.6 93.9 94.5 94.3 94.5 93.8 94.7 94.0

Q3 82.9 78.8 85.0 82.9 87.4 84.6 87.5 85.5 84.1 77.4 85.0 82.3 86.6 84.2 87.1 84.9 84.5 84.8 86.4 85.0 85.4 76.8 85.5 82.0 86.5 84.1 87.2 84.7 84.4 84.6 86.0 84.4 84.4 89.0 88.1 87.3 84.5 88.5 88.0 89.6 88.7 89.4 88.2 89.5 89.4 89.0 88.2 89.5 89.1

44.7 64.7 62.2 76.1 70.9 80.6 75.6 83.2 44.0 66.9 61.4 78.3 69.7 82.2 74.7 84.6 82.6 80.5 78.1 82.7 43.6 68.5 60.9 79.5 69.1 83.2 74.1 85.6 84.7 82.9 77.8 83.4 84.2 85.0 89.0 73.7 86.2 84.6 89.3 88.3 90.5 89.8 91.3 90.7 92.0 89.3 91.0 93.1 92.8

40.9 60.7 57.9 71.8 66.2 76.0 71.3 77.8 40.4 63.3 56.9 73.6 65.2 77.3 70.1 79.4 77.8 75.7 73.6 77.6 40.0 64.9 56.4 74.8 64.6 78.1 69.4 80.1 79.3 77.8 73.4 78.5 79.1 80.1 83.3 69.0 80.5 79.6 83.6 83.9 84.6 84.9 85.4 85.5 86.1 84.5 85.0 87.6 87.7

N. Balakrishnan and N. Kannan

450

.

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.

.

.

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o. ~ .- ~. ~. ~

o ~ o

o ~

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~z

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.

~. ~o~~~ ~- -~ ~~~ o- ~~ ~~ ~- õ-

7?7

.o

.

~ - o

~

~

Point and interval estimation for parameters of the logistic distribution

I I I I I I l l l l l l l l ~ l l l l l l l l l l

o~

~~

~ o ~ ~ ~ ~ ~ o

451

N. Balakrishnan and N. Kannan

452

©

©

©

ù~

©

I

l

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I

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l

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Point and interval estimation for parameters of the logistic distribution

~?~~~?~??~7~~~~

~~~??~?~~?~~??~~~~~~~~~~~

453

454

N. Balakrishnan and N. Kannan

-

~-#

P2-

~-#

P3 = ä - a

(42)

to be standard normal. Similarly, the asymptotic distribution of Qi, i = 1,2, 3, the pivotal quantities based on the approximate estimators, is also standard normal. Through Monte Carlo simulations, we simulate the probability coverages P(-1.65_
i=1,2,3

which we expect to be approximately 90% and 95%, respectively. We repeated this process for the pivotal quantities Qi as well. Tables 6 and 7 provide these results of the simulation for the MLEs and the approximate estimators. It is clear that when a is unknown, the probability coverages are extremely unsatisfactory especially when the effective relative sample size, viz., m / n is small. If a is known, the coverage probabilities for P2 and Q2 are close to the required levels. In most practical situations, however, « is unknown and hence using the asymptotic normal approximation for the corresponding pivotal quantities is not advisable. The distributions of the pivotal quantities have been observed to be extremely skewed, which render the normal approximation to be unsatisfactory. In Tables 8 and 9, we provide the unconditional percentage points for these pivotal quantities determined through Monte Carlo simulations. It is clear for small sample sizes, the percentage points are very different from what might be expected if the distributions were normal. These percentage points will be helpful to construct unconditional confidence intervals for # and a based on the pivotal quantities in (42).

8. Illustrative e x a m p l e

Consider the following progressively Type-II right censored data giving the logtimes to breakdown of an insulating fluid tested at 34 kV. These data are taken from Nelson (1982, Table 6.1), and have been used earlier by Viveros and Balakrishnan (1994). i

1

2

xl...... Ri

-1.6608 -0.2485 -0.0409 0.2700 1.0224 1.5789 1.8718 1.9947 0

3 0

4 3

0

5 3

6 0

7 0

8 5

For this example, n = 19, m = 8. Assuming that the data have come from the logistic distribution in (1), we obtain the MLEs of # and a as B=1.876

and

# = 1.637 .

Point and interval estimation for parameters of the logistic distribution

455

We simulated the percentage points for a 90% confidence interval to obtain P ( - 2 . 9 3 < P1 _< 1.42) = 0.90 and P ( - 3 . 7 2 _< P3 _< 1.02) = 0.90, from which we obtain the 90% confidence interval for # to be (1.24, 3.19) and the 90% confidence interval for a to be (1.14, 3.42). The approximate estimators of/~ and a are =1.867

and

ä=1.658

.

We simulated the percentage points for a 90% confidence interval to obtain P ( - 2 . 9 3 _< Q~ _< 1.39) = 0.90 and P ( - 3 . 7 4 < Q3 _< 1.12) =- 0.90, from which we obtain the 90% confidence interval for # to be (1.12, 3.45) and the 90% confidence interval for « to be (1.08, 3.59). Assuming that the data had come from a normal distribution, Balakrishnan et al. [2001] obtained the MLEs of # and a as : 1.882

and

&=1.615 .

The 90% confidence interval for /~ was (1.26,3.41) and the 90% confidence interval for a was (1.17,3.16). Since the logistic distribution is heavier tailed, the confidence interval given above for the scale parameter of the logistic distribution is marginally wider than the corresponding interval for the normal scale parameter.

9. Conclusions

In this article, we have considered point and interval estimation for parameters of the logistic distribution based on progressively Type-II censored data. An approximation of the cdf of the logistic distribution has been used to derive explicit approximate estimators of the parameters. A procedure based on generalized least squares is also discussed in order to present some initial estimators which may be used as efficient starting values in the computation of the MLEs. Results of a simulation study show that the approximate estimators and the MLEs are almost identical in terms of bias and variance. However, the BLUEs are not very efficient especially when the effective sample size is small. A simulation study was conducted to examine the coverage probabilities of the pivotal quantities based on asymptotic normality. The coverages obtained are extremely unsatisfactory especially when the effective sample size is small. Based on these observations, we recommend the use of simulated unconditional percentage points to construct confidence intervals for the parameters.

References Cohen, A. C. (1991). Truncated and Censored Samples." Theory and Applications. Marcel Dekker, New York. Balakrishnan, N. and A. C. Cohen (1991). Order Statistics and Inference: Estirnation Methods. Academic Press, San Diego.

456

N. Balakrishnan and N. Kannan

Cohen, A. C. and B. J. Whitten (1988). Parameter Estimation in Reliability and Life Span Models. Marcel Dekker, New York. Cohen, A. C. (1963). Progressively censored samples in life testing. Teehnometries 5, 327 329. Cohen, A. C. (1966). Life testing and early failure. Technometrics 8, 539 549. Mann, N. R. (1969). Exact three-order-statistic confidence bounds on reliable life for a Weibull model with progressive censoring. J. Am. Stat. Assoc. 64, 306-315. Mann, N. R. (1971). Best linear invariant estimation for Weibull parameters under progressive censoring. Teehnometrics 13, 521-533. Viveros, R. and N. Balakrishnan (1994). Interval estimation of parameters of life from progressively censored data. Technometrics 36, 84-91. Balakrishnan, N., N. Kannan, C. T. Lin and H. K. T. Ng (2001). Point and interval estimation for the normal distribution based on progressively Type-II censored samples. IEEE Trans. Reliab. (under revision). Balakrishnan, N. and R. Aggarwala (2000). Progressive Censoring: Theory, Methods, andApplications. Birkhäuser, Boston. Balakrishnan, N. (Ed.) (1992). Handbook of the Logistic Distribution. Marcel Dekker, New York. Gajjar, A. V. and C. G. Khatri (1968). Progressively censored samples from log-normal and logistic distributions. Teehnometrics 11, 793 803. Tiku, M. L., W. Y. Tan and N. Balakrishnan (1986). Robust Inference. Marcel Dekker, New York. Balakrishnan, N. and R. A. Sandlau (1995). A simple simulational algorithm for generating progressive Type-II censored samples. The Amer. Statistieian 49, 229-230. Balakrishnan, N. and C. R. Rao (1997). Large-sample approximations to the best linear unbiased estimation and best linear unbiased prediction based on progressively censored samples and some applications. In Advanees in Statistieal Deeision Theory and Applications, pp. 431~444 (Eds. S. Panchapakesan and N. Balakrishnan). Birkhäuser, Boston. Nelson, W. (1982). Applied Life Data Analysis. Wiley, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

•

i J

Progressively Censored Variables-Sampling Plans for Life Testing Uditha Balasooriya

Notation

n m L k L(p) B Xi,n Ri

total sample size total number of failures in a progressively censored sample of size n lower specification limit acceptance constant operating characteristic curve likelihood function ith-order statistic from a progressively censored sample of size n number of surviving items withdrawn from the life test at the time o f / t h failure

1. Introduction

It is often necessary to measure the reliability of a product to determine its ability to perform the intended purpose within specifications for the desired period of time. In evaluating the reliability of a product or system, an important quality variable is the lifetime of a product. Sampling plans designed to determine the acceptability of a product with respect to its lifelength are known as reliability sampling plans. For many products that are designed to operate for long periods of time without failure, testing under normal use conditions might be exceedingly time consuming and expensive. On the other hand, there is a need for rapid and upfront testing for products and systems to have them commercially available in the market-place as early as possible. Therefore, various procedures have been used in practice to reduce the duration of the total test time in life test experiments. In life and fatigue tests, it is common practice to cut short the experiment prior to failure of all test specimens by censoring. In the literature, various mechanisms of censoring are identified. In Type I censoring experiments, all specimens are put 457

458

u. Balasooriya

on test simultaneously and the test is terminated at a prespecified time, while in Type Il censoring experiments, the test is terminated when a prespecified number of failures is observed. Another approach used to shorten the test time of an experiment is to conduct accelerated tests. Generally, it is needed to extrapolate the results of accelerated life tests to obtain the parameter estimates of lifelength random variable at normal use conditions. In some life tests, in addition to failure time data, reliability engineers are interested in obtaining degradation-related information. This has become increasingly important particularly for complicated high reliability products with low failure rates. Degradation data, in some instances, can provide more precise information on the process of deterioration than failure-time data. One method to obtain degradation information is to remove some test specimens at different stages of the experiment and then expose them for detailed laboratory experiments, etc. When a prespecified number of test specimens is removed at the time of some failures, the procedure is known as Type II progressive censoring. In the present work, we discuss the design of reliability sampling plans by variables to determine the acceptability of a product whose lifetime distribution is a location-scale distribution. Sampling plans for the exponential and Weibull (extreme-value) distributions are discussed in detail. Our development of sampling plans is based on Type II progressively censored data. Under this method of life testing, functioning items that are removed at different stages of an experiment can provide valuable information on both degradation process and reliability. Further, while the procedure allows for observation of extreme lifelengths, it also releases test facilities for other use (For further details refer to Cohen, 1963; Viveros and Balakrishnan, 1994). Note that the usual complete and Type Il censored sampling methods are special cases of progressive censoring scheme. The progressive censoring scheme has been used by Montanari and Cacciari (1988) in the study of aging carried out on XLPE-insulated cables. The objectives of the study were to evaluate the endurance of cables to electrical and combined thermal-electrical stresses, and to evaluate aging mechanisms. In addition to failure data, aging and other necessary information were obtained from specimens which were removed at specified time points and/or at the time of some failures. Mann (1969, 1971) discussed some applications of progressively censored life test experiments in industrial situations. Halperin et al. (1989) discussed the construction of distribution-free confidence intervals for a parameter of Wilcoxon-Mann-Whitney type under progressive censoring. As an alternative to sampling by variables, orte can employ an attribute sampling for determining the acceptability of a product. Under this approach, a unit which does not satisfy the required reliability specification is classified as non-conforming. If the number of non-conforming items among the n test specimens is less than the acceptance number then the lot is accepted, otherwise, it is rejected or possibly sent for rework. In general, if the quality characteristic under consideration is a quantitative variable such as lifelength of a product then variables-sampling plans require a smaller sample size than attribute sampling

Progressivelycensoredvariables-samplingplansfor life testing

459

plans. The fact is variable sampling plans make use of more sample information than attribute sampling plans. For the case where quality characteristic of interest follows a normal distribution, the design and performance of variables-sampling plans have been discussed in great detail in the literature (see Owen 1964, 1969). Das and Mitra (1964) studied the effect of non-normality on sampling inspection using CornishFisher approximation to compute acceptance and rejection probabilities for specified quality levels (Ps and p/~) for non-normal distributions with known skewness. Rao et al. (1972), using Edgeworth expansion, examined the effect of non-normality on tolerance limits. When the underlying distribution is the betadistribution, Schneider and Wilrich (1981) investigated the OC curve of a variable sampling plan. Kocherlakota and Balakrishnan (1984) discussed two-sided acceptance sampling plans based on modified maximum likelihood estimators. A three-class procedure for acceptance sampling was discussed by Newcombe and Allen (1988). A comparison of methods for univariate and multivariate variablessampling was carried out by Hamilton and Lesperance (1995).

2. Design of variables-sampling plans for progressively censored life-test experiments Suppose the lifetime of a test specimen when measured on a suitable scale (e.g., logarithmic scale) has a location-scale distribution with pdf of the form

f(x;li, a)=lg~--2~-~),

xcRx

,

(2.1)

where - e c 0 and Rx denotes the range of the lifelength variable X. Let L denote the one-sided lower specification limit imposed on the lifelength of the product under consideration. That is, products with lifelength less than L are classified as non-conforming. Thus, the fraction of non-conforming items, p, is given by the probability P(X < L). Note that specification limits in general can be single (lower or upper) or double (lower and upper) limits. As our interest is mainly on life tests, we consider a one-sided lower specification limit throughout this work. Suppose a random sample of n items from a lot is chosen and put on test simultaneously. A life test is conducted under progressive censoring plan until a prespecified number of failures is observed. For deciding the acceptability of a lot, we apply the well-known Lieberman and Resnikoff (1955) procedure in which a lot is accepted if fi - kô- > L ,

(2.2)

where/~ and 6- are estimators based on sample data taken from the lot and k is known as the acceptance constant. Lots with fraction of defectives p, less than Pc are presumed to be good, and the consumer accepts them with a probability of at least 1 - ~ where c~is the producers risk. On the other hand, lots with fraction of

U. Balasooriya

460

defectives, p, greater thanp~ are rejected with a probability of at least 1 - / ~ where B is the consumer's risk. Thus it is required that operating characteristic curve (OC) of the plan passes through the points (Pc, 1 - c~) and (p~,/~). In acceptance sampling, pc is called the acceptance quality level while p~ is known as the limiting quality level. Note that for the case where « is known, the procedure is essentially the same as in the case of it being unknown. In fact it is somewhat simpler in this case as we need to estimate only one parameter. We proceed as above and for a given specification limit L, a lot is accepted if B-ka>L

,

(2.3)

where/~ is the estimator of #. In this study, we consider a progressively censored life test experiment in which m failures (m < n) are observed with Ri (Ri _> 0) functioning items removed at the time of the/th failure. Let X I , , . . ,Xm,n denote the progressive order statistics of a random sample of size n from (2.1) and qi = R i / n for i = 1 , . . ,m. For determining sampling plans, the two points (p~., 1 - e) and (pB,/~) on the OC curve and degrees of censoring, ql,...,qm, are prespecified. The results discussed in the following sections are based on our presentations in Balasooriya (1995), Balasooriya and Saw (1998) and Balasooriya et al. (2000).

2.1. The two-parameter exponentiaI distribution One and two-sided sampling plans for the exponential distribution were obtained by Kocherlakota and Balakrishnan (1986). Balasooriya (1995) considered the failure-censored reliability sampling plans under the situation where specimens are to be tested in batches of fixed size. Variables-sampling plans under Type II progressive censoring were discussed by Balasooriya and Saw (1998). In the case of exponential distribution, the function 9(') in (2.1) is given by

g(u) = exp(-u),

u> 0 .

(2.4)

Note that Qp = ( L - # ) / a - - -ln(1 - p ) where Qp is the pth quantile of (2.4). From (2.2), we have

~-~ (7

k_~ L - ~ G

C7

where fi and ~ are the maximum likelihood estimators (MLEs) given by 1

fi=Xl,n

and

m

~ = m ~ . ,2 S i ( X=i , , - X l , n )

and S/-- Ri + 1, i = 1 , .., m. Regardless of the progressive censoring configuration, these two estimators are independent and their sampling distributions are given by 2n(~ - # ) / « ~ Z~2)and 2m(6/«) ~ Z22m-2(c.f., Viveros and Balakrishnan, 1994).

Progressively censored variables-samplingplansfor life testing

461

Following the arguments of Guenther et al. (1976), Balasooriya and Saw (1998) have shown that the OC curve of an exponential random variable when k > 0 is given by L(p)-

(1-P)= (1 + k) m-'

'

0
,

where m = n - ~i'-1Ri. Solving the two equations corresponding to the given two points (p~, 1 - e) and Coß,fi) on the OC curve and the degree of censoring, ql,..., qm, we obtain

1/(m-l) k=

-1

or k

[(l ~ßPB)'_] l/(m-1)_l '

and

= [ ln/|__-~/_-ln; m

1

Lln( 1 - p c ) - ln(1 - p/~)J

For the case k < 0, the OC curve is given by

{ 1 ~r - (1 ['2m(l+k)ln(1-p)'~ r ~ p)~[ 2(m 13~ ~ ) - - ' 2 ( m -{2mln(1-p)'~ 1)~~), L(p)=

1-I2m(-2mln(1-p)), (l-p)" 1 +~E1-H(p/1-~2
k> -1, k=-l,

1~(~),

k< -1,

where

m--2

H(p) = Z e m(k+l)ln(1 P)/k[--m(k j-0

-I- 1)ln(1 -p)/k]J/j! ,

and/2~(') denotes the incomplete gamma function ratio, given by

ha(w)=

fO w w c-le-w/2

2cr(c~)

dw.

In this case, we need to find m and k by iteratively solving the two equations corresponding to the given specifications. It is noted that L(p) does not depend on the specific progressive censoring configuration chosen but only on the total degree of censoring. The variables-sampling plans discussed here are directly applicable to complete (m = n) and Type II right censoring (Ri = 0 for i = 1 , . . , m - 1 and Rm > 0) cases.

462

U. Balasooriya

In some situations, as described by Johnson (1964), there are advantages in grouping the test specimens into assemblies and then running each assembly simultaneously until the occurrence of a first failure in each group. In this set-up, failure of a specimen in a group is considered as an assembly failure. In a life test of ball bearings, for example, instead of testing 80 bearings with replacement until a prespecified number of failures, an experimenter could test them in groups of 8 each, until the failure of a first bearing occurs in each set. Note that if X(1)l < )((1)2 < . . "X(1)m denote the ordered set of first-order statistics of m samples of size n from the two-parameter exponential distribution, then X(0x _< X(1)2 ~ '' 'X(1)m correspond to an ordered sample from f ( x ( 0 ; # , a ) = (1/«*)exp[-(x(1) - f t ) / « * ] ,

x_> fr, a* > 0 ,

(2.5)

where «* = a/n (see Balasooriya, 1995). Thus, one could also use the above sampling plans with straightforward modifications when specimens are to be tested in assemblies of fixed size, and failure of an item is considered as an assembly failure.

2.2. The Weibull (extreme-value) distribution Fertig and Mann (1980) discussed life-test sampling plans for the Weibull distribution (and extreme-value distribution). For Type II censoring, failurecensored variables-sampling plans for lognormal and Weibull distributions were studied by Schneider (1989). Variables-sampling plans under Type II progressive censoring were considered by Balasooriya et al. (2000). Note that if random variable Z has a Weibull distribution with pdf

f(y;c~,fl)

C~

;3-1exp -

,

y > 0 ,

(2.6)

where c~and/~ are, respectively, the scale and shape parameters of the distribution, then X = in Z has an extreme-value distribution with location parameter # = in and scale parameters cr = 1/il. Thus the function 9(.) in (2.1) is given by

9(u) = euexp{-e"},

- o o < u < oo .

(2.7)

I f / i and ~ denote the MLEs, then using the large sample theory

B -- ke ~ N(}t - ko-, [711 (Y/) -~- k2722(n) - 2k712(n)]a2/n)

,

where 7ij(n)'s are the variance and covariance factors in the asymptotic covariance matrix of (/1, ä)' given by (in terms of Fisher information matrix)

F(n)

[y1](~ )

y12(n) 1

0.2, [«ll(n ) «12(n)]-1 Lv21(n) v22(n)J = n LJ21(n) J22(r/) J

'

463

Progressively censored variables-sampling plans f o r life testing

where 0-2 //8 2 in 5o'~ " - / " / = 7 ~ ~ w s[- ù . ) - /

x-'m S.e~* ù ,

(3"2 / ' ~ 2 l n ~ ~

J12(ù/= J21(ù/= 7 E t ~ T õ 7 ) ~ 0-'

/ ~ 2 In y ' ~

~m=l&eu'(1 + # i )

ù

q,

~,'."-, {13Æ(~2o + 2~, - ~~]S~e", - ~ 3 n

and q = min. The likelihood function £0, given a Type II progressively censored sample Zl,n,..-,Zm,n (with corresponding censoring numbers R 1 , . . , R m ) from Weibull distribution in (2.6), is given by m

i-1

S = --~ H [ n - Z i-1

SJ]g(Yi,n)[G(Yi#)]R* '

j-1

where y~,ù = (Xz,ù - bt)/0-, xi,~ = lnz~,~,E(Y~,~) =/~»i = 1 , . . ,m with 9(') given by (2.7) and G(y) = exp{-eY}. Note that in obtaining the above approximate variance-covariance factors, we considered a first-order Taylor approximation by expanding g'(y~,~)/g(y~,~) and 9(y~,ù)/G(yz,ù) around the actual mean #z of the standardized order statistic Y~,ù.Thus the standardized variate - k « - (~ - k0-) (0-/V/~) [711 (n) -}-/¢2722(n ) -- 2kT12(n)]

is asymptotically distributed as N(0, 1). For given two points (p~, 1 - c) and (p~,fi), and degrees of censoring ql,. • .~ qm, using approximate OC curve, we find that the acceptance constant k and the sample size n are, respectively, given by k = yp«Zl fi - ypvZ~ z« -- Z l - f l

(2.8)

and the sample size n is to be obtained iteratively by

[z.

~-Zl

fi

1'

n = Lyp~ -Yp~J

['~ll(rt) -}-k2~/22(n) -- 2kT~2(n)l ,

(2.9)

where, for 0 < ~ < 1, y~ - in I- ln(1 - ~)] and z~ - q~-I (~) with ~b(.) denoting the standard normal distribution function. A simulation study carried out by Balasooriya et al. (2000) showed that the proposed procedure provides reasonably reliable results for practical purposes. Note that the total degree of censoring and the choice of progressive censoring pattern are important factors when using these large sample results. For the case of Type II censoring, in fact, one can obtain the approximate Fisher information matrix of the maximum likelihood estimators fi and ä without using Taylor approximation to expand g'(Yi,n)/9(Yi,n) and g(Yi,n)/G(yi,n) around

U. Balasooriya

464

the mean/~i. Let p[= (n - m)/n] denote the degree of censoring at the right. In this case, for obtaining F(n) in Section (2.2), one could use the variance-covariance factors given below (see Harter, 1970; Schneider, 1989) that do not require the computation of moments of order statistics: Jll(n) = 1 -p,

J12(n) = J21 (n) = F'(2; - lnp) - p l n p . l n ( - lnp), J22(n) = - ( 1 - p) -

2[r'(1; -

lnp)] + F"(2; - lnp) +

2r'(2;

- lnp)

- p l n p . l n ( - lnp)[2 + l n ( - lnp)] , where U(1;-lnp)

__f

lnp

ln(t) exp(-t)dt,

--J0

U(2; - lnp) - f - l n p t ln(t) exp(-t)dt, --J0

and F"(2; - lnp) =

f

- lnp

t ln(t) 2 e x p ( - t ) d t .

d0

In the computation of the above integrals, one can use the appropriate subroutines of NAG, IMSL or the program given by Escobar and Meeker (1986).

2.3. Step-by-step procedure for determining (n, k) The main steps involved for obtaining variables-sampling plans under progressive censoring, are given below.

Step 1. Choose two points (p~, 1 - ~) and (p~,/~) on the OC curve and determine the total degree of censoring. Step 2. Choose an appropriate progressive censoring configuration, and determine the degrees of progressive censoring ql,. • •, qm. Step 3a. For the exponential case, compute the acceptability constant k and the sample size n using the appropriate formulae given in Section 2.1 or by solving the two equations iteratively. Step 3b. For the Weibull (extreme-value) case, compute k and n using the formulae given in Section 2.2. Note that n is to be obtained iteratively and in this regard, a computer program is required to compute the moments of the progressively censored order statistics. Step 4. Round oft the value of n to the nearest integer and the desired sampling plan (n, k) is obtained. 2.4. Remark It is clear from the above discussion that orte could obtain variables-sampling plans along the same lines for any location-scale distribution. Although we have

Progressively censored variables-sampling plans for life testing

465

used the M L E s one could use, for example, best linear unbiased estimators (BLUEs) or any other suitable estimators of/~ and «. Note that from the generalized least-squares theory, the BLUEs of/~ and a of (2.1) are given by

B = _ I«TFx and 1 = Ä 1TFX , where X = 0(1,n,X2,n,... , X.m,ù)T, Y = ( X - #l)/a, « = E(Y), 12 = Var(Y), 1 is an m × 1 column vector of l's, F is a skew-symmetric matrix defined as F=E I ( I « T - - « I T ) ~ ] -I and A = (ŒT~-I«)( 1T~2-1 1 ) - (ŒT~2 11)2 . Further, Var(fi) ~ o-2(«T~-lŒ)/A,

Var(8) ~ o-2(1TE-11)/A

and

Cov(]2, ~) ~ --O'2(«Tz ll)/A (c.f., Balakrishnan and Rao, 1997). The details on the use of BLUEs in obtaining variables-sampling plans for the lognormal (normal) distribution are available in Balasooriya and Balakrishnan (2000).

3. Illustrations EXAMPLE 1. Herd (1956) discussed the estimation of mean life of gyroscopes using progressively censored data. Suppose we are interested in determining reliability sampling plans for life testing of gyroscopes. Assume that lifelengths follow the two-parameter exponential distribution and the plan requires that there is at least a 1 - c~= 95% (1 - / 3 = 90%) chance of accepting (rejecting) a batch of gyroscopes when p~ = 0.0209 (p~ = 0.0742) where p denotes the proportion of nonconforming gyroscopes in the batch. Note that here we have chosen a (p~, p~) value from M I L - S T D - 1 0 5 D (US Department of Defense, 1963). Under the given specifications, we find k = 0.0217 and m = 42. Thus, if 60% of total censoring is required, then n = m/0.60 = (2.5)m = 105. EXAMPLE 2. In an example considered by M a n n and Fertig (1973), referred to data taken during the early development of F-100 fighter aircraft, 13 aircraft components were placed on test under Type II censoring with the test terminating at the time of the tenth failure. It was fairly established that the failure data follow a two-parameter Weibull distribution. For illustration, suppose that a progressively censored sampling plan is desired given p~ = 0.001, 1 - c~ = 0.99, p~ = 0.10,/3 = 0.01, ql = 0.3 and qm = 0.2. Using

466

U. Balasooriya

f o r m u l a e (2.8) a n d (2.9), w e o b t a i n k = 4 . 5 7 8 8 as t h e a c c e p t a n c e c o n s t a n t a n d n = 25 as t h e s a m p l e size. T h u s , t o i m p l e m e n t t h e s a m p l i n g p l a n , w e t a k e a r a n d o m s a m p l e o f 25 a i r c r a f t c o m p o n e n t s a n d r e m o v e a t o t a l o f 13 = [25 x 0.5] components

s u c h t h a t R1 = 8,R2 = R3 . . . .

R l l = 0, a n d R12 = 5.

References Balakrishnan, N. and C. R. Rao (1997). Large-sample approximations to the best linear unbiased estimation and best linear unbiased prediction based on progressively censored samples and some applications. In Advances in Statistics Decision Theory and Applications, pp. 431~444 (Eds. S. Panchapakesan and N. Balakrishnan). Birkhäuser, Boston. Balasooriya, U. (1995). Failure-censored reliability sampling plans for the exponential distribution. J. Stat. Comput. Simul. 52(3), 337 349. Balasooriya, U. and N. Balakrishnan (2000). Reliability sampling plans for the lognormal distribution based on progressively censored samples. IEEE Trans. Reliab. 49(2), 199-203. Balasooriya, U. and S. L. C. Saw (1998). Reliability sampling plans for the two-parameter exponential distribution under progressive censoring. J. Appl. Star. 25 (5), 707 714. Balasooriya, U., Saw, S. L. C. and V. G. Gadag (2000). Progressively censored reliability sampling plans for the weibull distribution. Technometrics 42(2), 160 167. Cohen, A. C. (1963). Progressively censored samples in life testing. Technometrics 5(4), 327-339. Das, N. G. and S. K. Mitra (1964). The effect of non-normality on sampling inspection. Sankhya 26, 169 176. Escobar, L. A. and W. Q. Meeker (1986). Elements of the information matrix for the smallest extreme value distribution and censored data: Algorithm as 218. Appl. Star. 35, 80 86. Fertig, K.W. and N . R . Mann (1980). Life-test sampling plans for two-parameter Weibull populations. Technometrics 22(2), 165 177. Guenther, W. C., S. A. Patil and V. R. R. Uppuluri (1976). One-sided/3-content tolerance factors for the two parameter exponential distribution. Technometrics 18(3), 333 340. Halperin, M., M. I. Hamdy and P. F. Thall (1989). Distribution-free confidence intervals for a parameter of Wilcoxon-Mann Whitney type or ordered categories and progressive censoring. Biometrics 45, 509 521. Hamilton, D.C. and M.S. Lesperance (1995). A comparison of methods for univariate and multivariate acceptance sampling by variables. Technometrics 37(3), 329-339. Harter, H. L. (1970). Order Statistics and Their Use in Testing and Estimation, vol. 2. US Government Printing Office, Wahington, DC. Herd, G. R. (1956). Estimation of the parameters of a population j?om a multi-censored sample. Ph.D. Thesis, Iowa State College. Johnson, L. G. (1964). Theory and Technique of Variation Research. Amsterdam, Elsevier. Kocherlakota, S. and N. Balakrishnan (1984). Two-sided acceptance sampling plans based on MML estimators. Commun. S t a t . - Theory Meth. 13, 3123-3131. Kocherlakota, S. and N. Balakrishnan (t986). One- and two-sided sampling plans based on the exponential distribution. Naval Res. Logist. Quart. 33, 513-522. Lieberman, G. J. and G. J. Resnikoff (1955). Sampling plans for inspection by variables. J. Am. Stat. Assoc. 50, 457 516. Mann, N. R. (1969). Exact three-order-statistics confidence bounds on reliable life for a Weibull mode1 with progressive censoring. Technometrics 64, 306-315. Mann, N. R. (1971). Best linear invariant estimation for Weibull parameters under progressive censoring. Technometrics 13(3), 521 533. Mann, N. R. and K. W. Fertig (1973). Tables for obtaining confidence bounds and tolerance bounds based on best linear invariant estimates of parameters of the extreme-value distribution. Technometrics 15(2), 87 102.

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Montanari, G. C. and M. Cacciari (1988). Progressively-censored aging tests on XLPE-insulated cable models. IEEE Trans. Reliab. 23, 365-372. Newcombe, P.A. and O. B. Allen (1988). A three-class procedure for acceptance sampling by variables. Technometrics 30, 415421. Owen, D. B. (1964). Control of percentages in both tails of the normal distribution. Technometrics 6, 377-387. Owen, D. B. (1969). Summary of recent work on variables acceptance sampling with emphasis on non-normality. Technometrics 11, 631-637. Rao, J. N. K., K. Subrahmaniam and D. B. Owen (1972). Effect of non-noramlity on tolerance limits which control percentages in both tails of normal distribution. Technometrics 14, 571 575. Schneider, H. (1989). Failure-censored variables-sampling plans for lognormal and Weibull distributions. Technometrics 31(2), 199-206. Schneider, H. and P. T. Wilrich (1981). The roubstness of sampling plans for inspection by variables. In Computational Statistics, pp. 281-295 (Eds. H. Büning and P. Naeve) Walter de Gruyter, Berlin, New York. US Department of Defense (1963). Sampling Procedures and Taóles for Inspection by Attributes: MIL-STD-IO5D. US Government Printing Office, Washington, DC. Viveros, R. and N. Balakrishnan (1994). Interval estimation of parameters of life/'rom progressively censored data. Technometrics 36(1), 84-91.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. Ali rights reserved.

1

KY

Graphical Techniques for Analysis of Data From Repairable Systems

P e r A n d e r s A k e r s t e n , B e n g t KlefsjÖ a n d B o B e r g m a n

The TTT-plot (TTT = Total Time on Test) and its theoretical counterpart, the scaled TTT-transform, are well known and useful tools in the reliability work. These concepts were first introduced by Barlow and C a m p o in 1975. Since then several applications based on the TTT-plotting technique and intended for data from non-repairable units have been presented. In this paper we discuss how the TTT-plot and a similar plot can be used when analyzing data from a repairable system. We discuss two different graphical approaches useful to see if a power-law process is a suitable model for the data.

Notation

R(t) p(t) z(t) xj x0? wj vj

survival function = 1 - F(t) intensity function of the failure process failure rate of a non-repairable unit, i.e. f(t)/R(t) time after repair number (j - 1) to failure number j, jth inter-event time; local time jth ordered inter-event time, i.e. x(1) _< x(2) _< .-. _< x(n) time to the jth failure, i.e. Xl + x2 + ... + xj; global time

tj/tn --ln(tn-j/tn), j = 1,2, . . , n - - 1 total time on test at the jth failure time

Sj/Sù;j = O, 1, ..,n

(i/n, uj) plotting positions to get a TTT-plot

~0(u)

the scaled TTT-transform of a life distribution F(t)

1. Introduction

The TTT-plot (TTT = Total Time on Test) is an empirical and scale independent plot based on failure data. It was introduced by Barlow and C a m p o in 1975, 469

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P. A. Akersten, B. Klefsjö and B. Bergman

together with the corresponding asymptotic curve, called the scaled TTT-transform. Primarily these tools were developed for model identification purposes. Later on the tools have proven to be very nseful in several applications within reliability. Some examples of practical applications are analysis of aging properties, maintenance optimization and burn-in optimization. The TTT-plot and the scaled TTT-transform have also been used in the design of test statistics for particular purposes and for the study of their properties. In this paper, the usefulness and clarity of the TTT-plotting technique and some related graphical techniques are studied. Applications as simple and clear graphical tools for analysis of failure time data from repairable systems are described. The graphical methods described are primarily used for the illustration of goodness-of-fit of the power-law process, one of the most used models for reliability growth of complex systems and for reliability of repairable systems.

2. Some concepts for repairable systems Suppose that we are studying a repairable system. Let xj denote the time after repair n u m b e r j - 1 to failure n u m b e r j and let tj denote the time to thejth failure, i.e. tj = x t + x2 + • • • + xj; see Figure 1. In the study of repairable systems, one characteristic of great interest is the intensity p ( t ) of the failure process. The intuitive interpretation of p ( t ) is as a measure of the probability that a failure will occur during the next At units of time in the sense that P{a failure occurs in the time interval It, t + A t ) } ~ p ( t ) A t

.

Accordingly, in the case of an increasing intensity, the successive times between the failure events tend to become shorter and shorter. In the case of decreasing intensity they will tend to successively increase. We want to emphasize here that the intensity of the failure process for a repairable unit should not be confused with the failure rate z ( t ) for a nonLoeal time times)

xl

x2

tl

x3

t2

t3

Global time (Event epochs)

Fig. 1. Some time concepts for data from a repairable system.

Graphical techniquesfor analysis of data from repairable systems

471

repairable unit. The failure rate z(t) is a conditional measure that a unit, which has survived up to t will fail during the next A t units of time in the sense that P{the failure of the unit occurs in the time interval It, t + At)]survival up to time t} ~ z(t)At . For a thorough discussion of these concepts, see Ascher and Feingold (1984). The intensity of the failure process depends on the past history of the system. This history incorporates information about failure event epochs, measured by the global time t, and times between failures, measured by the local time x, the time elapsed since the previous repair. If the intensity of the failure process is only a function of the local time x, we have a renewal process. This means that the times between failures are independent and with the same life distribution. This situation occurs if the unit is repaired to an "as-good-as-new"-state after each failure or, equivalently, replaced at failure with a new, identical unit. This model is in most situations in practice not a realistic one for repairable units. If you repair your old car after a minor failure it is certainly not in "as-good-as-new"-state. I f the intensity is constant, say equal to 2, the failures occur according to a renewal process in which the times between the successive failures are exponentially distributed with the same parameter 2. This means that the failures occur according to a homogeneous Poisson process with intensity 2. When the intensity of the failure process is a function only of the running time t, the failures may occur according to a non-homogeneous Poisson process. This situation is obtained if, for instance, we use a minimal repair policy, which means that upon failure the system is restored to the state it had just before the failure occurred. The concept of minimal repair is described for instance in Barlow and Hunter (1960). For an excellent general discussion on analysis of failure data from repairable systems and the abuse of the renewal process assumption, see Ascher and Feingold (1984). One of the most useful non-homogeneous Poisson process models is the one where the intensity function p(t) is a power of t, i.e. it can be written in the form

~/~/ ~(0) ~-~

~~~/

with parameters 0 > 0 and fl > 0. This model covers monotone increasing intensities (fi > 1) as well as monotone decreasing intensities (fi < 1). When B = 1 we get the homogeneous Poisson process. This model has been discussed by several authors, see e.g. Crow (1974), Moller (1976), Bain and Engelhardt (1986), Bain (1991), Rigdon (1989), Rigdon and Basu (1989, 1990), Park and Seoh (1994) and Crétois and Gaudoin (1998). M a n y authors include "Weibull" in the name of this process due to the fact that the intensity function in (2.1) has the same form as the failure rate z(t) of a Weibull life distribution. However, including "Weibull" in the name causes a lot of confusion since it gives an incorrect impression that the times between failures

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472

follow a W e i b u l l d i s t r i b u t i o n (for a discussion, see e.g. Ascher, 1981). W e prefer to call this process the power-law process.

3. The T T T - p l o t I f we have a c o m p l e t e o r d e r e d sample 0 = x(0) _< x(~) < --- < x(n) o f times to failure f r o m n identical a n d i n d e p e n d e n t n o n - r e p a i r a b l e units, the TTT-plot o f these o b s e r v a t i o n s is o b t a i n e d in the following way: 1. C a l c u l a t e the T T T - v a l u e s

Sj = nx(1) + (n - 1)(x(2) - x(~)) + . . . + (n - j +

1)(x(j) - x(j_l) )

(3.1)

for j = 1 , 2 , . . , n (for convenience we set So = 0). 2. N o r m a l i z e these T T T - v a l u e s b y calculating

uj = Sj/S,,

for j = 0, 1 , . . , n

.

(3.2)

3. P l o t (j/n, uj) for j = 0, 1 , . . , n. 4. J o i n the p l o t t e d p o i n t s by line segments. The a c r o n y m " T T T " in the n a m e " T T T - p l o t " m e a n s " T o t a l Time on Test". The r e a s o n for this n a m e is t h a t if all the units are p u t into test at the same time, then Sj is the t o t a l test time for all the units at time x(j); see F i g u r e 2. W h e n the s a m p l e size n increases to infinity the T T T - p l o t converges ( u n i f o r m l y a n d with p r o b a b i l i t y one; see L a n g b e r g et al., 1980) to a curve n a m e d the scaled TTT-transform o f the life d i s t r i b u t i o n F(t) f r o m which the s a m p l e has come. M a t h e m a t i c a l l y the scaled T T T - t r a n s f o r m is defined as

1 2 3 j-1

1

J i"1-1

ù. x

1

J

I

1

l

I ...

X(n)

Fig. 2. The reason for the acronym "TTT" is that if all the units are put into test at the same time, then Sj is the total test time for all the units at time xo>

Graphical techniques for analysis of data ßom repairable systems

z

1-[F-~(t) R(u)du

473

(3.3)

where R(t) = 1 - F(t) is the survival function and/~ is the mean. In this paper we only use the simply proved fact that the scaled TTT-transform of any exponential distribution, independently of the scale parameter, coincides with the diagonal in the unit square. Different types of deviations of the TTT-plot from the diagonal means different deviations from the exponential distribution. For instance, it is well known (see Barlow and Campo, 1975) that the failure rate of F(t) is increasing (decreasing) if and only if the scaled TTT-transform is concave (convex). Scaled TTT-transforms of a few life distributions are given in Figure 3. If the TTT-plot is based on a sample from an exponential distribution then Ul,U2,..,un, has the same distribution as an ordered sample from a uniform distribution on [0, 1], i.e. F(t) = t, 0 < t < 1 (see e.g. Epstein, 1960). A consequence of this is that the TTT-plot then tends to wriggle around the diagonal of the unit square. The scaled TTT-transform and the TTT-plot were first presented by Barlow and Campo (1975). They used these concepts for model identification by comparing the TTT-plot with scaled TTT-transforms of different life distributions. Since then several other applications have appeared. Among these are the analyses of different aging properties and for optimization when studying different

|

I

~,,

i

i

|

ù11

Fig. 3. Scaled T T T - t r a n s f o r m s of various life distributions: (1) N o r m a l (# = 1, a - 3); (2) g a m m a , with shape p a r a m e t e r 2.0; (3) Exponential (4) L o g n o r m a l (/~ = 0, er - 1 for the log(r.v.)); (5) Pareto with R(t) -- (1 + 0 -2, t > 0 (from Klefsjö and K u m a r , 1992).

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P. A. Akersten, B. Klef~jö and B. Bergman

replacement problems of non-repairable units. For more information about the TTT-tools and different applications of these tools, see e.g. Bergman and Klefsjö (1984, 1988, 1998), Klefsjö (1991), Westberg and Klefsjö (1993) and references in these papers. In this paper we shall use the TTT-plot and another similar graph for model identification purposes in connection with the power-law process in the analysis of data from a repairable system.

4. Plotting inter-event times If we study a single repairable unit up to time T the data normally provided consists of epoches of failure events, measured on an operation time scale or a calender time scale. In some cases the duration of repair is reported as well. However, in this paper repair times are excluded from the discussion. The sequence of event epochs, i.e. times to failures, q, t2,.. • (to = 0), gives rise to a sequence of inter-event times, i.e. times between successive failures, Xl = tl - to,

X2

=

t2 --

tl,...

(4.1)

A c o m m o n error in the analysis of data from repairable systems is to consider the set of inter-event times as a set of independent and identical distributed (i.i.d) random variables. Only in the case of a renewal process, stopped at a pre-determined number of renewals, this is a correct approach. This situation occurs when we repair the system up to the "as-good-as-new"-state after each failure. This type of repair is equivalent to a replacement with an identical unit. In other cases this approach may lead to misinterpretations and erroneons conclusions. As an illustration of this type of misinterpretation let us study the TTT-plot based on the following (fictive) times between failures (from Ascher, 1981): 43 15 177 65 27 51 32 The TTT-plot is given in Figure 4. As the plot wriggles around the diagonal the plot in no way indicates that a homogeneous Poisson process model is incorrect. N o w suppose that the times between failures instead occur in the following order: 177 65 51 43 32 27 15 I f we make a TTT-plot using these values, we obviously get the same plot as before since we use the same observations just in a different order. However, it is obvious that we now do not have observations from a homogeneous Poisson process, but from a process with an increasing intensity function since the times between failures are becoming shorter and shorter. Under fairly general assumptions concerning the failure intensity the set of unordered inter-event times from a non-homogeneous Poisson process has the same asymptotic distribution as a sample from a distribution having decreasing

Graphical techniquesfor analysis of data from repairable systems

475

o

Fig. 4. The TTT-plot based on a ficitive data set from Ascher (1981).

failure rate (DFR); see Akersten (1991). Simulation results give strong support to the use of the asymptotic result even for a very small number of observations. This means that a usual TTT-plot based on the inter-event times from a non-homogeneous Poisson process is expected to behave convexly. In other words, this means that if the inter-event plot does not show a convex tendency, a non-homogeneous Poisson process may be an unsuitable model. Another conclusion is that if the inter-event times from a non-homogeneous Poisson process are treated as independent and identically distributed, the resulting TTT-plot can erroneously be interpreted as supporting a hypothesis of decreasing failure rate.

5. Plotting event epochs Suppose we are studying a repairable system during a period of time T. If the failures occur according to a Poisson process, homogeneous or non-homogeneous, then the normalized failure times

wj=tj/T,

j=l,2,..,n

,

(5.1)

where ty is the time to failure number j, will have the same distribution as an ordered sample of size n from a distribution with cumulative distribution function F(t) = A(t)/A(T), 0 <_ t < T, where A(t) is the mean value function

A(t) =

1/0~

p(s)ds ,

(5.2)

i.e. E(N(t)) = A(t) (see e.g. Feigin, 1979). For a power-law process with intensity function according to (2.1), the mean value function is A(t) = (t/O) ô. This means that the conditional distribution of the normalized failure times wl, w 2 , . . , wn in

P. A. Akersten, B. Klefsjö and B. Bergman

476

this situation is the distribution of an ordered sample of size n from a distribution with

F ( t ) = t ~,

0
1 .

(5.3)

This in turn means that we,w2~,..,w~ß will have the same distribution as an ordered sample from a uniform distribution. A consequence of this is that if the failures occur according to a power-law process with power fi then the points O'/n, w~) will be scattered around the diagonal. This gives in theory a graphical possibility to check whether we have a power-law process or not. If we in particular have a homogeneous Poisson process, i.e. fl = 1, then Wl, w 2 , . . , wn will have the same distribution as an ordered sample from a uniform distribution. One way to graphically analyze if the failures have occurred according to a homogeneous Poisson process is therefore to plot j/n against wj, connect the point with line segments and compare the resulting plot to the diagonal. I f the plot deviates too rauch from the diagonal this is an indication that a homogeneous Poisson process assumption is unsuitable.

6. Combination of the two plots In the case of a homogeneous Poisson process both the usual TTT-plot and the plot based on the points (j/n, wj) - the wj-plot, for short - will rend to wriggle around the diagonal. If the failure history instead can be modelled by a nonhomogeneous Poisson process we expect the TTT-plot based on the inter-event times to have a convex shape and the wj-plot to have a concave or convex shape, depending on whether the intensity function is increasing or decreasing, respectively. If the TTT-plot based on the inter-event times has a concave shape, a non-homogeneous Poisson process seems to be an incorrect assumption. This is illustrated in Figure 5.

7. Some useful test statistics Suppose that we want a test for testing if a homogeneous Poisson process assumption is realistic or not. It is well known that the cumulative TTT-statistic n-1

V= Zu j

(7.1)

j--1

is a suitable test statistic to test whether the inter-event times from a renewal process is from an exponential distribution. This is easily understood since V is a linear function of the area A below the TTT-plot (see e.g. Klefsjö and Kumar, 1992). In fact A

1 n-1

( ~ ) V+

(7.2)

Graphical techniquesfor analysis of data from repairable systems

477

o; 1

1

0 0 1

1~

0

1

Fig. 5. TTT-plots (to the left) and wFplots (to the right) based on inter-event times from simulated data from: (a) homogeneous Poisson process (first row); (b) non-homogeneous Poisson process (middle row); (c) a superposition of three power-law processes with different values of ~ (last row).

O n e o f several a d v a n t a g e s w i t h this statistic is t h a t it is, even for m o d e r a t e values o f n, a p p r o x i m a t e l y n o r m a l l y d i s t r i b u t e d w i t h m e a n ( n - 1)/2 a n d v a r i a n c e (n - 1)/12 (see e.g. B a r l o w et al., 1972). A c c o r d i n g l y

478

P. A. Akersten, B. Klefsjö and B. Bergman

W --

V -

(n -

,/(n-

1)/2

1)/12

(7.3)

is asymptotically N(0, 1)-distributed. If the failure process follows a homogeneous Poisson process, also wl, w 2 , . . , wn have the same distribution as an ordered sample from a uniform distribution. Accordingly, also WE = ~j~~j'~~wj- ( n - 1)/2

,/(~- 1)/12

(7.4)

is asymptotically N(0, 1)-distributed in this case. This test statistic is closely related to what is called Laplace's trend test; see e.g. Cox and Lewis (1966). Furthermore, Akersten (1991) has proven that the event epochs and the unordered set of inter-event times are asymptotically independent under a homogeneous Poisson process assumption. A consequence of this is that W and WL are asymptotically independent. This means that C = W2 + WI2

(7.5)

is a suitable test statistic for testing the hypothesis that the observations are from a homogeneous Poisson process. It also follows that if the sample is from a homogeneous Poisson process, then C is approximately z2-distributed with two degrees of freedom. The test statistic V has been used in other situations in e.g. Barlow et al. (1972) and Hollander and Proschan (1975, 1980). The statistic C was introduced and studied by Akersten (1991).

8. An exalnple The data regarding project 1 in Musa (1975) contain 136 times between failures (execution time in seconds) from a software development program. The corresponding TTT-plot and the wj-plot are shown in Figure 6. The pattern of the TTT-plot is convex as is expected when the observations are from a non-homogeneous Poisson process. Furthermore, the wj-plot indicates a decreasing intensity function since it behaves convexly. In order to check whether a power-law process assumption might be suitable, we use the fact that a time transformation with parameter /3 gives the problem of checking whether the transformed data follow a homogeneous Poisson process. Accordingly, we try different values of ]? in order to see whether the transformed data are in agreement with a homogeneous Poisson process by calculating the corresponding value of the test statistic C. The choice/~ = 0.43 of the shape parameter gave the smallest value of the test statistic C = W2 ÷ W2 and a significance level less than 5%.

Graphical techniquesfor analysis of data fro m repairable systems 1

479

1

0

-

0

1

Fig. 6. The TTT-plot of the inter-event times (to the lefl) and the wj-plot (to the right) based on 136 times between failures (execution times in seconds) from a software development program. The observations are from Musa (1975).

F i g u r e 7 illustrates the T T T - p l o t a n d the wj-plot based on t~,t~2,..,t~ for B = 0.43. B o t h these plots show g o o d a g r e e m e n t with a h o m o g e n e o u s P o i s s o n process a s s u m p t i o n , which indicates t h a t a p o w e r - l a w process is a plausible m o d e l for the original data.

9. A TTT-plot based on transformed data

To t r a n s f o r m by t r i a l - a n d - e r r o r m i g h t be a little tedious, even if we use c o m p u t e r software. Therefore, we p r e s e n t a n o t h e r g r a p h i c a l p r o c e d u r e b a s e d on the

1

1

0 0

Fig. 7. The TTT-plot (to the left) and the wiplot (to the right) of the inter-event times of the Musa data after using the transformation t 0'43.

480

P. A. Akersten, B. Klefsjö and B. Bergman

same two plots as before for testing a power-law process assumption of the failure history of the system. The difference is that we transform the data before plotting. Suppose that we observe a power-law process with intensity function

and that failures have occurred at q, t2, • • •, t~. I f we use the transform vj = - l n ( t n _ j / t ù ) ,

j = 1,2,..,n-

1

(9.2)

then the set/)1, v2, Vn-1 is distributed as an ordered sample of size n-1 from an exponential distribution with mean 1/fi, i.e. F(t) = 1 - exp(fit), t _> 0. This result is due to Moller (1976). This means that we can use a TTT-plot based on Vl _< v2 _< ... _< vn-1 (instead of x(1) _< x(2) _< .-- _< x(n)) for checking the power-law process assumption. Note also that the vTvalues correspond in a sense to the tj-values in opposite order. I f our original data are in agreement with a power-law process, the TTTplot based on the vj-values is expected to wriggle around the diagonal since the TTT-plot is an estimate of the corresponding scaled TTT-transform, which in this situation is the diagonal of the unit square. This graphical idea was presented by Klefsjö and K u m a r (1992). In Figure 8 the TTT-plot based on the vFvalues for the data from Musa (1975) illustrates a fairly good agreement with the diagonal, i.e. a power-law process seems appropriate for modeling the original data. Note that this procedure is independent of the parameter fl since the TTT-plot and the scaled TTT-transform are tools which are independent of scale. •

•

,

'

j

0 0 Fig. 8. TTT-plot, based on the Musa data, after using the vj-transformation.

Graphical techniquesfor analysis of data from repairable systems

481

10. Another example As a further i l l u s t r a t i o n o f the two a p p r o a c h e s p r e s e n t e d to analyze d a t a f r o m a n o n - h o m o g e n e o u s P o i s s o n process we use a s i m u l a t e d d a t a set used in Klefsjö a n d K u m a r (1992). It is b a s e d on s i m u l a t i o n f r o m a p o w e r - l a w process o f size n = 52 with fi = 0.5 o r i g i n a t i n g f r o m C r o w a n d Basu (1988). F i g u r e 9 illustrates the usual T T T - p l o t a n d the wFplot b a s e d on the original d a t a f r o m C r o w a n d Basu (1988). T h e t e n d e n c y is the same as for the d a t a f r o m M u s a (1975) in F i g u r e 3. A triala n d - e r r o r investigation with different values o f fi gave t h a t the test statistic C was

1

1

0

Fig. 9. The TTT-plot (to the left) and the w;plot (to the right) based on a simulated data set of size 52 from a power-law process with fi - 0.5 (data from Crow and Basu, 1988).

1

0 Fig. 1Õ. The TTT-plot (to the left) and the wj-plot (to the right) of the Crow and Basu data, after using the transformation t°54.

482

P. A. Akersten, B. Klefsjö and B. Bergman

minimized when fl = 0.54. The TTT-plot and the wj-plot for the data transformed b y t 054 are presented in Figure 10 and, as is expected, they are both wriggling around the diagonal. As a comparison, the TTT-plot and the wj-plot of the Crow and Basu data, using the transformation with the correct value fi = 0.5 are given below. The two TTT-plots in Figures 10 and 11 are very close to each other. Likewise the two wiplots look very much the same. Figure 12 presents the TTT-plot for the vfvalues based on the simulated power-law process data from Crow and Basu (1988). As expected, the agreement with the diagonal is good in Figure 12 as well as in Figures 10 and 11.

1T

0 0 Fig. 11. TTT-plot (to the left) and ws.-plot (to the right) of the Crow and Basu data after using the transformation t°5°.

Fig. 12. The TTT-plot based on the Crow and Basu data after using the vKtransformation.

Graphical techniques for analysis of data from repairable systems

483

11. Conclusions and comments

Klefsjö and Kumar (1992) noticed that given a set of data from a renewal process (in particular with a small coefficient of variation) it was difficult to outrule a power-law process by using a TTT-plot based on the vj-values. However, if we combine the TTT-plot based on the vj-values with the combination of inter-event times TTT-plot and event epochs wj-plot, we will have a more sensitive tool. It can be used to illustrate the adequacy of the power-law process model and show the effect of different choice of model parameters.

References Akersten, P. A. (1991). Repairable systems reliability studied by TTT-plotting techniques. Ph.D. dissertation, Division of Quality Technology, Linköping University of Technology, Sweden. Ascher, H. E. (1981). Weibull distribution versus Weibull process, in Proceedings from Annum Reliability and Maintainability Symposium, pp. 426429. IEEE, Piscataway. Ascher, H. E. and H. Feingold (1984). Repairable Systems Reliability. Marcel Dekker, New York. Bain, L. J. (1991). Statistical Analysis of Reliability and Life-testing Data, 2nd edn. Marcel Dekker, New York. Bain, L. J. and M. Engelhart (1986). On the asymptotic behaviour of the mean time between failures for repairable systems. In Reliability and Quality Control, pp. 1-7 (Ed. A. P. Basu). Elsevier, Amsterdam. Barlow, R. E., D. J. Bartholomew, J. M. Bremner and H. D. Brunk (1972). Statisticallnference Under Order Restrietions. Wiley, New York. Barlow, R. E. and R. Campo (1975). Total time on test processes and applications to failure data analysis. In Reliability and Fault Tree Analysis, pp. 451481 (Eds. R. E. Barlow, J. Fussell and N. D. Singpurwalla). SIAM, Philadelphia. Barlow, R. E. and L. Hunter (1960). Optimum preventive maintenance policies. Oper. Res. 8, 9~100. Bergman, B. and B. Klefsjö (1984). The total time on test concept and its use in reliability theory. Oper. Res. 32, 596 606. Bergman, B. and B. Klefsjö (1988). Total time on test transforms. In Eneyelopedia of Statistical Sciences, Vol. 9, pp. 297-300. Wiley, New York. Bergman, B. and B. Klefsjö (t998). Recent applications of the TTT-plotting technique. In Frontiers in Reliability, pp. 47-61 (Eds. A. P. Basu, S. K. Basu and S. Mukhopadhyay). World Scientific, Singapore. Cox, D. R. and P. A. W. Lewis (1966). The Statistieal Analysis ofSeries ofEvents. Methuen, London. Crétois, E. and O. Gaudoin (1998). New results on goodness-of-fittests for the power law process and application to software reliability. Int. J. Reliability, Quality, Safety Eng. 5, 249 267. Crow, L. H. (1974). Reliability analysis for complex, repairable systems. In Reliability and Biometry, pp. 379410 (Eds. F. Proschan and R. J. Serfling). SIAM, Philadelphia. Crow, L. H. and A. P. Basu (1988). Reliability growth estimation with missing data - II. In Proeeedings from Annual Reliability and Maintainability Symposium, pp. 248553. IEEE, Piscataway. Epstein, B. (1960). Tests for the validity of the assumption that the underlying distribution of life is exponential. Part I. Teehnometrics 2, 83-101. Feigin, P. D. (1979). On the characterization of point processes with the order statistics property. J. Appl. Prob. 16, 297 304. Hollander, M. and F. Proschan (1975). Tests for mean residual life. Biometrika 62, 585 593. Hollander, M. and F. Proschan (1980). Tests for mean residual life. Amendments and corrrections. Biometrika 67, 259.

484

P. A. Akersten, B. Klefsjö and B. Bergman

Klefsjö, B. (1991). TTT-plotting a tool for both theoretical and practical problems. J. Stat. Plann. Inf 29, 99 110. Klefsjö, B. and U. Kumar (1992). Goodness-of-fit tests for the power-law process based on the TTT-plot. IEEE Trans. Reliab. 41, 593-598. Langberg, N. A., R. V. Leone and F. Proschan (1980). Characterization of nonparametric classes of life distributions. Ann. Prob. 8, 1163-1170. Musa, J. D. (1975). A theory of software reliability and its applications. IEEE Trans. Software Eng. 1, 312-327. Moller, S. K. (1976). The Rasch-Weibull process. Scand. J. Stat. 3, 107-115. Park, W. J. and M. Seoh (1994). More goodness-of-fit tests for the power-law process. IEEE Trans. Reliab. 43, 275-278. Rigdon, S. E. (1989). Testing goodness-of-fit for the power law process. Comm. Star. Part A - Theory Meth. 18, 46654676. Rigdon, S. E. and A. P. Basu (1989). The power law process - a model for the reliability of repairable systems. J. Qua•ty Technol. 21, 251-260. Rigdon, S. E. and A. P. Basu (1990). The effect of assuming a homogeneous Poisson process when the true process is a power law process. 3". Quality Technol. 22, 1 t 1-117. Westberg, U. and Klefsjö, B. (1993). TTT-plotting for censored data based on the piecewise exponential estimator. Int. J. Reliab. Quality Safety Eng. 1, 1-13.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

1 ~7 /

A Bayes Approach to the Problem of Making Repairs

Gary C. McDonald

The question to which this paper is addressed is simply: ifa given item fails to meet specified requirements as determined by an appropriate test, what repairs (assumed finite in number) should be made in order to insure its passing at minimum average cost? The knowledge of an a priori distribution, i.e., the frequency of need for the various repair combinations, is assumed known. Thus, Bayes decision procedures are developed which map an observation, the output of an appropriate test, into the collection of all possible repair combinations. Two numerical examples are worked in detail - one in which the observation is univariate, another in which the observation is bivariate. Another related question also discussed in this paper is: How much is the test actually worth in terms of reducing the overall risk?

1. Formulation of the problem An attempt is made here to present a meaningful and valid decision theory formulation for certain testing and subsequent repair problems. The question to which this formulation is addressed is simply: if a given item fails to meer specified requirements as determined by an appropriate test, what repairs should be made in order to insure its passing at minimum average cost (or loss)? Throughout this discussion the items under consideration are assumed homogeneous within classes (states of nature); and hence, given the state of nature, the data resulting from the tests may be regarded as independent identically distributed realizations on an underlying random variable (or possibly random vector). Our goal then is to formulate good decision procedures which map an observation into the set of all possible repair combinations. Let R designate the set of all possible repairs which may be made on the item and let the number of such repairs be k, i.e., R = { r l , . . . , r k } , where ri is t h e / t h possible repair. A typical repair ticket would then list several of these repairs which should be made, i.e., a repair ticket is one of the 2k possible subsets of R. Let S = {Sl,S2,.. ,s2k} denote the collection of all possible subsets of R, so a repair ticket is now simple an element of S. 485

G. C. McDonald

486

The state of nature will be denoted by Q = { 0 1 , . . » 02k }. The state Oi will be interpreted as the item does in fact require only those repairs which are specified by the subset si. The action space will be designated by A = {a 1 , .., a2k} and the action ai is that action which repairs the item according to the exact specifications ofsi. Besides the costs of parts, labor and retests required in making repairs, one may also incur other losses in such situations. Less tangible losses might be an inconvenience of storage required in making certain repairs, a loss of prestige among contemporaries due to making too many (or too few) repairs, etc. This composite loss of utility if 0i is the true state of nature and one takes action aj is denoted by L(Oi, aj), 1 <_ i,j <_2 k. A random vector X = ( X 1 , . . ,Am) is defined to be the characterizing vector for that class of items under consideration. One observation on this random vector is obtained from each item on its first test and is denoted by x = (xl,... ,xm). The space of all possible observations will be designated by Z and is a subset of n-dimensional Euclidean space. The conditional probability density function of the random vector X given the state of nature 0 E Q will be given by f(x]0). A decision procedure is now simply a mapping from the space Z to the space A; notationally, if d is a decision procedure then d : Z -+ A. The collection of all such decision procedures is designated by D. In Section 2, Bayes procedures will be described and discussed while Sections 3 and 4 will be devoted to examples. In Section 5 the components of a Bayes decision procedure will be discussed from a practitioner's point of view. An excellent elementary development of decision procedures, in particular Bayes decision procedures, may be found in Chernoff and Moses (1959). A more advanced approach is contained in the books by Ferguson (1967), D e G r o o t (1970), and Berger (1985).

2. Bayes decision procedures The expected loss that one incurs if 0i is the true state of nature and a decision procedure d E D is being used is given by

~(Oi, d) = E[L(Oi, d(X)] = fzz L(Oi, d(x))f(x]0i)dx

m

(2. 1)

Let 9 be any discrete probability mass function defined over the states of nature ~2; i.e., the stare of nature is being considered as a random variable O with mass function 9 ( 0 i ) = P ( O - Oi), i= 1 , 2 , . . , 2 k. The risk associated with g and d, r(g, d), is given by

r(g, d) = Æg[~(o, d(X))l 2~

= Z S(Oi,d(X))~(Oi) i=1 2k

= Z fzz L(Oi, d(x))f(x[O,)g(O,)dx . *=1

(2.2)

A Bayes approach to the problem of making repairs

487

Let d* c D be a decision function such that

r(g, d*) = minr(g, d) = r*(g) .

(2.3)

dcD

Then it is said that d* is a Bayes decisionfunction (procedure) against the a priori distribution g, and r* (g) is called the Bayes risk. Taking the summation inside the integral in Eq. (2.2), the risk r(g, d) becomes

~,~~,: ~~/~~,0~

0~,~,0~,/dx

,~4,

Therefore, a decision procedure d that minimizes this risk can be found by minimizing the summation in (2.4) contained in the braces for each fixed x E Z. In other words a Bayes decision function d* against g can be constructed as follows: for each value x E Z, ler d* (x) be any decision d C D which minimizes the sum

~ L(O, «(, ))f(xlO)g(O ) ,

(2.»)

(2

where ~ a means the summation is taken over all possible states of nature. REMARKS. (1) In some instances the experimenter might wish to consider not the losses but the regrets, the loss of utility over and above the minimum loss for a given state of nature. Specifically, the regret of taking action ag. when Oi is the true state of nature is given by h(Oi, aj) = L(Oi, aj) -minacA L(Oi, a). However, it is an irrelevant consideration in the sense that the same Bayes strategy(ies) will emerge when using regrets as when using losses for a fixed a priori distribution. Numerically, it is usually more convenient to work with the regrets. (2) It is offen of interest to determine how much the test or experiment is worth. In other words, how much should one be willing to pay in order to obtain the observation(s) which then determines the appropriate action to be taken. Let f(g) be the minimum average loss associated with the actions ai c A when no observations are available. Then the difference C = r(g) - r* (g) is the reduction in risk obtained by taking one observation, and is an upper bound on the " w o r t h " of the experiment or test required to obtain the observation. The cost of the test is assumed independent o f b o t h the a priori distribution and the outcome of the test. The three quantities required to compute a Bayes strategy are: (i) the loss function, (ii) the conditional distribution of the observations given the state of nature, (iii) an a priori distribution over the states of nature.

3. A univariate observation example The number of possible repairs is 2, rl and r2. The elements of S are then: $1: do nothing to the item; s2: implement repair tl;

G. C. McDonald

488

s3: implement repair r2; $41 implement both rl and i"2. The states of nature denoted collectively by f2 are now 01: 02: 03: 04:

item item item item

requires requires requires requires

no changes; only repair rl; only repair r2; both repair rl and ra

The losses (which are actually regrets in this example) are given by: Loss o f utility,

L(O, a)

State of nature

01 02 03

04

Action

al

a2

a3

a4

0 1 1 1

1 0 1 1

1 1 0 1

1 1 1 0

Take the a priori distribution to be: g(01) = 0.1,

g(02) = 0.3,

g(03) = 0.4,

g(O4) =

0.2.

Finally, ler the f ( x l 0 ) be simply a univariate normal distribution where 0 determines the mean and the variance is fixed at 1.

State of nature

01

02

03

04

M e a n off(xlO )

4.0

4.5

5.0

4.75

With the above structure, it is easy to compute Action

(27C)1/2 S L(O, a)f(xlO)g(O ) o

al a2 a3 a4

0.3e (x 4.»)2/a + 0 . 4 e (" »0)2/2 -}-0.2e -(x-4"75)2/2 0 . l e (x 4.0)2/2 + 0 . 4 e (~ 50)2/2 +0.2e -(x-4-75)2/2 0.1 e -(x-4'O)z/2 + 0.3 e -(x-4"5)2/2 + 0.2e (x-4"75)2/2 0.1 e (x 4.o)2/2 + 0.3e (x 4.5)2/2 + 0.4e-(,-5.0)2/a

The above expressions are tabulated in T a b l e 1 for x = 0.0(0.2)10.0 and graphically displayed in Figure 1 for al, a2 a n d a3. The expression for a4 is never the minimum, so it has been omitted from the figure. The Bayes strategy is now evident (approximately); namely,

A Bayes approach to the problem o f m a k i n g repairs

{

al

d*(x)=

489

ifx_< 2.0,

a2 i f 2 . 0 < x _ < 4 . 0 , a3

(3.1)

ifx>4.0.

It is interesting to note that in this example the Bayes strategy never takes the all-inclusive action a4. The Bayes risk r*(9 ) can be obtained from Eq. (2.2); in this example, a convenient form of the risk is given by

r*(9) = Z

P(d*(x)

=

ai

and 0 = Oj)

(3.2)

.

1«_i,j«4 i¢j,4

Making use of the normality of f(x]O) and letting ~b(.) denote the cumulative distribution function of the standard normal distribution, the first term in this summation is seen to be

P(«*(x)

=

a 1 and 0 ----02) = PO( <_ 2.010 = 02)g(02) = 0.3~(-2.5) = 0.001863 .

(3.3)

Continuing in this manner the Bayes risk is evaluated as r * ( 9 ) - 0.57. In this example the Bayes strategy is given in (3.1) and is the best possible strategy in the sense of minimizing the expected loss of utility. Any other strategy employed in this problem would lead to an expected loss per item larger than 0.57, the Bayes risk. To determine the worth of the test, the so-called "no-data problem" must first be solved. A convenient tabular presentation of the problem is as follows: Stare of nature

Regrets

A priori probabilities

al

a2

a3

a4

Ol 02

0 1

1 0

1 1

1 1

0.1 0.3

03

1 1 0.9

1 1 0.7

0 1 [Ö~

1 0 0.8

0.4 0.2

04

P(9, a)

The quantity r(g, a) = EgaL(O, a)g(O) is the expected loss using action a, and ~ ( g ) - mina~A P(g,a)= 0.6. Hence, the Bayes action for the no data problem would be action a3 with a Bayes risk of 0.6. The Bayes strategy with one observation reduced the risk to 0.57, so the reduction in risk is C = 0.60 - 0.57 ~- 0.03 which is an upper bound on the worth of the test. The units of this bound are the utility (or regret) units and it is assumed the cost of the experiment or test is fixed in advance of the outcome and does not depend on the a priori distribution.

G. C. McDonald

490

Table 1 Computations of Eq. (2.5) to determine Bayes strategy for the univariate observation example

E L(O,a)f(x[O)g(O)

Minimizing action

al

a2

a3

~4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0000064 0.0000157 0.0000370 0.0000841 0.0001837 0.0003859 0.0007799 0.0015166 0.0028375 0.0051084 0.0088499

0.0000150 0.0000333 0.0000715 0.0001477 0.0002946 0.0005672 0.0010547 0.0018948 0.0032912 0.0055296 0.0089905

0.0000192 0.0000433 0.0000942 0.0001973 0.0003985 0.0007755 0.0014546 0.0026301 0.0045841 0.0077023 0.0124763

0.0000188 0.0000423 0.0000920 0.0001928 0.0003894 0.0007585 0.0014251 0.0025831 0.0045181 0.0076274 0.0124303

ai

2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

0.0147542 0.0236719 0.0365526 0.0543235 0.0777071 0.1069924 0.1418007 0.1809038 0.2221624 0.2626354

0.0141511 0.0215689 0.0318406 0.0455274 0.0630489 0.0845510 0.1097675 0.1379053 0.1675905 0.1969100

0.0194830 0.0293308 0.0425675 0.0595523 0.0803078 0.1043815 0.1307548 0.1578399 0.1835923 0.2057413

0.0195593 0.0297204 0.0436153 0.0618232 0.0846487 0.1119598 0.1430465 0.1765436 0.2104549 0.2423021

a2

4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0

0.2988818 0.3274237 0.3452897 0.3505222 0.3425302 0.3221987 0.2917281 0.2542425 0.2132636 0.1721734 0.1337753 0.1000287 0.0719764 0.0498366 0.0332028 0.0212836 0.0131260 0.0077878 0.0044449 0.0024404 0.0012887 0.0006546 0.0003198 0.0001502 0.0000679

0.2235697 0.2451649 0.2595264 0.2650750 0.2611076 0.2479411 0.2268753 0.1999789 0.1697481 0.1387172 0.1091080 0.0825834 0.0601396 0.0421300 0.0283875 0.0183957 0.0114635 0.0068689 0.0039573 0.0021919 0.0011672 0.0005975 0.0002940 0.0001391 0.0000632

0.2221094 0.2309609 0.2333041 0.2230743 0.2071503 0.1852002 0.1593928 0.1320447 0.1052820 0.0807842 0.0596483 0.0423772 0.0289664 0.0190481 0.0120496 0.0073322 0.0042914 0.0024158 0.0013079 0.0006810 0.0003410 0.0001642 0.0000760 0.0000338 0.0000145

0.2693972 0.2892026 0.2997163 0.2998026 0.2893936 0.2695119 0.2421064 0.2097376 0.1751822 0.1410427 0.1094366 0.0818152 0.0589217 0.0408697 0.0272981 0.0175545 0.0108668 0.0064744 0.0037121 0.0020479 0.0010870 0.0005550 0.0002726 0.0001287 0.0000585

a3

A Bayes approach to the problem ofmaking repairs

491

~L(O,a)f(xlO)g(O)

9.2 9.4 9.6 9.8 10.0

Minimizing action

al

a2

a3

a4

0.0000295 0.0000123 0.0000049 0.0000019 0.0000007

0.0000276 0.0000116 0.0000047 0.0000018 0.0000007

0.0000060 0.0000024 0.0000009 0.0000003 0.0000001

0.0000255 0.0000107 0.0000043 0.0000017 0.0000006

a3 /

|

+

4. A bivariate observation example Let S, f2 a n d g(') be as in the p r e v i o u s example. T h e loss o f utility will be c h a n g e d to reflect the (hypothesized) fact t h a t it is twice as b a d to m a k e no r e p a i r if s o m e t h i n g is r e q u i r e d or to m a k e the w r o n g r e p a i r t h a n it is to d o t o o rauch. Loss o f utility, L(O, a) State of nature 01 02 03 04

Action al

a2

a3

a4

0 2 2 2

1 0 2 2

1 2 0 2

1 1 1 0

F i n a l l y , let f ( x l 0 ) be a b i v a r i a t e n o r m a l d i s t r i b u t i o n , where 0 determines the m e a n v e c t o r a n d the v a r i a n c e - c o v a r i a n c e m a t r i x is fixed at the identity matrix. L e t the m e a n vectors be given as follows: State of nature

01

02

03

04

Means of Xl,X2

2.1, 22.0

2.4, 23.0

2.7, 20.0

3.0, 30.0

W i t h the a b o v e structure, the relevant expression given b y (2.5) can be w r i t t e n explicitly for each o f the f o u r actions; namely,

Action

EL(O, a)f(x[O)g(O)

al a2 a3

0.6f(x]02) + 0.8f(x]03) + 0.1 f(xl01 ) + 0.Sf(x[03) + 0.1f(xl0 0 + 0.6f(x]02) + 0.1f(xl01) --0.3f(xl02) +

a4

0.4f(x]04) 0.4f(x]04) 0.4f(xl04) 0.4f(x[03)

(4.1)

492

G. C. McDonald

~"L(O,a)f(xl O)g(O) v s .

x

f2

ù

'"",7\

0.1

a3

0.01

', f

0.001

0.0001

/ I /

__@__

a

o.oooo, « - 7

~' . . . . . .

I_.

~-7-. ' ~.'~ 7---.->~,-" . . . . . . .

a

-,~

3 .....

r

Fig. 1. Graphical display of computations in Table 1 determining the Bayes decision function.

where f(x]01) = (27c)-1e -[(xz-2"l)2+(x2-22"O)2]/2, f(xl02) = (2~) le-[(xl-2"4)2+(x2 230)2]/2, f(x]03)

=

(27z)-le -[(x'-27)2+(x2-2°°)21/2,

f(xl04 ) = (27r)-le-[(x~-3.0)z+(xe-30.0)~l/2 .

(4.2)

A Bayes approach to the problem ofmaking repairs

493

34 33 32 31 30

Take Action a 4

29 28

27 26 25 X2 24

Take Action a 2

23 I

22 21 20 19

Take Action a 3

18

17 16 15

J

0

i

1

i

2

i

3

i

4

5

i

6

i

7

L

8

i

9

10

i

11

[

12

i

13

14

XI Fig. 2. Graphical display of Bayes decision function for the bivariate observation example.

The expressions given in (4.1) have been evaluated for various values ofxl and x2. These computations lead to the construction of Figure 2 which yields (approximately) the Bayes decision function. In Figure 2 the (xl, x2) space is divided into several regions. The lower region (denoted by RL) has the property that for each point x in this region,

~--~ L(O, a3)f(xlO)g(O ) = min ~-~ L(O, a)f(xlO)g(O ) . Q

(4.3)

G. C. McDonald

494

The middle and upper regions ( R M and Ru) have the same interpretation with a3 replaced by a2 and a4, respectively. In other words, the (approximate) Bayes decision function is given by

d* (x) =

{

a4

if x ~ upper region, Ru

a2

if x E middle region, RM

a3

if x E lower region, RL .

(4.4)

In this example the Bayes strategy never takes the "do nothing" action ai. The boundary curve between the a2 region and the a3 region, in this case, is a straight line expressed as x2 = 0. lXl q- 21.341. The boundary curve between regions a2 and a4 appears linear but actually has some mild curvature. The Bayes risk r*(9) can be obtained as in the previous example, i.e.,

r*(g) = ~

L(Oj,ai)P(d*(x)

=

ai

and 0 = Oj)

.

(4.5)

l
ißj, i

In order to simplify the evaluation of the Bayes risk, (and hence obtain an upper bound to the true value), the regions in Figure 2 determining the Bayes procedure will be simplified to RÜ

:

{(X1,x2): 26.3 < X2 < OC},

R~ = {(xl,x2) : 22.0 _< x2 < 26.3},

(4.6)

R[ = {(x,,x2):x2 < 22.0} . The simplified regions are independent of xl, and consequently ease the computation of the terms appearing in (4.5). In effect, we are replacing the Bayes strategy d* (x) given in (4.4) by the following:

d'(x) =

{

a4

if 26.3 _< x2 < oo,

a2

if 22.0 < x2 < 26.3,

a3

ifx2 < 22.0 .

(4.7)

Since d*(x) is Bayes, it follows that

ù*(g) _
L(Oj, ai)P(d'(x)

=

ai

and 0 = Oy) .

(4.8)

l
iCj, 1

This summation is evaluated in the same manner as (3.2) in the previous example. The first term in the summation is seen to be

L(Oa, a2)P(d'(x) = a2 and 0 = Õ~) - P ( 2 2 . 0 < X2 < 26.310 = 01)g(01) = 0.1[~(4.3) - ¢b(0)] = 0.05 .

A Bayes approach to the problem ofmaking repairs

495

Continuing in this manner the risk r(g , d') is evaluated as r(g, d I) - 0 . 2 1 4 , and hence, r*(g) <_0.214. To determine the worth of the test, the no-data problem must be solved. As before State of nature

Regrets

A priori probabilities

al

a2

a3

a4

01 02 03 04

0 2 2 2

1 0 2 2

1 2 0 2

1 1 1 0

PO, a)

1.8

1.3

1.1

~ß

0.1 0.3 0.4 0.2

The quantity r(g,a)= ~aL(O,a)g(O) is the expected loss using action a, and P(g) = minacA P(g,a)= 0.8. Hence, the Bayes action for the no data problem would be action a4 with a Bayes risk of 0.8. The Bayes strategy with one bivariate observation reduced the risk to (at least) 0.214, so the reduction in risk is C = 0.8 - 0.214 = 0.586 which is a measure of the worth of the test. The units of the worth are the utility units (which quite conceivably could be dollars) and it is assumed the cost of the test is fixed in advance of the outcome and does not depend on the a priori distribution.

5. Further comments on Bayes strategies As noted earlier, the three quantities required to compute a Bayes strategy are (i) the loss function, (ii) the conditional distribution of the observations given the state of nature, (iii) an a priori distribution over the states of nature. The loss function represents the loss of utility associated with taking a given action when the underlying state of nature is fixed. The loss of utility concept may, in practice, be the objective dollar loss taking into consideration the parts and labor required to make the various repairs, the necessity and cost of a retest, etc. However, the utility concept may be expanded to consider a much broader notion of loss one that takes into account less quantified losses such as inconvenience, reputation or credibility, etc. Excellent elementary discussions of these ideas may be found in Luce and Raiffa (1957) and Raiffa (1968). Perhaps the most difficult item to handle is the conditional distribution of the observations given the state of nature. Testing a sufficiently large number of items which belong to a known state of nature and then fitting a probability density (or mass) function to the resulting observations is one method of obtaining an approximation to the true conditional distribution. To know the state of nature of an item may require taking good items and misadjusting them so that the required repairs (and hence the state of nature) are in fact known. Fitting a probability

496

G. C. McDonald

density, in most cases, will involve estimating the parameter(s) in certain well-known density functions, i.e. beta, normal, multivariate normal, etc. In instances where it is not possible to estimate the conditional density of the observations given the state of nature according to the above methods, another approach might be feasible. The space of all possible observations Z is partitioned into a finite number of regions W 1 , . . , Wq such that Uiq_l ~ = z and IgiiN Wj = qb for i ¢ j. In other words these partitioning regions are disjoint (have no points in common) and cover the observation space. Instead of estimating the density given the state of nature, one now estimates the probability that the observation belongs to each region given the state of nature. The effect of this "discretizing" the conditional density is to replace the integral in (2.1), (2.2) and (2.4) by a summation over the q regions. In many cases this will ease the computation of the Bayes strategy and Bayes risk, the strategy now assigning an action to be taken for each partitioning region of Z. With regard to the a priori distribution, the frequency of occurrence of the state of nature, some work has appeared in the literature on the sensitivity of Bayes procedures to this distribution. Pierce and Folks (1969) and Berger (1984) discuss aspects of this problem and provide many references to related work. Analytical results on the sensitivity of Bayes procedures to a priori distributions are sparce, and most conclusions will be based on numerical methods. Berger (1985) provides an excellent discussion of Bayesian robustness (see Section 4.7) to possible misspecification of the prior distribution. The no-data problem and one-observation problems have been discussed in some detail. The difference in the Bayes risks for the best no-data strategy and best one observation strategy yields a measure of the worth of obtaining the one observation. There may arise instances where one has the option of obtaining two or more independent observations on a given item before stating which repairs should be made. Bayes strategies based on these independent observations can then be developed as before. The details can be found in the books by Chernoff and Moses (1959), Ferguson (1967), and D e G r o o t (1970). Again the worth of taking additional observations can be measured by the reduction in the corresponding Bayes risks. It should be noted that the effect of the a priori distribution is a diminishing function with an increasing number of observations. That is, the Bayes strategy and Bayes risk will tend to be insensitive to slight errors in estimating the a priori distribution when larger numbers of observations are taken. Once the repairs have been made as dictated by the Bayes strategy, the item is once again put on test with a positive probability of failure. Hence, there is need to develop appropriate Bayes strategies for several levels of retest, i.e., what repairs should be made given that an item has failed a retest in order to insure its passing at minimum average cost with the next retest. This problem is identical to our original formulation and the Bayes strategy can be developed as before; however, it should be expected that the a priori distribution and the conditional distribution of the observations given the state of nature will differ from initial test to first retest, to second retest, and so on.

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497

References Berger, J. O. (1984). The robust Bayesian viewpoint (with discussion). In Robustness of Bayesian Analysis (Ed. J. Kadane). North-Holland, Amsterdam. Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd edn. Springer, New York. Chernoff, H. and L. Moses (t959). Elementary Decision Theory. Wiley, New York. DeGroot, M. (1970). Optimal Statistical Decisions. McGraw-Hill, New York. Ferguson, T. (1967). Mathematieal Statistics: A Decision Theoretic Approach. Academic Press, New York. Luce, R. and H. Raiffa (1957). Games and Decisions. Wiley, New York. Pierce, D. and J. L. Folks (1969). Sensitivity of Bayes procedures to the prior distribution. Oper. Res. 17, 34zP350. Raiffa, H. (1968). Decision Analysis: Introductory Lectures on Choices Under Uncertainty, AddisonWesley, Reading, MA.

N. Balakrishnan and C. R. Rao, eds., Handbook ofStatistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserve&

]

& t.3

Statistical Analysis for Masked Data

Betty J. Flehinger ~, Benjamin Reiser and Emrnanuel Yashchin

Notation General

g N k icg no ni ng t'lg,i

hg+ = ~ i = 1 Hg,i

hg rt~ = rt i + ~ g » i rtg,i

¢i

indexes each masked group the number of systems tested the number of competing causes for system failure means that cause i is contained in masked group g the number of systems that do not fail; for lifetime data this is the number of censored cases the number of failed systems which are found in stage 1 to be due to cause i the number of failed systems with masked failure causes restricted to group g the number of system failures restricted to masking group g in stage 1 and identified with cause i in stage 2 the total number of system failures restricted to masking group g in stage 1 that are identified in stage 2 the number of system failures restricted to masking group g in stage 1 that are not taken to stage 2 for further identification (n + + hg = hg) the total number of system failures identified with cause i in either stage 1 or 2 the probability that a system failure is due to cause i

For pass]fail data Fi Pi

the probability that cause i induces a system failure the /th identification probability, is the probability that system failure induced by cause i is correctly identified in stage 1

499

500

B.J. Flehinger, B. Reiser and E. Yashchin the masking probability, is the probability that a system failure induced by cause i is restricted to masking group g in stage 1 the diagnostic probability, is the probability that a failure restricted to masking group g in stage 1 is actually induced by cause i

Lifetime data t}i),j - 1 , . . , n i t¢a),j : 1 , . . , n e tJc),j = 1 , . . , n o t j , j = 1, . . , N - no

the lifetimes of the ni failures which are identified at stage 1 to have cause i the lifetimes of the ng failures which are identified at stage 1 to be associated with the masking group g the censored lifetimes the failed lifetimes, identified or masked ~ - no = ~/k= 1

ne -t- ~g hg)

B(t),si(t) f ( t ) , S(t)

tJ

t~i*) t~g,i)

the probability density functions and survival functions associated with cause i the probability density function and survival function associated with the system the j t h event time that corresponds either to a failure or a censoring time; j = 1 , .. N the j t h failure time among all failure times that were identified with cause i in either stage 1 or 2 the j t h failure time among all unresolved failure times in masking group g the j t h failure time among the ng,i failure times resolved in stage 2 to be caused by risk i

1. Introduction Consider a computing module consisting of k components (chips), mounted on a ceramic substrate. The leads of each chip are attached to pads on the substrate and electrical connection is obtained between chips through wires running in the substrate. Each device is subject to an extensive test. A failure in any component causes the module to fall. If a module fails, failure analysis procedures restrict the cause of module failure to some subset of the components. If this subset consists of more than one component, it is called a masked group. This can occur as a consequence of the lack of proper diagnostic equipment, cost and time constraints, and the destructive nature of certain component failures making exact diagnosis difficult. Standard procedure is to perform a "multiple pull" that replaces the entire masked group. This is costly; leading to waste of chips and

Statistical analysis for masked data

501

labor. Given such masked data we would like to: provide inference on the component reliabilities and for a given masked set try to infer which component is the culprit. An example with data for this type of pass/fail situation is presented in Flehinger et al. (1996). Gupta and Gastaldi (1996) discuss how certain inspection strategies can result in masking. Masking can also arise when system life length (failure or censoring time) is available. Reiser et al. (1995) discuss data of this type arising from the testing of a particular type of IBM PS/2 computer. Here the system is viewed as being composed of three components in a series (1) the planar (motherboard), (2) the direct access storage device and (3) the power supply. As a further example of masking consider a hard drive with a built-in test facility which produces codes on hard drive failure. Due to the complexity of the situation certain codes do not identify the failure cause but rather restrict it to a subset of causes. An analogous problem arises in biomedical settings concerned with survival. In clinical trials and epidemiological studies it is not uncommon to have missing information on the cause of death. Andersen and Ryan (1998) discuss a study on colon cancer in which the cause of death was masked for 25% of the deaths. Lapidus et al. (1994) in a study of motorcycle fatalities found that 40% of death certificates are missing information. This type of problem can also occur in animal bioanalysis (Kodell and Chen, 1987). The masking problem is a generalized form of the classical competing risks problem (David and Moeschberger, 1978; Kalbfleisch and Prentice, 1980) in which a system failure can be due to only one of k competing causes. A system can be a man-made machine with failure being a malfunction and cause of failure being a machine component or it can be a biological system with failure being death and cause of failure being a disease. If the cause of failure is exactly known, then standard methods of analysis are available. For the analysis of masked data a substantial literature has developed under various parametric and nonparametric settings, using both frequentist and Bayesian methodology (see, for example, Usher and Guess 1989; Miyakawa, 1984; Usher and Hodgson, 1988; Guess et al., 1991; Doganaksoy, 1991; Reiser et al., 1995; Aboul-Seoud and Usher, 1996; Albert and Baxter, 1995; Goetghebeur and Ryan, 1995; Usher, 1996; Mukhopadhyay and Basu, 1997; van der Laan and McKeague, 1998; Basu et al., 1999). All of the above papers make the strong symmetry assumption, sometimes only implicitly, that the probability of masking does not depend on the true failure cause. Lin and Guess (1994) and Guttman et al. (1995) show how orte must beware this assumption. Dinse (1986) discusses masking both with and without the symmetry assumptions. He develops a likelihood-based procedure based on grouping the data which leads to non-identifiability and breaks down if the hazard functions of the competing risks are proportional. Frequently, engineering and medical considerations require that some subset of masked failures be subject to a second stage of study to arrive at a definitive resolution of the failure cause. This diagnosis occurs in failure analysis labora-

502

B.J. Flehinger, B. Reiser and E. Yashchin

tories or on autopsy tables. This stage-2 data, appropriately collected, can be used to improve statistical procedures and do not require artificial assumptions such as symmetry. Recently, a series of papers (Flehinger et al., 1996, 1998, 1999; Reiser et al., 1996) have examined this approach. In this paper we review and further develop the use of the two-stage experimental procedure. For most of the paper we assume that the failure causes operate independently and that each system failure is caused by one and only one of the competing causes (risks). At the initial testing stage (stage 1) various constraints can result in masking, either complete (cause of failure unknown) or partial (cause of failure is restricted to some subset of possible causes). In stage 2 a sample of cases from each masked group found in stage 1 is definitively diagnosed. We suppose that the probability with which the sample is selected depends only on the data observed in stage 1 and not on the unknown parameters of interest. In Section 2 we consider the pass/fail case in which failure times are not available. In this case our analysis focuses on inference about the probabilities that various causes will induce a system failure, the probabilities that a failure due to a given cause will result in a masked set (masking probabilities), and probabilities, given a masked set, that the failure is due to a specific cause (diagnostic probabilities). In Section 3 we consider the situation in which the lifetimes of individual failures and censoring times are available. In this case the inference is focused on survival curves of the individual causes of failure and on masking parameters. We first consider the nonparametric case in which the hazards of the underlying competing risks are assumed to be proportional to each other. The analysis in this case is greatly simplified by the fact that it can be decomposed into the analysis of lifetimes and analysis of failure counts; the diagnostic probabilities turn out to be independent of the failure times. Next we consider a more general case where the hazards are not assumed to be proportional. We treat this case parametrically, with the emphasis on the Weibull lifetime distributions for the individual hazards. The Weibull is a particularly useful choice due to its flexibility. We illustrate use of the Weibull model based on the example arising in the process of hard drive manufacturing and compare the results obtained by applying this model with those obtained under the proportional hazards approach. Finally, in Section 4 we consider the case in which the competing risks are not assumed to be independent.

2. Pass/fail case

In order to illustrate the statistical analysis of the masked pass/fail case we examine the data on a five-component system introduced by Flehinger et al. (1996) which is reproduced in Table 1. Our statistical analysis for this data proceeds via construction of the likelihood function. Since we are assuming that a system failure is due to one and only one component, we can state that F~Fj ~ 0, for every i , j and, furthermore, that the probability for system failure is F = ~/k_l Fi (Reiser et al., 1996 consider a more

Statistical analysis for masked data

503

Table 1 Data from a two-stage experiment of 5000 systems of 5 components Test results

Stage- 1

Functional systems

4560

Defective component 1

16

2 3 4 5

22 53 58 57

Masking group 1,2 1, 3 4, 5 1,2,3

Stage-2 resolution 31 28 143 32

1

2

2 4

5

No stage-2 3

4

-

-

-

10

19

7

-

2

1

0

5

24 22 114 24

general model that allows for m o r e than one c o m p o n e n t to have failed). A system under test either (a) does not malfunction, (b) fails due to c o m p o n e n t i and is identified as such in stage 1, (c) fails and in stage 1 is classified as belonging to the m a s k e d group g while c o m p o n e n t i is diagnosed as responsible in stage 2 or (d) is classified as belonging to the m a s k e d group g and is not investigated further. Each of the four possible events occurs with probability (a) (1 - ~ ~ - 1 El) = 1 - F , (b) F~P,-, (c) F,.PgI,, (d) ~~cgF,.PgL» Consequently, the likelihood function obtained on observing i = l, 2 , . . , k and various m a s k e d groups g, is

( ~ )~0H HH

L =

1-

Fi

(F~P,.)n*

i=I

i=1

H(~

\ iCg

for

/ ~~

F.pgl~

(F~Pgl~)n0,,

i=1 gDi

no, ni, ngl~,bs,

(2.1)

/

with

Pi+~P
i=l,...,k

.

(2.2)

gDi

The u n k n o w n p a r a m e t e r s of the likelihood function are the Fi,Pi,Pgli for i = 1 , 2 , . . . , k and observed m a s k e d groups g. It is convenient to reparametrize this likelihood function. D e n o t e by 4>i the probability that given a failure the cause i is responsible; clearly, Oi = Fi/F. Let 7i = P/¢i be the probability that system failure is due to cause i and is identified in stage 1 and Bg = ~ i c ~ Pvl*~i be the probability that a failure is m a s k e d and is restricted to group g in stage 1. In the pass/fail situation the diagnostic probability can be written as ~'Pgli

_

_

B.J. Flehinger, B. Reiser and E. Yashchin

504

It is easy to see that for every g (2.3a) iCg

and for every i (2.3b)

(9i = 7i q- ~ ßgI-[i]g oci

and k

~-~~7i+ Z B g = 1 . i=1

(2.3c)

g

Rewriting (2.1) in terms of F, Ói, Bg and 7Zilgresults in the likelihood function L = Ls x L1 × L2 ,

(2.4)

where Ls = F ~ ni+~ n«(1 - F) n°,

(nil)

L1 =

7~'

\i=1

g

B;g ,

(2.5)

g

\'Cg

In this form the likelihood function is intuitively appealing breaking up into three separate multinomial pieces with Ls dealing solely with the results at a system level, Lx expressing the stage 1 data while L2 is concerned only with stage 2. From (2.4) and (2.5) maximum likelihood estimators of the "new" parameters can be immediately written down as B =

~ i ni -r- ~-~~gn«

Se ni + ~ ~ ng + no

N - no -

ni -ni ~i = E i Æ i _ } _ E g n 9 N - n o ^

__

Y/g,i

__ Æ9,i

Bi = ni + ~ o » i ngng,i/n+ N - no

Bgi

_

ng

N - no

-

-

N

'

(2.6a)

(2.6b) '

(2.6c)

(2.6d)

(2.6e)

Statisticalanalysisfor maskeddata

505

Retransforming back to the original parametrization and taking into account the fact that n s = ng+ + hg gives

1

N,

i = 1,..,k

N (n; + n,),

,

(2.7a)

where h~ is the number of masked system malfunctions that were not resolved but that are allocated to the ith component by the m a x i m u m likelihood procedure. This allocation is intuitively satisfying because it allocates the masked cases in the same proportion in which diagnosis was made in those masked cases brought to resolution in the second stage. The identification probabilities are

Pi = ni/(n; + h;)

(2.7b)

and the masking probabilities are

Bgli = ( ng'i q- fig ]~jCg ng'ino,j/]-~/(n~ + h:) "

(2.7C)

These formulas can be readily applied to the data of Table 1 resulting in the estimates given in Table 2. The F / p r o v i d e information on component reliability and are useful in monitoring and improving product quality. For more detailed statistical inferences likelihood ratio tests based on (2.1) can be obtained and profile likelihoods can be computed and numerically inverted to provide approximate confidence bounds. In particular, for any specified model parameter of interest (i.e., any of the Pi, Pgli, Oi o r F/) say 7, let L* (7) denote the maximal value of L obtained under the constraint that the value of 7 is fixed. L* (7) is termed the profile likelihood of ?. The log-likelihood ratio associated with this value is ~(~) = -2{logL*(7) - l o g Q

,

(2.8)

Table 2 Estimates of failure and identification probabilities for individual components based on data in Table 1. The third column contains the upper 95% confidence bounds for F/ obtained by the profile likelihood method Component i

Failure probability (~)

Upper 95% bound for ~

Identification probability (/5/)

1 2 3 4 5

0.0095 0.0088 0.0181 0.0215 0.0301

0.0131 0.0116 0.0219 0.0268 0.0357

0.337 0.498 0.587 0.540 0.378

506

B.J. Flehinger, B. Reiser and E. Yashchin

asymptotically, at the hypothesized value o f 7, this value has a chi-squared distribution with one degree of freedom. A p p r o x i m a t e confidence limits for 7 can be obtained by finding the values o f 7 such that 7'(7 ) is equal to the required percentile point o f this distribution. F o r example, 95% upper confidence bounds on the c o m p o n e n t failure probabilities F / o b t a i n e d via the profile likelihood m e t h o d are given in Table 3. C o m p u t a t i o n a l details and further examples are given by Flehinger et al. (1996). The estimates o f the diagnostic probabilities can be used as aide to decision when system malfunctions occur. F o r example, from Table 3, we note that if a masked g r o u p c o m p o s e d o f c o m p o n e n t s (1, 2, 3) is observed it is m o s t likely that c o m p o n e n t 3 is at fault. Inference for diagnostic probabilities is based on stage-2 information. F o r an observed masking subset 9, the relevant data consist o f the hg,/for all i c 9. These values follow the multinomial distribution n~!

1-1. iln~,

I~icgno,i! lic9 l

(2.9)

ilg "

The likelihood function in (2.9) is obtainable as a profile likelihood from (2.4). Standard procedures for tests o f hypothesis or confidence intervals for multinomial parameters apply. Flehinger et al. (1996) provide further details in the use o f diagnostic probabilities.

3. Life time data: Independent c o m p e t i n g risks

This case parallels the pass/fail situation with the addition o f system lifetimes which are either failure times or censoring times (for the non-malfunctioning systems. As a motivating example consider a scenario in which a c o m p a n y m a n u f a c t u r i n g hard drives for computers is trying to analyze causes o f failures of a certain sub-assembly. Following a failure, the hard drive is typically returned to the manufacturing plant, with one o f several dozen error codes such as " c a n n o t settle d o w n on the track", " c a n n o t read" or " c a n n o t write". Typically, the underlying causes can be subdivided into a n u m b e r o f categories, such as " h e a d failure", " h e a d flyability", " c o n t a m i n a t i o n " , "failed electronic c o m p o n e n t " (such Table 3 Estimates of masking and diagnostic probabilities for masked groups corresponding to data in Table 1 Masking group g (1, 2) (1, 3) (4, 5) (i, 2, 3)

Masking probabiIity 1

2

0.186 0.393

0.502 0.103 0 0.310

0.084

3

Diagnostic probability 4

5

1

0.622

0.286 0.714 0.667 0.333 0.125 0 0.875

0.460 -

2

3

4

5

0.345

0.655

Statistical analysis for masked data

507

as a card), "microcode problem", and so forth. Some of these causes, such as "head failure", are related to components, but others (e.g. "particle contamination") are not; in this application, we do not treat these separately and refer to them simply as causes of failure. To simplify the presentation, in this section we will focus on three causes of failure: "electronic card" (cause 1), "head flyability" (cause 2) and "head/disk magnetics" (cause 3). We assume that these causes act independently and in series. If a true underlying cause of failure is "electronic card", it can manifest itself through a number of error codes - sometimes these codes, together with some additional diagnostics are sufficient to identify this cause as a culprit. However, it can also manifest itself through error codes that, even with additional diagnostic procedures, do not enable one to exclude the possibility that the actual culprit is "head/disk magnetics" - only the stage 2 laboratory analysis can tell these two causes apart. In this case the stage 1 analysis results in a masked group (1, 3). Finally, the "electronic card" cause can manifest itself in a way that makes it impossible (without going into stage 2 analysis) to exclude the possibility that either "head flyability" or "head/disk magnetics" are the actual culprits - such failures result, after stage 1 analysis, in a masked group (1, 2, 3). If the true cause of failure is "head flyability" it will either be definitively apparent from the error codes and stage 1 analysis, or it would not be possible, without stage 2 analysis, to exclude the possibility that causes 1 or 3 are responsible for the failure, i.e., stage 1 can result in a masked group (1, 2, 3). However, masked groups (1, 2) or (2, 3) are impossible - for example, if the true cause of failure is "head flyability", it is not possible that the first stage analysis will declare that the failure could also be due to "electronic card", but not to "head/disk magnetics" or that the failure could also be due to "head/disk magnetics", but not to "electronic card". Finally, if the actual cause of failure is "head/disk magnetics", there is a possibility that this fact will be revealed based on error codes and stage 1 analysis. Otherwise, stage 1 will result in masked failure (1, 3) or (1, 2, 3), but never in (2, 3). Now consider the scenario in which four years ago 10,000 drives were manufactured and since then information about failures was collected in a database. The number of failures observed in this period was 172. Some of the failures were masked and a selected number of those were analyzed to complete resolution in the defect isolation laboratory. The resulting data are given in the first four columns of the Table 4 (other columns correspond to the output of estimation procedure described later in the text). Note that these data are disguised and, therefore, conclusions based on subsequent analysis do not reflect the actual performance of the hard drives manufactured by IBM. The first two columns give the sequential number and failure time, respectively. The third column (outcome) gives the cause of failure if it was either identified immediately or resolved in stage-2; the value - 1 corresponds to unresolved failures. Finally, the fourth column gives the masking information. Consider, for example, failure #6 for which the entry in the third column is 1 and the entries in the fourth column are 2, 3. This failure originally corresponded to the masked group (1, 2, 3) but then was resolved as a failure due

B . J . Flehinger, B. Reiser and E. Yashchin

508

Table 4 Life time data with masking # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

FaiI time

Outcome

0.0183 0.0357 0.0427 0.0447 0.0735 0.119 0.131 0.143 0.171 0.261 0.316 0.334 0.368 0.475 0.484 0.489 0.494 0.594 0.604 0.664 0.697 0.712 0.743 0.743 0.749 0.752 0.789 0.831 0.833 0.867 0.874 0.890 0.955 1.03 1.04 1.10 1.11 1.17 1.21 1.30 1.3i 1.31 1.33 1.35 1.35 1.37 1.42 1.46 1.49 1.53

1 1 -1 1 -1 1 -1 -1 1 -1 1 1 2 1 2 2 -1 1 1 2 1 1 1 2 -1 2 2 1 3 -1 2 -1 3 -1 3 3 1 -1 -1 1 1 3 3 -1 -1 1 3 3 1 1

Masking

Weibull diagnostic prob.

3 1

0.992

3

0.008

1 2 1 i 3 1 3

2 3 2 3 3

0.901

0.099

1

3

0.783

0.217

2

3

3 2 1 I

3 3 2

3

0.444

0.262

0.294

2 1 1

3 2 2

3

0.410

0.253

0.336

1

3

0.604

0.396

1

3

0.553

0.447

1 3 1 1 2

2 2 2 3

1

2

1 1 3 1 1 2

2 2

3

3

0.765

0.217

0.017

3

0.718 0.957

0.245

0.037 0.043

3 3

0.340 0.332

0.230 0.227

0.430 0.441

3 3

0.305 0.305

0.217 0.217

0.478 0.478

Statistical analysis for masked data ,

# 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

509

Fail time

Outcome

Masking

Weibull diagnostic prob.

1.57 1.58 1.58 1.61 1.61 1.63 1.64 1.66 1.67 1.68 1.68 1.68 1.71 1.72 1.73 1.75 1.82 1.88 2.00 2.03 2.04 2.04 2.05 2.05 2.07 2.09 2.12 2.15 2.18 2.22 2.23 2.26 2.26 2.27 2.29 2.30 2.35 2.37 2.39 2.40 2.41 2.41 2.41 2.43 2.49 2.52 2.57 2.64 2.66 2.67

-1 -1 -1 -1 -1 -1 3 3 -1 1 3 2 3 3 1 2 3 -1 -1 -1 -1 3 1 1 -1 3 3 2 3 3 1 3 -1 -1 -1 1 -1 -1 1 2 -1 3 2 -1 3 -1 -1 3 -1 3

1 1 1 1 1 1 1

3 3 3 3 2 2 2

3 3

0.400 0.398 0.397 0.392 0.266 0.263

0.199 0.198

0.600 0.602 0.603 0.608 0.535 0.539

1 3 1 1 1 1

2

3

0.257

0.195

0.548

3 2 2

1 1 1 1 1 1 3 3 1 1 1

2 3 3 2 3 2

1

2

1 1 1

3 2 2

3 3

0.281 0.193 0.192

0.161 0.161

0.719 0.646 0.648

1 1

2 2

3 3

0.186 0.185

0.158 0.157

0.656 0.658

1 1

3 3

1 1 1 1 1 1 1

2

3

0.181

0.154

0.665

3 2

3

0.250 0.170

0.148

0.750 0.682

2

3

0.164

0.144

0.692

3

3

0.338 0.318 0.215 0.313

0.174

0.662 0.682 0.611 0.687

0.693

0.307

2

0.737

0.263

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Table 4 (Contd.) #

Fail time

Outcome

Masking

Weibull diagnostic prob.

101 102 103 104 105 106 i07 108 I09 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 13I 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150

2.70 2.74 2.75 2.75 2.76 2.79 2.82 2.83 2.85 2.86 2.87 2.95 2.96 2.98 3.03 3.04 3.07 3.07 3.08 3.11 3.12 3.13 3.15 3.15 3.16 3.16 3.18 3.19 3.21 3.24 3.25 3.27 3.28 3.37 3.42 3.43 3.46 3.51 3.63 3.65 3.65 3.65 3.67 3.67 3.70 3.72 3.77 3.77 3.77 3.78

-1 -1 -1 3 3 3 3 -1 3 3 -1 2 -1 3 -1 1 1 1 -1 2 -1 3 3 -1 3 2 3 3 -1 3 2 -1 -1 -1 2 -1 1 3 3 3 3 -1 -1 1 3 1 3 3 -1 3

1 1 1 1 1

3 3 3

0.232 0.223 0.227

0.768 0.772 0.773

3

0.220

0.780

1 1 1 1 1

2

3

0.152

0.136

0.712

1

2

3

0.146

0.133

0.721

1

3

2 2 1 1 1 1 1 1 1

3 3 2 3 3

1 1 1 1

0.204

3

3 2

2

1 1 1 1 1 3

3 2 2 3 2

1 1 1 1 1 3 i

2 2

1 1 1 1

2 2 2

0.140

0.796

0.129

0.731

0.196

0.804

0.194

0.806

3

0.134

0.125

0.741

3 3

0.186 0.130 0.127

0.123 0.120

0.814 0.747 0.753

3

0.124

0.118

0.758

3 3

0.163 0.162

3

0.111

0.837 0.838

0.109

0.780

Statistical analysis for masked data

#

Fail time

Outcome

Masking

151 152 153 154 155 156 157 158 159 160 1öl 162 163 164 165 166 167 168 169 170 171 172

3.79 3.79 3.80 3.81 3.81 3.83 3.83 3.84 3.84 3.85 3.86 3.86 3.87 3.89 3.91 3.91 3.93 3.95 3.95 3.96 3.98 3.98

-1 -1 1 3 -1 1 -1 -1 2 3 -1 -1 -1 -1 -1 -1 3 3 -1 3 3 3

1 1 2 1 1

3 2 3 2 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1

2 2 3 2 2 3 2 3 3 3 2 2 2 2

511

Weibull diagnostic prob.

3

0.155 0.111

0.109

0.845 0.781

3

0.110

0.108

0.782

3 3

0.109 0.109

0.108 0.108

0.783 0.783

3

0.109 0.152 0.108 0.150 0.150 0.149

0.107

0.784 0.848 0.785 0.850 0.850 0.851

0.105

0.105

3

3

0.107

0.790

1

Devices censored at 4 years: 9828

to cause 1. Failure #20 was immediately identified as one related to cause 2. In cases of unresolved failures column 4 contains the complete masking group; for example, it is still unknown whether failure #3 is due to cause 1 or 3. One can see that the only masked groups contained in the data are (1, 3) and (1, 2, 3). Other groups do not appear because of the properties of stage 1 analysis described above. We must note, however, that in practical cases there is no need to establish a priori which masked sets are possible - the estimation procedures described below behave as if the only possible masked sets are those that actually appear in the data. We will discuss two approaches to the analysis of this type of data. The first developed in Flehinger et al. (1998) is non-parametric but assumes that the hazard functions of the competing risks are proportional to each other. This assumption is frequently made when analyzing competing risks, e.g. Yip and Lam (1992), Goetghebeur and Ryan (1990, 1995) and Dewanji (1992). The second approach due to Flehinger et al. (1999) assumes a parametric model which does not require the proportional hazards assumption. In both these approaches we assume that the identification probabilities (Pi) and masking probabilities (Pgli) do not depend on time and that the failure causes operate independently, i.e., the survival function S(t) of the system is determined from the survival functions Si(t) corresponding to causes i = 1, 2 , . . . , k by means of the formula

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B.J. Flehinger, B. Reiser and E. Yashchin k

s(t) :- IIs,(t)

.

i--1

In parallel with (2.1) the likelihood function can be written as

j=l

"=

J=

)< 1-I H I ' I i=l gDi j=l

xH

"

]iß

g,i)

Sl

PgF~f~ ~o) H S,

g j=l Lrcg

g,i)

g)

(3.1)

l#r

In the above representation the terms of the first line relate to censored observations and to the failures that were not masked. The second line relates to failures that were originally masked and then resolved in stage 2. The last line relates to failures that were masked in stage 1 and remain unresolved. In practical applications the survival functions of individual causes are either known up to some parameters that have to be estimated from the data (parametric models) or are estimated non-parametrically. The data analysis is simplest in the case of proportional hazards that we consider hext.

3.1. Proportional hazard models Assuming that the hazards corresponding to individual causes of failure are proportional implies that

Si(t) = [S(t)l ó~, i= 1, ..,k

(3.2)

f i ( t ) = ~ßi[S(t)] ¢i Il(t)

(3.3)

and

,

where

k

ZOi=

1 .

(3.4)

i=l The exponent ¢i can be interpreted as the probability that a failure is caused by component i: Ói does not depend on time and measures the relative contribution of the components to the overall risk of failure. Substituting (3.2)-(3.4) into (3.1) and some algebraic manipulation enables one to represent the likelihood in terms of the parameters 7i,Bg and 7Zilgdefined in Section 2 L = Lst x L1 × L2 ,

(3.5)

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513

where Lst is the likelihood function of lifetime data at the system level,

H f(tj) ~s {~~/~~~,/} ~-~°

(3.5a)

j=l

and L1,L2, the contributors related to first and second stage data, are given by (2.5). Note that Lst is the usual likelihood associated with K a p l a n - M e i e r survival estimation; it plays a role analogous to that of Ls in the pass/fail case. F r o m Lst K a p l a n - M e i e r estimates S(t) are available, while from L1 and L2 the estimates ~)i,/}g,/cil0 and consequently ~bi are obtained exactly as in Section 2. Note, however, that though the expressions of the basic parameters in terms of qS~ remains the same as in Section 2, ~bi here is the basic model parameter, while in the pass/fail case it was a derived parameter defined by ~bi = Fi/F. Finally, estimates of the survival function of the individual risks are obtained as

Si(t)=[S(t)] ~',

i= 1,..,k

.

(3.6)

For some inference problems, such as estimation of masking probabilities, it is convenient to represent the count-dependent part Lc = L1 × L2 of the likelihood (3.5) in a somewhat different form, k

/

Lc H (Pi@) i=l

\n~ k

Pg[ißi g \ iCg

H H (P9Iißi)n~« /

i=1 gDi

Profile likelihoods can be obtained for the q5i and Si(t) which can be inverted numerically to provide approximate confidence intervals. Algorithms for these computations and examples on data can be found in Flehinger et al. (1998). These computations indicate that only a small proportion of masked cases are required to be taken to stage-2 in order to provide effective inference. Inference for diagnostic probabilities is based on stage 2 information, as discussed in Section 2 for the pass/fail case. For the lifetime data in Table 4, the estimates of ~bi and their associated 95% confidence bounds are presented in Table 5. This table indicates that cause 3 is the most likely to be the source of system failure. In Table 6 we give the estimates of masking and diagnostic probabilities for every masking group. Table 5 Estimates of probabilities 0i that a failure is due to cause i, for the lifetime data in Table 4 Parameter

MLE

95%bounds

01 02 03

0.318 0.144 0.537

0.233 0.090 0.439

0.414 0.216 0.632

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B. J. Flehinger, B. Reiser and E. Yashchin

Table 6 Estimates of masking and diagnostic probabilities for the lifetime data in Table 4 Masking group

1, 3 1, 2, 3

Masking probs.

Diagnostic probs.

1

2

3

1

2

3

0.402 0.324

0.476

0.454 0.427

0.344 0.257

0.171

0.656 0.571

Table 6 suggests, for example, that if a failure is caused by cause 1 then the estimated probability that this failure will result in a masking group (1, 3) is 0.402 and the probability that it will result in a masking group (1, 2, 3) is 0.324. Consequently, the estimated probability that no masking will occur is 1 - 0.402 - 0.324 = 0.274. If a failure is due to cause 2 it will result in a masking group (1, 2, 3) with probability 0.476 and no masking will occur with probability 1 - 0.476 = 0.524. Furthermore, given a masked (1, 3) failure, the estimated probability that it is due to cause 1 is 0.344 and the probability that it is due to cause 3 is 0.656. In Table 7 we give the confidence bounds (for individual causes o f failure) for probabilities to survive 1, 2, 3 and 4 years. The estimated survival functions under the proportional hazards assumptions are given in Figure 1. 3.2.

WeibulI model

In m a n y practical situations the assumption o f proportional hazards cannot be justified. Analysis o f such situations generally tends to be far more complex because (a) the probability q5i that a failure is due to cause i depends on the time of failure and (b) the likelihood function c a n n o t be represented as a p r o d u c t o f two functions, one depending s o M y on the failure counts and another depending solely on failure times, as in (3.5). One possibility is to assume a fully parametric model for c o m p o n e n t failure times distributions. Since the Weibull assumption is m o s t frequently encountered in practical applications o f this type, in this article we limit our attention to the Weibull case only. In this case Table 7 Values of Si(t) and corresponding 95% lower confidence bounds, for the lifetime data in Table 4 Years

1 2 3 4

Cause 1

Cause 2

Cause 3

Prob.

Bound

Prob.

Bouild

Prob.

Bound

0.9989 0.9978 0.9964 0.9945

0.9985 0.9971 0.9952 0.9929

0.9995 0.9990 0.9883 0.9975

0.9993 0.9985 0.9976 0.9964

0.9982 0.9963 0.9939 0.9907

0.9976 0.9953 0.9924 0.9888

S t a t i s t i c a l analysis f o r m a s k e d d a t a

1.000

515

- ~ ' ~ - . " - ! ...... -- . . . . . . . . . " . . . . . . . . . . . . i .... -"----. ....... !........ ..,..: .............

-

0.995

,,

,,

i ! ........... :

~-.-.,..

,

.,,

0.990

0.985 -- System .... Cause 1 - -- - - Cause 2 --. -Cause 3

0.980

\

i

0

i

1

i

2

3

F i g . 1. E s t i m a t e s o f r e l i a b i l i t y ( p r o p o r t i o n a l

Si(t) = e x p [ - ( t / O i ) ~ i ] ,

i

4

hazards model).

(3.8)

i = l,.. . k ,

and the 2-stage log-likelihood function can be represented in the form k

N

log L = - Z

~~_~(t-j/Oi) ~i

i=1 j = l

+~

nilog 4 + Z ng,ilog Pgli

i=l

÷ Z

gDi

n~log((Si/Oi)4-((5i-

i=1

i*)/Oi

j=l

÷ ZZlog g

1)Zlog

j=l

ZPglr(6r/Or)@a)/Or)

6r-1

.

(3.9)

Lrcg

The diagnostic probabilities now depend on fime and can be written as

Hilg(t)

=

P~I~(~i/O~)(t/Oi) ~~-~ ~rcgpglr(6~/O~)(t/Õ~)a,.-1

(3.10)

The analytical formulas for the MLEs developed in the pass/fail case and proportional hazards case no longer apply and (3.9) needs to be optimized numerically. Algorithms for carrying this out and computing profile likelihood based confidence intervals are described in Flehinger et al. (1999). When applying the Weibull model to the data in Table 4, the resulting MLE estimates of the shape and scale parameters are (61, 3 2 , 3 3 ) = (0.691, 1.006, 2.151)

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B . J . Flehinger, B. Reiser and E. Yashchin

and (01, 02, 03) = (7570, 1570, 34.9). Figure 2 gives the estimated survival functions for the individual causes o f failure and for the system. This figure suggests that, t h o u g h most o f the system failures are caused by cause 3, the early failures are more likely to be due to cause 2. This contrasts with the conclusions from the nonparametric p r o p o r t i o n a l hazards model, which implies that the probability (~bi) that a failure is due to cause i remains constant over time, for all i (see Figure 1). The estimates o f masking probabilities for the Weibull case are given in Table 8. Orte can see that these estirnates are quite close to those obtained under the assumption o f p r o p o r t i o n a l hazards given in Table 6. The estimated diagnostic probabilities IIilg(~ «)) are shown in the last 3 columns o f the Table 4. F o r example, in the unresolved (1 3) masked case corresponding to failure #3 the estimated probability that the failure due to cause 1 is 0.992. In a similar masked case #166, however, this probability is only 0.149 (note that under the p r o p o r tional hazards assumption this probability is 0.344 independently o f the time o f failure, see Table 6). Table 9 presents point estimates and lower 95% confidence b o u n d s for the survival probabilities at certain selected times.

1.000

0.995

.

.

.

.

,

.

?.................... ?<-.............. ._...... ,

"¢.2,2.,-,

. . . . . .

.

ù.~

.-,"~ 0.990

.....................................................

"_.~__

0.985 .... --System ........................................... . . . . Cause 1 ---- - Cause 2 ----- Cause 3 0.980 ~ ~ I 0 2

I 4

Years Fig. 2. Estimates of reliability (Weibull model). Table 8 Estimates of masking probabilities (Weibull model) for the lifetime data in Table 4 Masking group

1, 3 1, 2, 3

Masking probs 1

2

3

0.412 0.310

0.469

0.446 0.436

Statistical analysis for masked data

517

Table 9 Values of Si(t) and corresponding 95% lower confidence bounds for the lifetime data in Table 4 obtained under Weibull model. The last two columns provide estimates for the system Years

1 2 3 4 5

Cause 1

Cause 2

Cause 3

System

Prob.

Bound

Prob.

Bound

Prob.

Bound

Prob.

Bound

0.9979 0.9966 0.9956 0.9946 0.9937

0.9970 0.9954 0.9940 0.9927 0.9915

0.9994 0.9988 0.9882 0.9975 0.9969

0.9988 0.9980 0.9971 0.9962 0.9952

0.9996 0.9979 0.9949 0.9906 0.9848

0.9991 0.9968 0.9934 0.9883 0.9808

0.9968 0.9933 0.9987 0.9828 0.9756

0.9958 0.9918 0.9967 0.9801 0.9713

F i g u r e 3 gives the e s t i m a t e d h a z a r d functions o f the i n d i v i d u a l causes o f failure, hi(t) ----f i ( t ) / S i ( t )

-- 6i toi-10i -g)i

(3.11)

a n d those o f the system. One i m p o r t a n t q u e s t i o n is w h e t h e r the d a t a in T a b l e 4 c o n t r a d i c t the h y p o t h e s i s t h a t the u n d e r l y i n g h a z a r d s are in fact p r o p o r t i o n a l . A s s u m i n g the W e i b u l l model, testing this h y p o t h e s i s is equivalent to testing the equality o f the shape p a r a m e t e r s for the k causes, i.e., /'/ : 61 = 62 . . . . .

6k •

I n the d e r i v a t i o n o f a l i k e l i h o o d ratio test o f H it is necessary to o b t a i n the m a x i m a l value o f the l i k e l i h o o d f u n c t i o n u n d e r this h y p o t h e s i s (we d e n o t e the c o m m o n , a n d u n k n o w n , value o f the shape p a r a m e t e r b y 6) a n d the u n c o n i

0.006

~q

i

i

i

- System . . . . Cause 1 . . . . . Cause 2 ---- - Cause 3 -

i

i

i

-

~

/

/

//./..

-

0.004

ZZ

/f/.-// k

j ù

..J

.~-

0.002 ..... «--

0

J"

- .....................................................

~ ~ , , ~ _ _". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

I

1

I

I

2

I

I

3

Years Fig. 3. Estimates of the hazards (Weibull model).

I

4

B . J . Flehinger, B. Reiser and E. Yashchin

518

strained maximal value of the likelihood function,/~. The value of L is obtained from (3.9), by substituting the parameter MLEs. Given H, the survival function of cause i is Si(t) = expl-(t/Oi)~l and the system survival function can be written as

S(t) = exp[-(t/O) ~] ,

(3.12)

where the system scale parameter 0 is defined by

o;(~~) -1~'~:~

(3.13)

and, correspondingly, the probability density function for the system lifetime is

f ( t ) = 6t~-10.6 exp[-(t/0) õ] .

(3.14)

The probability q5i that a failure is due to cause i can in this situation readily be seen to be q5i = (0/0~) ~ ,

(3.15)

independently of the time at which the failure occurs. After some algebra LH, the likelihood under H can be written as L/~ = Lst × Lc ,

(3.16)

where the expression for Lst is (3.5a) with S(t) and f ( t ) being computed using (3.12) and (3.14), respectively, and the expression for Lc is (3.7) with the values of q~i given by (3.15). Lst is the usual likelihood function for censored Weibull data and standard numerical optimization methods can be used to obtain 6 and Ô (e.g. see Meeker and Escobar, 1998). The estimates/3/,/5~1 i and ~i are obtained by maximizing Lc, resulting in formulae (2.7b), (2.7c) and (2.6d). Substituting these estimates into (3.16) results in LH. Consequently an asymptotic likelihood ratio test of H is obtained by considering - 2 l o g ( m a x L H / [ ) as a Z~-I variate. For the Table 4 data the estimated common shape parameter under H is 6 = 1.18 and the log-likelihood ratio statistic for the hypothesis H : 6 ~ =62 =63 is - 2 1 o g ( m a x L ~ / maxL) = 32.20. This value is clearly incompatible with the Z~ null distribution, indicating the unsuitability of the proportional hazards assumption. This conclusion is not surprising, in light of Figure 3.

3.3. Extensions One of the major differences between the Weibull and proportional hazards models is dependence of (bi, the probability that cause i is responsible for the system failure, on time. This suggests the construction of a semi-parametric model in which a parametric model linking the Ói to time would be used while no specific

Statistical analysis for masked data

519

parametric model for the probability density function corresponding to individual causes of failure would be assumed. One simple model of this type is k

log(~)=ai+bit,

i=l,2,..,k-l,

subject t o Z ~ i = l .

(3.17)

i=1

Similarly, one can obtain a semi-parametric model by imposing a certain form of time dependency on the diagnostic probabilities. Furthermore, our basic assumption that the masking probabilities are not functions of time could be weakened, both in parametric and nonparametric settings. Modifications of this type complicate the likelihood function and it would be necessary to develop suitable optimization methods. 4. Life time data: Dependent competing risks In Section 3 we have been assuming that the competing causes of failure operate independently. However this strong assumption cannot be tested from data comprised from only the system lifetimes and observed cause of failure. The difficulty is that different multivariate models can give raise to exactly the same hazard functions (Tsiatis, 1975; Lawless, 1982). In many problems some sort of dependence seems plausible. However, the choice of a particular parametric model for this dependency could be very difficult and misleading. Consequently a nonparametric approach which does not assume independence and emphasizes the estimation of cumulative incidence probabilities has appeared in the statistical literature (see, for example, Kalbfleisch and Prentice, 1980; Lawless, 1982; Gaynor et al., 1993) but does not appear to have influenced the reliability literature (but see Sun and Tiwari, 1997). The presence of masking along with dependence introduces a further complication which requires consideration. Let (T, C) denote the system lifetime and cause of failure. Let Äi(t)

lim F_Prob(t < T < t + At, C = ilT >_ t)l

At

~t-~~ L

J'

i

1,.. (4.1)

represent the cause-specific hazard rate and

Ii(t) = Prob(T < t, C = i) =

2

2i(u)S(u)du,

i = 1,.., k ,

(4.2)

denote the cumulative incidence probability or cause-specific failure probability resulting in

d ~i(t) = ä?r,(t)

2i~t)S(t)

(4.3)

as the cause-specific probability density function. In the above formulae S(t) represents the system survival function. Furthermore, the system hazard rate and probability density functions are

B.J. Flehinger, B. Reiser and E. Yashchin

520

k

k

2(t) = ~-~2i(t),

f(t) = Z#i(t)

i=l

(4.4)

.

i=1

When independence is not assumed the marginal survival functions are not identifiable and thus the Si(t) discussed in Section 3 are no longer estimable. For a fuller discussion of the meaning of cause-specific functions see Kalbfleisch and Prentice (1980) and Gaynor et al. (1993). Consequently, for data as described in Section 3 the resulting likelihood function is no

k

rti

(4.5) i=l g»i j = l

j = l Lrcg

Assuming that the cause-specific hazards are proportional, i.e., 2i(t) = qSi2(t), it can readily be seen that the proportionality constant q5i is in fact the probability that cause i is responsible for the system failure ~bi = Prob(C = i) = Ii(ec)

.

(4.6)

In addition, (4.7)

gi(t) = 4 i f ( t )

and //(t) = ~bi(1 - S(t))

.

(4.8)

Substituting (4.7) into (4.5) and some algebraic manipulation results in exactly (3.5), that is the same likelihood function considered under independence of causes of failure in Section 3, and the estimates ~ßi,S(t) are obtained as described there. Consequently, the cumulative incidence probabilities can be estimated by I~(0

= ~,,(1 - ~(t))

.

(4.9)

These can be used for data analysis when the assumption of independence is not considered to be reasonable. Though the usefulness of (4.9) is restricted by the requirement that the cause-specific hazards be proportional, one can point out important classes of practical situations in which this assumption can be justified. For example, consider the situation in which all the causes are affected by a common environment random vector V and the survival curves for individual causes of failure are k

Si(t,V)=[S(t,V)I

éi,

i= 1,..,k,

with ~-~~~bi = 1 , i=1

(4.10)

521

Statistical analysis for masked data

where S is the system survival function. I n this case the causes o f failure are c o n d i t i o n a l l y i n d e p e n d e n t , given V. D e n o t e by f the system survival density; then, one can see t h a t Ii(t) = Ev

//0 ~

~i[S(u, V ) ] 4 i - l f ( u , V) H [ S ( u , V)] 4j ldu j¢i

= 0z × {1 - E v [ S ( u , V ) ] }

/ (4.11)

i n d i c a t i n g t h a t we have a p r o p o r t i o n a l h a z a r d s m o d e l with d e p e n d e n t causes o f failure. The system survival density a n d h a z a r d are given b y g(t) = E v f ( t , V), g(t)

(4.12)

2(t) -- EvS(t, V) a n d can be e s t i m a t e d f r o m the data; however, w i t h o u t a d d i t i o n a l a s s u m p t i o n s , the d a t a does n o t c o n t a i n i n f o r m a t i o n a b o u t a n u m b e r o f o t h e r i m p o r t a n t quantities, such as VarvS(t, V). M o r e general n o n p a r a m e t r i c m e t h o d s for the case o f d e p e n d e n t causes o f failure in the presence o f m a s k i n g w i t h o u t the h a z a r d p r o p o r t i o n a l i t y a s s u m p t i o n still need to be developed.

Acknowledgement W e a p p r e c i a t e S t e p h e n K. M a g i e ' s ( I B M , Sau Jose) help in the c o n s t r u c t i o n o f the lifetime d a t a e x a m p l e in Section 3.

References Aboul-Seoud, M. H. and J. S. Usher (1996). The effect of modular system structures on component reliabilityestimation. In Fifth Industrial Engineering Research Conference Proceedings, Vol. 464-467. Albert, J. R. G. and L. A. Baxter (1995). Applications of the EM algorithm to the analysis of life length data. Appl. Stat. 44, 323 341. Andersen, J. W. and L. M. Ryan (1998). Analysis of competing risk data with missing failure type when missingness depends on covariates. In t'nvited Paper, Presented at the Joint Statistical Meetings. Dallas, TX. Basu, S., A. Basu and C. Mukhopadhyay (1999). Bayesian analysis for masked system failure data using non-identical Weibull models. J. Stat. Plann. Infer. 78, 255-275. David, H. A. and M. L. Moeschberger (1978). The Theory of Competing Risks. Griffin, London. Dewanji, A. (1992). A note on a test for competing risks with missing failure type. Biometrika 79(4), 855-857. Dinse, G. E. (1986). Nonparametric prevalence and mortality estimators for animal experiments with incomplete cause-of-death data. J. Am. Star. Assoc. 81, 328-336. Doganaksoy, N. (1991). Interval estimation from censored and masked system failure data. IEEE Trans. Reliab. 40, 281~285. Flehinger, B. J., B. Reiser and E. Yashchin (1996). Inference about defects in the presence of masking. Technometrics 38, 247 256.

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B . J . Flehinger, B. Reiser and E. Yashchin

Flehinger, B. J., B. Reiser and E. Yashchin (1998). Survival with competing risks and masked causes of failures. Biometrika 85, 151-164. Flehinger, B. J., B. Reiser and E. Yashchin (1999). Parametric modeling for survival with competing risks and masked failure causes. I B M Research Report R C 21595. Gaynor, J. J., E. J. Feuer, C. C. Tan, D. H. Wu, C. R. Little, D. J. Straus, B. D. Clarkson and M. F. Brennan (i993). On the use of cause-specific failure and conditional failure probabilities: examples from clinical oncology data. J. Amer. Star. Assoc. 88, 400~409. Goetghebeur, E. and L. Ryan (1990). A modified log tank test for competing risks with missing failure type. Biometrika 77, 207 2ll. Goetghebeur, E. and L. Ryan (1995). Analysis of competing risks survival data when some failure types are missiug. Biometrika 82, 821-833. Guess, F. M., J. S. Usher and T. J. Hodgson (1991). Estimating system and component reliabilities under partial information on the cause of failure. J. Stat. Plann. Infer. 29, 75-85. Gupta, S. S. and T. Gastaldi (1996). Life testing for multi-component systems with incomplete information on the cause of failure: a study on some inspection strategies. Comput. Stat. Data Anal. 22, 373-393. Guttman, I., D. K. Lin, B. Reiser and J. S. Usher (1995). Dependent masking and system life data analysis: Bayesian inference for two-component systems. Lifetime Data Anal. 1, 87-100. Kalbfleisch, J. D. and R. L. Prentice (1980). The Statistical Analysis ofFailure Time Data. Wiley, New York. Kodell, R. K. and J. J. Chen (1987). Handling cause of death in equivocal cases using the EM algorithm. Commun. Star. A Theory Meth. 16, 2565-2603. van der Laan, M. J. and I. W. McKeague (1998). Efficient estimation from right-censored data when failure indicators are missing at random. Ann. Star. 26(1), 164-182. Lapidus, G., M. Braddock, R. Schwartz, L. Banco and L. Jacobs (1994). Accuracy of fatal motorcycle injury reporting on death certificates. Accident Anal. Prevention 26(4), 535-542. Lawless, J. F. (1982). Statistical Models and Methodsfor Lifetime Data. Wiley, New York. Lin, D. K. J. and F. M. Guess (1994). System life data analysis with dependent partial knowledge on the exact cause of system failure. Microelectron. Reliab. 34, 535-544. Lo, S.-H. (1991). Estimating a survival function with incomplete canse-of-death data. J. Multivariate Anal. 39(2), 17-235. Meeker, W. Q. and L. A. Escobar (1998). Statistical Methodsfor Reliability Data. Wiley, New York. Miyakawa, M. (1984). AnaIyses of incomplete data in competing risk model. IEEE Trans. Reliab. 33, 293~96. Mukhopadhyay, C. and A. P. Basu (1997). Bayesian analysis of incomplete time and cause of failure data. J. Star. Plann. Infer. 59, 79-100. Reiser, B., B. J. Flehinger and A. R. Conn (1996). Estimating component defect probability from masked system success/failure data. IEEE Trans. Reliab. 45, 238~43. Reiser, B., I. Guttman, D. K. J. Lin, J. S. Usher and F. M. Guess (1995). Bayesian inference for masked system life time data. Appl. Stat. 44, 7940. Sun, Y. and R. C. Tiwari (1997). Comparing cumulative incidence functions of a competing risks model. IEEE Trans. Reliab. 46, 247-253. Tsiatis, A. (1975). A nonidentifiability aspect of the problem of competing risks. Proe. Natl. Acad. Sci. 72, 20-22. Usher, J. S. and F. M. Guess (1989). An iterative approach for estimating component reliability from masked system life data. Qual. Reliab. Eng. Int. 5, 257-261. Usher, J. S. and T. J. Hodgson (1988). Maximum likelihood estimation of component reliability using masked system life test data. IEEE Trans. Reliab. 37, 550-555. Usher, J. S. (1996). Weibull component reliability - predictiou in the presence of masked data. IEEE Trans. Reliab. 45, 229~32. Yip, P. and K. F. Lam (1992). A class of non-parametric tests for the equality of failure rates in a competing risks model. Commun. Stat. A - Theory Meth. 21(9), 2541-2556.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

| C] l

J

Analysis of Masked Failure Data under Competing Risks

Ananda Sen, Sanjib Basu and Mousumi Banerjee

Consider a system with K components which can be viewed as either multiple failure modes or different risk factors acting on the system. A system failure occurs at the earliest onset of any one of these risk factors. Under this "competing risks" framework, when the exact cause of the system failure can be identified, a detailed Failure Mode Effect Analysis (FMEA) can be carried out in a routine manner. In reliability applications, one however frequently encounters system life-data, where the cause of failure cannot be exactly identified, but can only be narrowed down to a subset of the K potential failure modes. In statistical literature, such data are termed as maskedfailure data. Masking is often the manifestation of an attempt to expedite the process of repair by replacing the entire subset of components responsible for failure instead of carrying out a second-stage resolution or "autopsy" that can be prohibitively expensive and time consuming. In this article, we present an extensive review of the statistical models and methodologies used in analyzing such data. The interrelationships between various models and the assumptions that underlie their development are discussed in detail. A special emphasis is given to the Bayesian methodologies used in this context, that have proved to be extremely useful and promising thus far. Although the focus of the article is in reliability application, we shall occasionally draw analogies from cancer clinical trials, another major area where masked data arise quite frequently.

1. Introduction

Analyzing failure data of a multi-component system has received considerable attention over the years in statistical literature. In an industrial application, the components typically represent multiple modes (causes) of failure for complex units or individual subsystems that comprise an entire system. Occurrence of a system failure is caused by the earliest onset of any of the failure modes. The framework is that of a system with components connected in a series, more commonly referred to as a "competing risks" scenario. In the reliability community, a great deal of research has been pursued under the general heading of Failure Mode 523

524

A. Sen, S. Basu and M. Banerjee

Effect Analysis (FMEA) when the exact causes of failure are identified. In engi-

neering applications, one, however, frequently encounters data where the cause-offailure is not completely known. Reiser et al. (1995) provide an example of an IBM PS2 system, the failures of which were attributed to the failure of the motherboard, disk-drives, or malfunction of the power supply. In three of the eight failures observed, the cause of failure could not be narrowed down to a single source. In the statistical literature, such data are termed as m a s k e d failure data. Masking is often the manifestation of an attempt to expedite the process of repair by replacing the entire subset of components responsible for failure instead of further investigation towards identifying the specific component which is the culprit. Failure data with masked cause-of-failure are also quite common in medical applications. The sources of failure (death) in a medical context typically refer to various potential risk factors for a patient observed in a clinical study. Available applications include patients in a heart transplant study (Greenhouse and Wolfe, 1984) and breast cancer patients (Cummings et al., 1986) observed longitudinally over several years. Whether one is confronted with masked failure data arising from an engineering or a medical application, the objectives of the investigation typically revolve around very similar issues. Given a collection of potential risk factors for a failure, a quantity of interest to both physicians as well as engineers is the estimation of the likelihood of a cause-specific failure. Reiser et al. (1996) call it the diagnostic probability. If various prognostic factors are observed during the study, such as in biomedical applications, interest may also 1le in the assessment of their effects on the overall survival. The sections in this article are organized as follows. Section 2 provides a detailed overview of the parametric models and methods applied in the context of analyzing masked system failure data. Both frequentist and Bayesian models are discussed at great length and illustrative examples are presented to supplement the theoretical findings. Section 3 reviews the relatively less studied non-parametric models investigated thus far. Finally in Section 4, we outline various areas that have either been unexplored or have been studied to a limited extent. Also in this section, we indicate certain related investigations that have been carried out in the context of a clinical trial.

2. Parametric inference procedures In this section, we provide a review of the parametric models and associated inference methods explored in the context of failure data with partially masked cause-of-failure. One of the earlier studies towards this end has been carried out by Miyakawa (1984). He assumes a simplified scenario of two failure modes and the data comprise failure times for which the failure-causing mode may or may not have been identified. Specifically, let (X/1,X/2), i = 1 , . . ,n denote the component lifetimes of the n systems under study. Let m(_< n) denote the number of systems for which the failure mode has been identified out of which r failures have

Analysis of masked failure data under competing risks

525

been attributed to mode 1 (the mode of interest, say). Under the assumption that (X/1,X/2) are i.i.d., where X/1 m exp(21) and )(/2 ~ exp(22) independently of each other, Miyakawa (1984) derives B1 - -

r

n

m-- ~i~=iTi '

^, _

F

"~1

m-

n-

1

~i=l -T,

to be the maximum likelihood and the uniform minimum variance unbiased estimators of 21, respectively, where T~s denote the observed system lifetimes. While simple, Miyakawa's (1984) approach has two shortcomings, namely, (i) ignoring censored data and (ii) assuming m to be fixed, although treating r to be random. Guess et al. (1991) extend Miyakawa's (1984) model and present a more general framework that includes (a) random right censored data, (b) (K _> 2) failure modes, and (c) randomness in the number of failed units. The general scenario of Guess et al. (1991) is described next. Consider an ensemble ofn independent systems each comprising K >_ 2 components (failure modes) connected in series. Let Xi; be the lifetime associated with the jth failure mode in the ith system and T/be the random lifetime for the/th system. Assume further that X~;s are independent, and for each fixed j, (Xlj,X2;,.. ,)(nj) represents a random sample from a continuous distribution with the survival function and the probability density function denoted by B, j), respectively. The system failure time T~denotes the time of earliest onset of one of the K potential failure modes. Under partial masking, a setting is considered where the cause of failure for the/th system may or may not be exactly identified, but can possibly be narrowed down to a minimum random subset (MRS) Si = { i l , • • • , i l } of the K risk factors. If the exact cause-of-failure is known to be t h e / t h mode, then Si = {l}. On the other extreme, complete absence of knowledge on the cause-offailure leads to Si = { 1 , 2 , . . ,K}. In addition to the failed units, the data may consist of systems that have not failed until a (possibly random) censoring time. With C and D denoting the set of indices of the censored and failed units, respectively, the full likelihood for the n systems can be written as

L

H I Z ~ J ' ( t i ) H i~~(ti)P(Mi : Si]~i : ti,]i : J ) l ) ieD [. jES~ gCj ×

ti

.

(2.1)

Here Mi denotes the random variable for which Si is the observed value, and Ii is the index for the component causing the failure. Note that as long as (a) censoring time is independent of failure time and (b) censoring distribution does not share any life distribution parameters, it suffices to work with (2.1) which does not involve any explicit expression for the censoring distribution. The quantity Q(i, j) = P ( M i = SiITi = ti, [i = j) in (2.1) is typically difficult to handle. Under the assumptions that for each i ~ D, and for fixed j E Si, (C1) Q(i,j) does not depend on the life distribution parameters and (C2) Q(i,j) = Q(i,f) for f E Si,

526

A. Sen, S. Basu and M. Banerjee

working with (2.1) is equivalent to using the reduced likelihood

La = H

~ f j ( t i ) HFe(ti)

ieD I.je&

II

£7~j

H Fj(ti)

iEC [,j=l

•

(2.2)

)

While (C1) is analogous to a censoring distribution being free of life distribution parameters, (C2) can be viewed as a "symmetry" assumption that entails an equal chance of observing the same masking subset of risk factors irrespective of the true cause (within the masking subset) of failure. Guess et al. (1991) and Lin and Guess (1994) cite industrial examples where it is reasonable to assume (C1) and (C2) hold. Guttman et al. (1995), Reiser et al. (1996), and Flehinger et al. (1996, 1998) provide some relaxation of condition (C2). Denoting hj(t) = fj(t)/~(t) and Hj(t) = -logffj(t) to be the hazard function and the cumulative hazard function, respectively, for the lifetime distribution of the jth component, one can alternatively write the expression in (2.2) as

L=U

Zhj(ti)

iED jESi

exp -

~-~Hj(ti)

i=1 j=l

(2.3)

Expression (2.3) has the interesting feature that only the part involving hjs depends on the MRS Sis and hence on the underlying masked cause of failures. The part involving Hjs, on the other hand, is independent of any masking mechanism.

2.1. Frequentist work After Miyakawa's (1984) initial investigation with two-component systems, orte of the earliest frequentist study with exponentially distributed component lifetime is documented by Usher and Hodgson (1988) even before they have presented the general formulation in their 1991 article. They take J) to be the pdf of an Exp(2j) random variable and provide Picard's iterative algorithm for obtaining the maximum likelihood estimators for a three-component system. Under the same setup, an alternative algorithm is proposed by Lin, Usher and Guess (1993) who replace the non-linear system of likelihood equations by a single fourth-order polynomial that can be solved easily by standard softwares. Lin and Guess (1994) relax the condition in (C2) by requiring

Q(i,j) = cQ(i,f),

j,j' E Si ,

(2.4)

where c ¢ 1 is the dependence parameter that is implicitly a function o f f and j. To understand the effect of c on the life distribution parameters, consider the following simple example of a two-component system, where B(t) = )cje-)«t,j = 112, in the absence of any censoring. Since condition (C1) still prevails, it suffices to work with a simplified version of L in (2.1), namely,

L(c)= H (")qe-(21+22)ti)H (~2e-(21+22)ti) I-I {(~l@Ä2c)e-(21+22)ti} si={1} si={2} si={ 1,2}

'

(2.5)

Analysis of maskedfailure data undercompetingrisks

527

where « = P(M,. = {1,2}IT/= ti, Ki = 2)/P(Mi = {1,2}IT/= ti,Ki = 1). L(«) is easily seen to be maximized by

B1 = {--cn2 4. (1 -- «)n12 4- (1 - 2«)nl

÷~/ (cn2-(1-c)n12+nl )24.4(1-c )nln12} /[2T (1-c )]

,

B~2 = {nl 4. (1 -- C)nl24. (2 - c)n2

- \/(c(n2 4. n12) - (n14- n12))2 4- 4Cnlg} / [ 2 T ( 1 - c)] , n where T = ~i=1 ti is the total time on test and nl, n2, n12 are the number of failed systems due to components {1}, {2} and the masked set {1,2}, respectively. Lin and Guess (1994) present several plots of the maximum likelihood estimators (MLEs) as functions of c, where c varies from 0 (extreme dependence) to 1 (independence) for various configurations (nl,n2,nl2). It is apparent that the MLEs behave quite robustly when the units failed due to the masking subset is proportionately fewer compared to the rest. None of the investigations mentioned thus far has explored beyond exponentially distributed components, neither have they made any explicit use of any censored data in their samples. Under the single censoring sampling scheme, Doganaksoy (1991) presents interval estimation of 2j parameters for a three-component system with exponentially distributed component lifetimes. His treatment is based on an application of standard large sample maximum likelihood theory on the reduced likelihood in (2.2). Usher (1996) extends these findings to masked and censored system failure data where the components have (possibly different) Weibull distributed lifetimes. As an alternative to the traditional Newton-Raphson iteration, Usher (1996) presents an iterative algorithm for finding MLEs that is supposed to be numerically efficient and easily programmable.

2.2. Bayesian formulation In the area of system life-data with masked cause-of-failure, alongside the frequentist development, there has also been considerable work from a Bayesian viewpoint. Bayesian formulation provides an integrated framework under which (a) predictive analysis can be carried out in a routine manner, (b) the inference results (albeit numerical) can be based on finite sample sizes, and (c) knowledge of available information, e.g., expert's judgment about the failure pattern of a component, can be incorporated in the current analysis in a standard way. In the context of analyzing masked competing risks data, Bayesian methods may also provide a means for introducing dependence among the component lifetimes.

528

A. Sen, S. Basu and M . Banerjee

Reiser et al. (1995) provide a Bayesian analysis under the general framework of Guess et al. (1991) describe earlier in Section 2. The authors work with the likelihood LR in (2.2) and assume the p d f f j of the lifetime of the j t h component to conform to that of an exponential distribution with parameters 2j, i.e., B ( t ) - )ve k/t, t > 0. As a prior, Reiser et al. (1995) use the non-informative Jeffrey's prior

K rc1(2~,.., 2x) e( H 2 f 1

(2.6)

j=l

and show that for K = 2, the Bayes estimates for 21,22 equal

__ /71 (~/1 @/72-r-/712)

--///2

(/71@/72@/712.)

which are identical to the corresponding MLEs. Here, as before, /71,/72, /712 refer to the number of observed system failures caused by component 1, component 2, and n the masked set { 1,2}, respectively, and T = }-~«=1 ti equals the total time on test. Various other prior choices have been investigated under the exponentiality assumption for component lifetimes. Lin et al. (1996) use a step-function prior for )V of the form ~22(#) = ~j,/cI n d ( # E [ak-1, ak)) , where 0 = a0 < a~ < a2 < . " is a partition of [0, oc), Ind (.) denoting the indicator function, and ~~_lCg,k(ak--ak_~)= 1 satisfying the usual probability constraint. Motivation for such a prior sterns from the need to incorporate an engineer's degree-of-belief about the failure rates of the individual components in the analysis. Lin et al. (1996) provide a numerical example with K = 2 components and various degrees of masking. A third prior choice in this context has been entertained by Mukhopadhyay and Basu (1993). Borrowing idea from the work of Pefia and Gupta (1990), they define the quantities @ = 2 y 2 , j = l , . . . , K - 1 ; 2 = ~ f = 1 2 j and impose a Dirichlet prior

~ZD(Cl,... ,CK-1) OCl ~ cj

j=l

1 --

j=i

Cj

,

~1,

. . ,O~K > 0

on (cl,...,cx-1) independently of 2 and assume 2 to have a Gamma (c~,7) distribution. Consequently, ( 2 , . . , 2x) has a prior of the form:

723('~1''''' "~K)°c Ifl=1 2;J-ll "~~-~°e-27'

21 > 0,..,~K > 0

(2.7)

with c~0 = ~Y-I ~J. Note that a form of Jeffrey's prior different from re1 in (2.6) can be obtained from (2.7) by choosing cq . . . . . ŒK= 1/2, and ~ = y = 0.

Analysis of masked failure data under competing risks

529

Entertaining the only possibilities of either observing the exact cause-of-failure ({Sj} is a singleton) or complete masking ({Sj} = { 1 , 2 , . . , K}) for the jth failed system, Mukhopadhyay and Basu (1993) obtain closed-form Bayesian estimators for 2is. Denoting by T, nj, nD, the total time on test for all failed and censored units, the total number of units failing due to cause j, and the total number of uncensored units, respectively, the posterior of 2j/2 turns out to be a Beta random variable with parameters c~j + nj and ~o - c9 + m - nj, where m = ~ y « nj. Also, a posteriori, 2 has a Gamma (c~+ nD, 7 ÷ T) distribution. Consequently, the posterior mean 2j and the posterior mode 2~ of 4j are ~2j = ( ~ j + n j ) ( c ~ + n D )

,

(ej+nj--

1)(c~+nD--K)

(~0+m)(~+r)' 4)= (LTm~2~~Tr) respectively. Mukhopadhayay and Basu (1993) also obtain closed-form expressions for the posterior dispersion matrix of 4js and provide an analytic expression for the approximate 100(1- ~)% HPD credible set for the parameters. Very little investigation has been carried out in the Bayesian context towards relaxing condition (C2) described earlier in Section 2. For a two-component system with exponentially distributed component lifetimes, Guttman et al. (1995) use (2.4) as the dependence structure and obtain closed-form Bayes estimators of the parameters and system reliability under the prior assumption of (2.6). They provide a sensitivity analysis of the parameter estimators as a function of the dependence parameter c under various configurations of the data. The findings are in harmony with those of Lin and Guess (1994). Additionally, Guttman et al. (1995) present posterior analysis for the case where c is assumed to be a uniform random variable on (0, 1) independently of 21,22. Some of the above-mentioned work has been extended by Mukhopadhyay and Basu (1997) and Basu, Basu and Mukhopadhyay (1999) to the case where the jth component lifetime has a Weibull distribution with pdf j~(t) = Bjfljt3J -1 e x p ( - 4 f ; ) , 4j > 0,/~j > 0. Specific choices including that in (2.7) have been entertained for the 4j parameters, while the prior choice for /~/ is kept somewhat generic. While Mukhopadhyay and Basu (1997) provide an EM algorithm for the generation of posterior mode, Basu et al. (1999) take a more modern recourse in providing a simulation approach based on Markov Chain Monte Carlo (MCMC) methods. Basu et al. (2000) present a more general framework that encompass a large class of distributions including those discussed above and indicate a way to handle more complex sampling schemes. In the next section, we present the basic idea behind the approach taken by Basu et al. (2000).

2.3. A general Bayesian f r a m e w o r k

Towards a generalization of the aforementioned discussions, we first assume that the (possibly transformed) failure time for component j follows a

530

A. Sen, S. Basu and M. Banerjee

distribution with probability density function conforming to a location-scale structure

B(t, ~» ~~) = ~j~°(~j(t - ~ j ) ) ,

(2.8)

where fy0 is a pdf free of parameters. The statistical methodologies and inference procedures are quite rich for location-scale family of distributions. Elegant features of the family include, for example, (a) the particularly simple form (linear function of the parameters) of the percentiles and (b) existence of pivotal quanriffes. The class in (2.8) encompass a large family of distributions traditionally used to model life-data, such as exponential, smallest extreme value (log-Weibull), twoparameter exponential, lognormal, log-logistic, to name but a few. The development presented hefe does not require fj0 to be the same for all components of the system. This allows the flexibility in modeling different components to have different life distributions. Another advantage of entertaining different functional forms for the life distributions of different components is to avoid the danger of possible lack of identifiability of the parameters which is offen a problem with competing-risks modeling (see Basu and Ghosh, 1978, 1980, 1983). In Bayesian analysis, however, non-identifiability is not a formal problem, although it offen leads to concerns such as slow convergence of simulation algorithms. The prior choice in this general framework is pretty open. There are offen partially conjugate families of priors available for certain choice of distributions. On the non-informative side, of course, Jeffrey's prior of the form K 7"C(~, "C) = 7"C(]J1, . . . ,/~K, "el,""", "CK) (X I I "CJ 1 j=l

(2.9)

is well accepted for the general location-scale family of the form in (2.8) (c.f. Box and Tiao, 1973). In practice, one needs to check the validity of a specific model when applied to the given data at hand vis-a-vis its conformity to formal or graphical goodness of fit procedures. In the Bayesian paradigm, one offen settles on a model from a class of competing ones based on its predictive power calculated via Bayes factor. We shall estimate the posterior quantities ofinterest by M C M C samples drawn from a data-augmented posterior. To facilitate this analysis, we augment the observed data with a set of latent variables Il, i c D, where D, as before, denotes the set of nncensored observations. The latent Ii is a categorical variable identifying the cause of failure for the ith unit, i.e.,/~ = j, if t h e / t h unit failed due to cause j, j o S» We assume that all values of Ii in its support are equally likely a priori, i.e., P(Ij = j ) = 1/{• elements in Si},j E Si, although differing prior probabilities can also be entertained. In the Gibbs sampler implementation of the M C M C approach, one alternately simulates from the full conditional distribution of Ii and those of other parameters. Note that the likelihood and prior contribution can be combined to give the joint density in the hazard form as

Anaßsis of maskedfailure data under competing risks

531

p(t,p,,~,Ii = ji, i E D ) = I I { "cj,hO('cj,(ti - #j))} icD

~H°(zj(ti-#;))

xexp -

i=1 j=l

x r~(g, z)P(Ii = j) ,

(2.10)

where h°(.), H°(-) are, respectively, the baseline hazard and cumulative hazard fnnctions for thejth component lifetime, and t = (tl, • • •, th) is the entire ensemble of failure and censoring times. The full conditional distribution ofli given the rest is:

~;,ho,(~;,(t,- ~j,)) ji E S,. .

~(Ii = j;Idata, Il, z, I_i) =

(2.11)

Here, I_i = (I1,I2,.. ; I i - l , I i - - 1 , . . ,Zn). Note that (2.11) implies that Ii is distributed independently of I_» Interestingly, the result in (2.11) continues to hold eren when the component lifetimes are no longer independent. In that case, however, the hazard function h ° ( z j ( t i - l@) associated with component j is replaced by the conditional hazard function of the jth component evaluated at time ti given that the remaining components survived up to time ti (c.f. Kochar and Proschan, 1991). While no general elegant expression is available for the remaining conditionals, with mild requirements of log-concavity of fj°(.) and 7c(g, z) with respect to/~;, z j, one can achieve log-concavity of the conditionals that greatly facilitates the M C M C simulations. (c.f. Gilks and Wild, 1992).

2.3.1. Illustrative examples The hazard function version in (2.10) of the joint contribution of likelihood and prior is an illuminative expression for demonstrating the examples. For simplicity of exposition, we take in our illustrations the same base distribution for all component lifetimes, i.e., h ° ( . ) = h ° ( . ) and H ° ( . ) = H ° ( . ) , j= 1,2,..,K. 2.3.1.1. Exponential distribution In this case, #9 = 0 for all j = 1 , . . ,K and h°(t) = 1 for all t > 0. Consequently, using (2.11), the full conditional distribution of Ii assumes the simple form 7c(Ii jlrest) = zj/~-~~j'ESl Zj,,j E Si. Using the popular choice of independent Gamma (a,b) priors n(zj) c< z] 1 e x p ( - b @ on ~], Eq. (2.10) simplifies to =

p(t,~,z,I)

oc H z j i e x p

iED

"cjti I - [ r j

i=1 j=l

r a+n;-1 exp - z j = rl].zJ j=l

j=l

b+

ti "=

,

(2.12)

532

A. Sen, S. Basu and M. Banetjee

where nj, as before, equals the number of failures risk j is responsible for and ji is the index of the risk component causing the failure of unit i. It follows immediately from (2.12) that ~(rjlrest) ~ Gamma(a + nj, b + ~-~«~1ti) and thus implementing M C M C steps is exceptionally simple in this case. It', on the other hand, one uses the noninformative prior in (2.9) in this case, considerations similar to the previous paragraph yield 7r(rjlrest ) Gamma(nj, ~i'-1 tl). Reiser et al. (1995) work with this prior choice and demonstrate that the posterior calculations are messy eren for two or three components. The simulation based approach based on the M C M C consideration, by contrast is extremely simple. Further it is evident that in this case the full conditional of vj is improper unless nj _> 1. So, for the propriety of the posterior, we need to have every risk factor causing (either in a masked or unmasked way) at least one failure. 2.1.3.2. Smallest extreme value (SEV) distribution Logarithm of Weibull failure times are distributed as SEV with the base pdf f ° ( t ) = exp(t - et), - o c < t < oc. Consequently, h°(t) = H°(t) = et and the full conditional distribution o f / , is given as "cjexp('cj(ti -- #j)) 7c(Ii = j[rest) = ~j'~si zj, exp(zj,( ti - #j,) ) '

j E Si .

(2.13)

We first consider a standard prior choice. Specifically, ~ ( - # j l z j ) ~ log Gamma(v0, 70) and ~j ~ Gamma(a, b). Consequently, ;z(#j, zj) o~ z] exp [-~j(Voßj + b) - 6o exp(-#jzj) l •

~~~

(2.14)

We further consider the entire collection {(#1, z l ) , . . , (#K, rK)} to be a random sample from the distribution in (2.14). Upon some algebraic manipulations, the conditionals of #j and zj are obtained as: -#jlrest ~ z)-1 log Gamma v0 + nj, 70 +

exp(rjti)

(2.15)

and ~(zj]rest) 0 ( 5

exp i:I~-j - exp(-#jzj)

70 +

exp(zjti)

,

(2.16)

where nj is as before. The M C M C sample generations cycle between (2.13), (2.15) and (2.16). While generation from (2.13) and (2.15) are trivial, it is evident that (2.16) does not conform to the pdf of a standard distribution. It is, however, log-concave in rj and thus the Adaptive Rejection Sampling (ARS) algorithm of Gilks and Wild (1992) can be employed to sample from (2.16).

533

Analysis of masked failure data under competing risks

With the improper prior choice of (2.9), the conditionals for &, rj are extremely similar to (2.15) and (2.16). In fact, the conditional for yj is the same as (2.15) with both v0 and 7o replaced by zero. Much like the exponential case, this entails that the conditional of #j is proper as long as nj > 1. The expression for the full conditional for rj is

~/~,lrest/~ ~~ expl ~,~~,~, Z'~ L

k

i:ii=j

- exp(-#jvj)

J exp

ti

,

which is seen to be log-concave in zj provided nj >_ 1, the same condition needed for propriety of 7c(#jtrest ). The implementation of the general methodology presented in Section 2.3 is in no way restricted to formulations which yield tractable results. The examples chosen here serve merely as illustrations. The strength of the simulation approach can be efficiently exploited to expand the application regime to include, for example, mixture densities. This allows a more flexible choice of distributions for component lifetimes. For instance, one may choose the base distribution fj0 for the (possibly transformed) jth component lifetime to be t with vj degrees of freedom conforming to J)(xj) (x [1 + vTl'cj(x j - [2j)21-(vy+l)2. Although t-density is not log-concave, it is a scale mixture of normal densities which are log-concave. The M C M C approach invokes a new step involving generation of additional latent variables in order to handle such mixture structure. We note here that the proposed method of analysis results in the generation of a large number of latent variables and one should cautiously monitor the convergence of the resulting sampler. 2.3.2. Interval censoring Interval censoring occurs when time of failure is only known to fall within an interval. This arises quite naturally in longitudinal clinical trials, where subjects are monitored at periodic intervals. In industrial experiments, the premise is not only practical, but often time it is also the only feasible sampling scheine available to the experimenter. While multiple inspection at prespecified times per item is quite common, there are various examples of quantal response study in reliability literature, where each unit is inspected only once and its status (failed or running) recorded. Once again, we consider a life-testing experiment with n units and K competing risks acting on each unit. The observations are divided into three groups: (i) observation i ~ D if its time to failure is observed to be ti with cause of failure narrowed down to MRS Si, (ii) i E C if time to failure is right censored at time ti; and (iii) i E E if time to failure ~ is interval-censored in the interval [ai, bil with cause of failure narrowed down to MRS &. Of course, interval censoring is the most general kind of censoring in the sense that it includes the cases of (a) right

A. Sen, S. Basu and M. Banerjee

534

censoring (bi = oo); (b) left censoring (ai = --oo); (C) exact failure times known (ai = bi). We, however, separate the groups in our treatment since it helps understanding the distinct roles of each of D, C, and E. For the first two groups, i E D and i E C, the likelihood contributions, are, respectively, Li = ~j~si{fj(ti) I~e~j~(ti)} and Li = [I~=1 ~(ti), as before. On the other hand, we have, for i c E, Li

Z jcsi

I,

J)(t) H Æ l ( t ) dt . ~¢j )

(2.17)

Note that, with the location-scale structure discussed in this section, B(t) = zjfj° ( z j ( t - @)). The evaluation of Li, in turn, involves evaluating the integral in (2.17) which is, in general, analytically intractable. Basu et al. (2000) propose a simulation-based approach that exploits the structure in (2.17) and present a viable alternative to direct evaluation of Li in the general location-scale framework.

3. Non-parametric methodologies In contrast to the parametric models and methods investigated for analyzing masked system failure data, the non-parametric approaches have been explored to a much lesser extent, especially for reliability applications. Much of the initial investigation in the non-parametric framework concern bioassays for animal carcinogenicity. Masking often arises due to the disagreement among veterinary pathologists as to the reliability of cause of death information. In the basic framework, each animal's life history is modeled as a four-state stochastic process with an initial alive and tumor-free state, a transient alive and tumor-bearing state and two absorbing states (death with or without tumor). Dinse (1982), Kodell and Chen (1987) focus on non-parametric estimation of survival functions associated with the cause (tumor) of interest as well as the other cause(s) lumped in a single group. Racine-Poon and Hoel (1984) formally account for the uncertainty in diagnosing the exact cause of death by assigning a score on the diagnostic probability beilag correct. The score could be arrived at by a consensus among the pathologists or from past data. Under this setup, Racine-Poon and Hoel (1984) develop generalized Kaplan Meier estimators for the cause-specific and overall survival functions. Dinse (1986) obtains non-parametric MLEs of various quantities such as cause-specific hazard rates, disease prevalence rates, and expected proportion of deaths due to the disease among the entire collection of deaths where the disease was present. Dinse's (1986) treatment, however, involves the situation where deaths may occur due to a cause other than the disease of interest, even when the disease is present; a framework harder to conceive in the reliability framework. Although most of the development is illustrated for two competing risks, where one risk represents the key cause and the other combines everything else, Dinse (1986) indicates a generalization to the multi-component case and describes how EM algorithm can be used to generate the MLEs of the

Analysis of maskedfailure data under competing risks

535

quantities of interest in this case. Under the same setup, Dinse (1988) further presents non-parametric estimation of tumor incidence rate which corresponds to the rate of tumor onset among live tumor-free animals. Recently, Ebrahimi (1996) has noted the effect of misdiagnosing the cause of death under a competing-risks framework. In the reliability framework discussed in this article, Schäbe (1994) exploits the strong theoretical foundation of counting process methodologies to derive nonparametric estimators ofFj based on masked system life data. Let pij(t) denote the probability that at time t, failure of component i will lead immediately to a succeeding failure of component j. It is assumed that Pij(t) = Pji(t) = Pij ,

an assumption needed to prevent non-identifiability and is implicit in the symmetry condition (C2) stated in Section 2. Let Hi, as before, denote the cumulative hazard function for t h e / t h component. Weakly consistent estimators ofp~j and H/ are provided by

~,s

-

,q,(t)

~~s

nij + ni _}_ nj

= H * x("1

'

~ , x ~:t,«< v'"~°~' - , 2d/(tk) - Pij) --'t-7~~'~ ,

(3.1)

(3.2)

jTAi

where di(t») = number of units that fail due to risk i alone at time th and n . ( & ) = number of unfailed and uncensored systems just prior to tk. Here 0 < tl < t2 < . . . denote an enumeration of the distinct system failure times. Eqs. (3.1) and (3.2) are friendlier and more easily interpretable versions of Eqs. (6) and (7) of Schäbe (1994). A non-iterative estimator F/ of F/ can be constructed from the relation =

1]

( 1 - d~q~(tk)) ,

k:tk
where A/t,. denotes the jump heights o f / t / . The processes nl/2(Fi - F~) converge jointly to a multi-variate mean zero Gaussian process. While closed-form, the estimator F/ is not efficient since it does not use the complete information contained in the masking subsets of cardinality greater than 2. A more efficient, iterative estimator of Fi can be found by maximizing the full non-parametric likelihood function. Some recent advances have been made in the research of masked failure data in the situations where a second-stage resolution or "autopsy" can be entertained on a subset of failed units with masked cause-of-failure. The advantage of this approach lies in that it does not depend on any arbitrary symmetry assumption such as (C2). Statistical inference for this two-stage and masked competing risks framework has been developed in the context of one-shot experiments (Flehinger et al., 1996; Reiser et al., 1996) as well as continuous time-to-failure data

536

A. Sen, S. Basu and M. Banerjee

(Flehinger et al., 1998). Readers interested in this topic are referred to the comprehensive review article by Flehinger et al. (2001) in this volume.

4. Extensions and other directions

In this article, we have provided a survey of the models and methodologies employed in analyzing system failure data with masked cause-of-failure. The context of application is industrial life-testing experiments, although the overlap with survival analysis in medical applications is briefly alluded to. In this concluding section of the article, we identify five important areas in the masked system lifedata analysis which have either been little or incompletely explored.

4.1. Dependent causes In almost all treatments of masked system-failure data, an implicit assumption is the independence of component lifetimes or failure mechanisms of different modes. While this assumption simplifies the mathematical formulation to a great extent, often times it leads to quite unrealistic and oversimplified approximation of the truth. A strong case in point is a study on prostate cancer patients, where the causes of death are classified as (a) prostate cancer specific, (b) other cancer and (c) other non-cancer related. While the independence between cause (c) and the other two causes can be argued for, the medical community will probably unanimously object to the assumption of independence of causes (a) and (b), since cancer in another organ, which ultimately caused death for a specific patient, could very well have originated from the prostate. Similar examples with dependent failure modes are also common in reliability applications. This clearly constitutes an open area of research in the analysis of masked failure data. One obvious difficulty lies in choosing an appropriate dependence structure. Incorporating dependence through random "frailty"-type effects may be a promising avenue in this context.

4.2. Symmetry assumption Much of the likelihood based approach for masked failure data is built around some sort of symmetry assumption relating to the conditional probability of observing a random masking set of components given that the failure occurred due to a specific component. The relaxation of the symmetry condition used in the literature is not satisfactory either. The main difficulty with this quantity is that this is not directly estimable from the data, and consequently any assumption made on it is as arbitrary as another. One situation where this problem can be resolved in a meaningful way is when an "autopsy" can be performed on at least a subset of failed systems with a masked group of "responsible" components. Dealing with this masking probability in the absence of a second-stage resolution remains an issue of concern in the analysis of masked failure data.

Analysis of masked failure data under competing risks

537

4.3. Incorporating covariates Possibly the most obvious next step generalization to the treatment presented in this article is to incorporate covariate effects through regressors. In the prostate cancer study, age, face and prostate specific antigen (PSA) level are important covariates for overall survival. Externally controllable factors such as temperature, pressure are typical examples of covariates in an industrial experiment. Not only may these covariates affect quantities of importance such as failure rate or reliability, more importantly, it is conceivable that the extent and nature of their effects vary across different components of the system. In deciding on a model, the choice clearly lies between the various parameters (e.g., mean, variance, hazard) on which regression is to be performed. In the location-scale structure presented in Section 2.3, regression on the location-parameter is a common choice that generalizes the usual single-component life regression model. Note that multi-sample comparisons can also be formulated by choosing appropriate indicator covariates. In a survival analysis context, Goetghebeur and Ryan (1990, 1995) explore a semiparametric regression model for a two-component system. Their modeling is built on proportional hazards structure with the following assumptions: A1. The cause-specific hazards for the two components assume the form

hl ( t; X, Z) = hl ( t) exp{pX(t)}, h2(t; Z , X ) = h2(t) exp{q~Z(t)} , where hl, h2 are the respective baseline hazards, X, Z are the vector of possibly time-varying covariates that may or may not overlap, and p, q5 are the corresponding unknown regression coefficients. In the model, X, Z represent the history up to time t of the respective covariates. A2. The baseline hazards, otherwise unspecified, assume the proportionality structure h:(t) = exp(~)hl(t). A3. Given a unit with a specified failure time, the status of the failure (known or unknown) is independent of the failure component indicator. Although Goetghebeur and Ryan (1990, 1995) do not mention it, this assumption is precisely the regression framework extension of the symmetry condition in (C2). The parameters p, q~ and ~ are estimated by maximizing certain partial likelihood. The resulting estimators, suitably standardized, can be shown to follow an asymptotic multivariate normal distribution. The derivation is heavily based on the martingale theory associated with multi-variate counting processes. Goetghebeur and Ryan (1995) also construct score tests for the global hypothesis of no regression effect for a specific component or failure mode (e.g., H0:q5 = 0). Anderson et al. (1996) illustrate these inference methods with a group of patients undergoing a clinical trial for Hodgkin's disease.

4.4. Other sampling scenario In the present context with continuous time-to-failure data, the investigation thus far has been confined to multiple right censoring scheine. In reliability applica-

538

A. Sen, S. Basu and M. Banerjee

tions, another commonly used scheine follows inspection at prespecified time points, giving rise to interval censored data. This typically imposes an additional level of complexity for likelihood-based analysis. A simulation-based approach in handling interval censoring is indicated in Basu et al. (2000). The nuances of the exact analysis in specific instances of parametric assumption is yet unexplored. 4.5. G o o d n e s s - o f - f i t p r o c e d u r e s

The difficulty of applying formal goodness of fit procedures in assessing the adequacy of the component lifetimes orten lies in the paucity of data pertaining to a specific failure mode. One taust thus rely on an overall likelihood ratio (frequentist approach) or Bayes factor (Bayesian approach) criterion in determining between a set of competing models. Model selection issue under a generic setup has not been discussed formally in the masking literature. This, however, is a very important issue as the prediction of diagnostic probabilities or overall survival probability are heavily dependent on the choice of the model. Statistical analysis of failure data with partially masked cause-of-failure, while complex, deals with an important area that has applications in the engineering industry as well as the medical sciences. In the competing-risks framework, rauch research has been undertaken either under the scenario where the failure indicators are completely known or where the system is treated as a blackbox and one operates under complete lack of knowledge about the specific component defect. The scenario presented in this article is a compromise between the two when partial information is available on the failure of certain components. The main objective of the statistical analysis then is to use this information in an efficient manner for diagnostic and prediction purposes. With the recent explosion of computing power, it is conceivable that quite complex models and methodologies can be entertained to analyze masked failure data along similar veins of our presentation of an omnibus Bayesian framework in Section 2.3.

Acknowledgement The authors are most grateful to Lynette Folken of Oakland University for her diligent and expert manuscript preparation service.

References Andersen, J., E. Goetghebeur and L. Ryan (1996). Analysis of survival data under competing risks with missingcause of death information: application and implications for study design. In L(fetime Data." Models in Reliability and Survival Analysis, 13-19 (Eds. N. P. Jewell, A. C. Kimber, M. T. Lee and G. A. Whitmore). Basu, A. P. and J. K. Ghosh (1983). Identifiability results for k-out-ofp systems. Comm. Statist. Theor. Meth. 12, 199-205. Basu, A. P. and J. K. Ghosh (1980). Identifiabilityof distributions mlder competing risks and complementary risks models. Comm. Statist. - Theor. Meth. 9, 1515-1525.

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Basu, A. P. and J. K. Ghosh (1978). Identifiability of the multinormal distribution under competing risks models. J. Multivariate Analy. 8, 413~429. Basu, S., A. P. Basu and C. Mukhopadhyay (1999). Bayesian analysis of masked system failure data using non-identical Weibull models. J. Stat. Plan. Inference 78, 255-275. Basu S., A. Sen and M. Banerjee (2000). Bayesian analysis of competing risks with partially masked cause-of-failure. (Submitted). Box, G. E. P. and G. C. Tiao (1973). Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading, MA. Cummings, F. J., R. Gray, T. E. Davis, D. C. Tormey, J. E. Harris, G. G. Falkson and J. Arsenequ (1986). Tamoxifen versus placebo: double blind adjuvant trial in elderly woman with stage II breast cancer. Nat. Cancer Inst. Monographs 1, 119-123. Dinse, G. E. (1982). Nonparametric estimates for partially-complete time and type of failure data. Biometrics 38, 417-431. Dinse, G. E. (1986). Nonparametric prevalence and mortality estimators for animal experiments with Incomplete cause-of-death data. J. Am. Statist. Assoc. 81, 328-336. Dinse, G. E. (1988). Estimating tumor incidence rates in animal carcinogenicity experiments. Biometrics 44, 405~415. Doganaksoy, N. (1991). Interval estimation from censored and masked system-failure data. IEEE Trans. Reliab. R-40, 280-285. Ebrahimi, N. (1996). The effects of misclassification of the actual canse of death in competing risks analysis. Statist. Med. 15, 1557 1566. Flehinger, B. J., B. Reiser and E. Yaschin (1996). Inference about defects in the presence of masking. Technornetrics 38, 247-255. Flehinger, B. J., B. Reiser and E. Yaschin (1998). Survival with competing risks and masked causes of failures. Biometrika 85, 151 164. Flehinger, B. J., B. Reiser and E. Yaschin (2001). Statistical analysis for masked data. In Handbook of Statistics, Advanees in Reliability, Vol. 20, pp. 499 522 (Eds. N. Balakrishnan and C. R. Rao). Gilks, W. R. and P. Wild (1992). Adaptive rejection sampling for Gibbs sampling. Appl. Stat. 41, 337 348. Goetghebeur, E. and L. Ryan (1995). Analysis of competing risks survival data when some failure types are missing. Biometrika 82, 821-833. Goetghebeur, E. and L. Ryan (1990). A modified logrank test for competing risks with missing failure type. Biometrika 77, 207-211. Greenhouse, J. B. and R. A. Wolfe (1984). A competing risks derivation of a mixture model for the analysis of survival data. Comm. Stat. - Theor. Meth. 13, 3133-3154. Guess, F. M., J. S. Usher and T. J. Hodgson (1991). Estimating system and component reliabilities under partial information on cause of failure. J. Stat. Plan. Inference 29, 75 85. Guttman, I., D. K. Lin, B. Reiser and J. S. Usher (1995). Dependent masking and system life data analysis: Bayesian inference for two-component systems. Lifetime Data Anal. 1, 87-100. Kochar, S. C. and F. Proschan (1991). Independence of time and cause of failure in the multiple dependent competing risks model. Stat. Sin. 1,295-299. Kodell, R. J. and J. J. Chen (1987). Handling cause of death in equivocal cases using the EM algorithm. Comm. Statist. Theor. Meth. 16, 2565-2585. Lin, D., J. Usher and F. Guess (1996). Bayes estimation of component-reliability from masked systemlife data. IEEE Trans. Reliab. R-45, 233-237. Lin, D. K. J. and F. M. Guess (1994). System life data analysis with dependent partial knowledge on the exact cause of system failure. Microelectron. Reliab. 34, 535-544. Lin D., J. Usher and F. Guess (1993). Exact maximum likeiihood estimation using masked system data. IEEE Trans. Reliab. R-42, 631 635. Miyakawa, M. (1984). Analysis of incomplete data in competing risks model. IEEE Trans. Reliab. R-33, 293-296. Mukhopadhyay, C. and A. P. Basu (1997). Bayesian analysis of incomplete time and cause of failure data. J. Stat. Plan. Inference 59, 79 100.

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Mukhopadhyay, C. and A. P. Basu (1993). Bayesian analysis of competing risks: k independent exponentials. Technical Report #516, Department of Statistics, The Ohio Stare University. Pefia, E. A. and A. K. Gupta (1990). Bayes estimation for Marshall-Olkin exponential distribution. J. Roy. Statist. Soc. Ser. B 52, 379-389. Racine-Poon, A. H. and D. G. Hoel (1984). Nonparametric estimation of survival function when cause of death is uncertain. Biometrics 411, 1151-1158. Reiser, B., I. Guttman, D. K. J. Lin, F. M. Guess and J. S. Usher (1995). Bayesian inference for masked system lifetime data. Appl. Statist. 44, 79-90. Reiser, B., B. Flehinger and A. Conn (1996). Estimating component-defect probabiiity from masked system success/failure data. IEEE Trans. Reliab. R-45, 238-243. Schäbe, H. (1994). Nonparametric estimation of component lifetime based on masked system life test data. J. Roy. Statist. Soc. Ser. B 56, 251-259. Usher, J. S. (1996). Weibull component reliability-prediction in the presence of masked data. IEEE Trans. Reliab. R-45, 229-232. Usher, J. S. and T. J. Hodgson (1988). Maximum likelihood analysis of component reliability using masked system life-test data. IEEE Trans. Reliab. R-37, 550 555.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

'~(~

Warranty and Reliability D. N. P. Murthy and W. R. Blischke

For products sold with warranty, the cost of servicing the warranty depends on the reliability of the product. The paper deals with this and various related topics that link warranty and reliability. We focus on mathematical and statistical modeling and analysis for the study of these topics and the statistical analysis of relevant data from many sources, including test data on parts, components, and systems, and warranty field data.

1. Introduction Warranty has been a part of trade for over 4000 years. L o o m b a 1 (H, Chapter 2) gives a historical perspective on warranty in different civilizations over this period. The modern notion of warranty is a post-industrial concept. A warranty is a seller's assurance to a buyer that a product or service is or shall be as represented. It may be considered to be a contractual agreement between buyer and seller (or manufacturer) which is entered into upon sale of the product. The second half of twentieth century has seen dramatic changes in the role and importance of warranty in relation to product sales and services. The reasons for these are many. A detailed discussion is given in W, Chapter 1. The importance of warranty will become more significant in the new millennium as virtually all consumer products and nearly all industrial and commercial products are sold with warranty. There are many issues related to warranty. These include selection of a warranty p01icy, warranty cost and its relationship to produce reliability, management issues, many related areas, and the interactions between these. In the remainder of this section, we discuss some of the important aspects of warranty and reliability. 1Three sources that will be referred to a number of times are Blischke and Murthy (1994), Warranty Cost Analysis, Blischke and Murthy (1996), Product Warranty Handbook, and Blischke and Murthy (2000), Reliability: Modeling, Prediction and Optimization. These will be referred to hereafter as W, H, and R, respectively. 541

542

D. N. P. Murthy and W. R. Blischke

1.1. Role o f warranty

Warranties are an integral part of nearly all consumer and commercial and many government transactions that involve product purchases. In such transactions, warranties serve a somewhat different purpose for buyer and seller. 1.1.1. Buyer's point o f view From the buyer's point of view, the main role of a warranty in these transactions is protectional; it provides a means of redress if the item, when properly used, fails to perform as intended under normal use, or as specified by the seller. Specifically, the warranty assures the buyer that a faulty item will either be repaired or replaced at no cost or at reduced cost. A second role is informational. Many buyers infer that a product with a relatively long warranty period is a more reliable and long-lasting product than one with a shorter warranty period. 1.1.2. Seller's point of view One of the main roles of warranty from the seller's point of view is also protectional. Warranty terms may and often do specify the use and conditions of use for which the product is intended and provide for limited coverage or no coverage at all in the event of misuse of the product. The seller may be provided further protection by specification of requirements for care and maintenance of the product. In addition, the seller specifies the length and other terms of the warranty. A second important purpose of warranties for the seller is promotional. Since buyers often infer a more reliable product when a long warranty is offered, this has been used as an effective advertising tool. This is often particularly important when marketing new and innovative products, which may be viewed with a degree of uncertainty by many potential consumers. In addition, warranty has become an instrument, similar to product performance and price, used in competition with other manufacturers in the marketplace. 1.1.3. Warranty in government contracting In simple transactions involving consumer or commercial goods, a government agency may be dealt with in basically the same way as any other customer, obtaining the standard product warranty for the purchased item. Often, however, the government, as a large entity wielding substantial power as well as a very large consumer, will be dealt with considerably differently, with warranty terms negotiated at the time of purchase rather than specified unilaterally by the seller. The role of warranty in these transactions is usually primarily protectional on the part of both parties. In some instances, particularly in the procurement of complex military equipment, warranties of a certain type play a very different and important role, that of incentivizing the seller to increase the reliability of the items after they are put into service. This is accomplished by requiring that the contractor service the

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items in the field and make design changes as failures are observed and analyzed. The incentive is an increased fee paid the contractor if it can be demonstrated that the reliability of the item has, in fact, been increased. Warranties of this type are called reliability improvement warranties (RIW).

1.2. Warranty issues 1.2.1. Manufacturer's perspective It is apparent that different warranty issues taust be addressed by the manufacturer and the buyer. From the manufacturer's perspective, some of the issues are: (1) What warranty policies (terms, length, and so forth) would be appropriate for the product? (2) What are the costs associated with the various policies? (3) How are these costs related to product reliability? (4) How do these costs change as a function of the warranty parameters (length, amount of rebate, repair vs. replace options, etc.)? (5) How does orte decide on the optimal warranty choice when multiple business decisions are involved? (6) What is the optimal strategy for servicing warranty? (7) What data (historic, test, field, claims, etc.) are needed for warranty management and how are these obtained? (8) What is the impact of these decisions with regard to product design? (9) What is the impact with regard to process design?

1.2.2. Buyer's perspective Similarly, issues from the buyer's perspective are: (1) The buyer, of course, is also concerned about the cost of warranty. How much would this product cost if it were sold without warranty? Is the additional cost worth it? (2) What options (e.g., length of warranty) are available? (3) What warranties are various vendors offering and how do they compare? (4) What is the reliability of this product and how does it compare with competing products? (5) What is my assessment of the life cycle cost of this product? (6) Which terms, if any, are negotiable? (7) Is an extended warranty available, and at what cost? Is it worth this additional cost?

1.2.3. Analysis of warranties Addressing these issues is a difficult problem for both manufacturer and buyer. In addition, there are many other aspects of warranty and many disciplines are involved in the analysis of these issues. Warranties have been analysed from many viewpoints, as indicated below. The literature for each viewpoint is very

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large; many additional references may be found in the following list of perspectives: (1) Historical: Origin and use of the notion of warranty (Loomba, H, Chapter 2). (2) Legal: Court action; dispute resolution; product liability (Kowal, H, Chapter 5). (3) Legislative: Magnusson-Moss Act; Federal Trade Commission; warranty requirements in government acquisition, particularly military (Kelly, H, Chapter 4). (4) Economic: Theories of warranty; market equilibrium (Lutz, H, Chapter 25). (5) Behavioral: Customer reaction; influence on purchase decision; perceived role of warranty; claims behavior (Kelly, H, Chapter 16). (6) Consumerist: Protection, information (Burton, H, Chapter 28; Patankar and Mitra, H, Chapter 17). (7) Engineering: Design and manufacturing issues (Murthy, H, Chapters 21 and 23). (8) Statistics: Data acquisition and analysis; estimation of costs; data-based reliability analysis (Blischke, H, Chapter 8; Kalbfleisch and Lawless, H, Chapter 9; Lawless, 1998). (9) Operations Research: Modeling; optimization (Murthy and Blischke, 199 ic; Murthy, H, Chapter 6). (10) Accounting: Tracking of costs; time of accrual (Maschmeyer and Balachandran, H, Chapter 26). (11) Marketing: Assessment of consumer attitudes; assessment of the marketplace; use of warranty as a marketing tool. (Padmanabhan, H, Chapter 15). (12) Management: Integration of many of the previous items; cost analysis; determination of warranty policy; warranty servicing decisions (Mitra and Patankar, H, Chapter 32). (13) Society: Public policy issues (Mossier and Weiher, H, Chapter 27). The bibliography by Djamaludin, et al. (H, Chapter 33) contains over 1500 references dealing with various aspects of warranty. 1.3. Reliability issues

Product reliability has an impact, directly or indirectly, in nearly all of the areas listed above. In particular, offering a warranty results in additional costs to the manufacturer as a result of servicing claims over the warranty period. This cost depends on product reliability and this in turn is influenced by the design and manufacturing decisions of the manufacturer. Other factors that affect this cost are product usage and the maintenance efforts on the part of the buyer over the warranty period. Similarly, from the buyer's perspective, the total life cycle cost is a function of the warranty terms and the reliability of the product.

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1.4. Objectives and chapter outline In the remainder of this chapter, we focus our attention on warranty costs, their relationship with product engineering and management, and on the statistical aspects of the link between product reliability and warranty. We review the statistical models that have been developed to study some of issues listed above. The outline of the chapter is as follows. In Section 2, we briefly discuss a number of warranty policies commonly used in consumer, commercial and military applications. Section 3 deals with warranty cost analysis. Models, cost analyses, and related results are given for several of the policies discussed in Section 2, for both new and used products. Section 4 deals with engineering and management aspects of reliability in the context of warranty. Three topics that link warranty and product reliability äre discussed, with the focus on reducing warranty servicing cost through bettet design, control during manufacturing and servicing of warranty. In Section 5, we consider data aspects of warranty. Included are discussions of the various types of data relevant to the analysis of warranty costs, including historical data, test data, operational data, and so forth, the analysis of the data and use of the results in statistical estimation of warranty costs. We conclude with a brief discussion of the role of effective management of the interactions between product warranty and reliability.

2. Some common warranty polieies

We first consider warranty policies for new products, following which we discuss warranty policies for used products. 2.1. Warranties for new products A taxonomy of warranty policies is given by Blischke and Murthy (1991). In this, the first criterion for classification of warranties is whether or not the manufacturer is required to carry out further product development (for example, to improve product reliability) subsequent to the sale of the product as part of the warranty contract. Policies which do not involve such further product development can be further divided into two groups - Group A, consisting of policies applicable for single item sales, and Group B, policies used in the sale of groups of items (called lot or batch sales). Policies in Group A can be subdivided into two sub-groups, based on whether the policy is renewing or non-renewing. For renewing policies, the warranty period begins anew with each replacement, while for non-renewing policies, the replacement item assumes the remaining warranty time of the item it replaced. A further subdivision comes about in that warranties may be classified as "simple" or "combination". The free replacement (FRW) and pro-rata (PRW) policies (to be discussed later) are simple policies. A combination policy is a simple policy combined with some additional features or a policy that combines the terms of two or more simple policies. Each of these four groupings can be further subdi-

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vided into two sub-groups based on whether the policy is one-dimensional or twodimensional (or higher). A one-dimensional policy is one that is most often based on either time or age of the item, but could instead be based on usage. In contrast, a two-dimensional policy is based on two variables, e.g., time or age as well as usage. In the case of an automobile, one can either have a one-dimensional policy (e.g., 2 years) or a two-dimensional warranty (e.g., 2 years or 30,000 miles, whichever comes first). The latter type is the most common auto warranty. Policies belonging to Group B can also be sub-divided into two categories based on whether the policy is "simple" or "combination". As in grouping A, Group B policies can be further subdivided based on whether the policy is one-dimensional or two-dimensional. Finally, policies which involve product development subsequent to the sale are labeled Group C. Warranties of this type are typically part of a service maintenance contract and are used principally in commercial applications and government acquisition of large, complex items - for example, aircraft or military equipment. Nearly all such warranties involve time and/of some function of time as well as a number of characteristics that may not involve time, for example, fuel efficiency. We describe a few warranty policies from each of the three groups. Some of the policies from Group A (offered with most consumer durables and some industrial and commercial products) will be studied later in the chapter. For a more comprehensive list of policies from Groups A-C, see Blischke and Murthy (1994, 1996). The following notation is used: W length of warranty period Cb unit sale price (cost to buyer) II time to failure (lifetime) of an item 2.1.1. Group A policies One-dimensional policies:

We begin with one-dimensional policies. A one-dimensional warranty policy is characterized by an interval, called the warranty period, which is defined in terms of a single variable, e.g., time, age, usage. We define the three most commonly offered one-dimensional warranty policies. Later, we give a two-dimensional warranty policy. 2.1.1.1. Policy 1: One-dimensional non-renewing free replacement warranty (FRW) policy The manufacturer agrees to repair or provide replacements for failed items free of charge up to a time W from the time of the initial purchase. The warranty expires at time W after purchase. In the case of non-repairable items, should a failure occur at age X (with J2 < W), under this policy the replaced item has a warranty for a period (W - X), the remaining duration of the original warranty. Should additional failures occur, this process is repeated until the total service time of the original item and its replacements is at least W. In the case of repairable items, repairs are made free of charge until the total service time of the item is at least W.

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Typical applications of these warranties are consumer products, ranging from inexpensive items such as photographic film to relatively expensive repairable items such as automobiles, refrigerators, large-screen color TVs, etc., and expensive non-repairable items such as micro-chips and other electronic components as well. 2.1.1.2. Policy 2: One-dimensional non-renewing pro-rata rebate warranty (PRW) policy The manufacturer agrees to refund a fraction of the purchase price should the item fail before time W from the time of the initial purchase. The buyer is not constrained to buy a replacement item. The refund depends on the age of the item at failure (J0 and it can be either linear or a nonlinear function of ( W - X), the remaining time in the warranty period. Let q(x) denote this function. This defines a family of pro-rata policies which is characterized by the form of the refund function. Two forms commonly offered are as follows: 1. Linear function: q(x) = [ ( W - x)/Vv']C b. 2. Proportional function: q(x) = [c~(W- x)/WlCb, where 0 < c~ < 1. Typically, these policies are offered on relatively inexpensive non-repairable products such as batteries, tires, ceramics, etc. The policy is inherently nonrenewing. A renewing version of this that is commonly used is a warranty under which the buyer may purchase a replacement item at a reduced price, which is a function of g(x), on failure of the item during the warranty period, with the replacement item warrantied as a new item (i.e., the warranty clock begins anew). 2.1.1.3. Policy 3: One-dimensional non-renewing combination F R W / P R W The manufacturer agrees to provide a replacement or repair free of charge up to time W1 from the time of initial purchase; any failure in the interval W1 to W (where W1 < W) results in a pro-rated refund. The warranty does not renew. The proration can be either linear or nonlinear. Again, depending on the form of the pro-ration cost function, we have a family of combined free replacement and pro-rata policies similar to that for the PRW. Warranties of this type are sometimes used to cover replacement parts or components where the original warranty covers an entire system. They are also widely used in sales of consumer products. There are several renewing versions of this warranty: It may renew in the F R W sector of the warranty, the P R W sector, or both.

Two-dimensional policies: In the case of two-dimensional warranties, a warranty is characterized by a region in a two-dimensional plane, with one axis representing time or age of the item, and the other representing item usage. As a result, m a n y different types of warranties, based on the shape of the warranty coverage region, may be defined. We describe one such policy.

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2.1.1.4. Policy 4: Two-dimensional non-renewing F R W policy The manufacturer agrees to repair or provide a replacement for failed items free of charge up to a time W or up to a usage U, which ever occurs first, from the time of the initial purchase. W is called the warranty period and U the usage limit. The warranty region is the rectangle bounded by the time and usage axes and lines at usage = Uandtime = W. Note that under this policy, the buyer is provided warranty coverage for a maximum time period W and a maximum usage U. If usage is heavy, the warranty can expire well before W. On the other hand, if usage is very light, then the warranty can expire well before the limit U is reached. Should a failure occur at age X with usage Y, it is covered by warranty only if X is less than W and Y is less than U. If the item is replaced by a new one, the replacement item is warrantied for a time period (W - J0 and for usage (U - Y). Nearly all auto manufacturers offer this type of policy, with usage corresponding to distance driven.

2.1.2. Group B policies One-dimensional Group B policies are called cumulative warranties and are applicable only when items are sold as a single lot of n items and the warranty refers to the lot as a whole. The policies are conceptually straightforward extensions of the non-renewing FRWs and PRWs discussed previously. Under a cumulative warranty, the lot of n items is warranted for a total time of n W, with no specific service time guarantee for any individual item. Cumulative warranties would quite clearly be appropriate only for commercial and governmental transactions since individual consumers rarely purchase items by lot. In fact, warranties of this type have been used by airlines in the purchase of electronic equipment and have been proposed in the US for use in acquisition of military equipment. The rationale for such a policy is as follows. The advantage to the buyer is that multipleitem purchases can be dealt with as a unit rather than having to deal with each item individually under a separate warranty contract. The advantage to the manufacturer is that fewer warranty claims may be expected because longer-lived items can offset early failures. We describe one such policy; for additional examples, see Guin (1984). The following notation will be used: X~ = service life of item i,

i = 1,2~...,

n

S,, = ~ - ' ~ . it1

2.1.2.1. Policy 5: Cumulative F R W A lot of n items is warranted for a total (aggregate) period of n W. The n items in the lot are used one at a time. If Sn < nW, free replacement items are supplied, also one at a time, until the first instant when the total lifetimes of all failed items plus the service time of the item then in use is at least n W.

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This policy is applicable to components of industrial and commercial equipment bought in lots as spares and used one at a time as items fail. Examples of possible applications are mechanical components such as bearings and drill bits. The policy would also be appropriate for military or commercial airline equipment such as mechanical or electronic modules in airborne units.

2.1.3. Group C policies Warranty policies in G r o u p C are also called R I W policies. The basic idea is to extend the notion of a basic consumer warranty (usually the F R W ) to include guarantees on the long-term reliability of the item and not just on its immediate or short-term performance. This is particularly appropriate in the purchase of complex, repairable equipment that is intended for relatively long use. The intent of RIWs is to negotiate warranty terms that will motivate a manufacturer to continue improvements in reliability after a product is delivered. Under RIW, the contractor's fee is based on bis ability to meet the warranty reliability requirements. These often include a guaranteed M T B F (mean time between failures) as a part of the warranty contract. The following policy from G a n d a r a and Rich (1977) illustrates the concept: 2.1.2.2. Policy 6: Reliability improvement warranty The mänufacturer agrees to repair or provide replacements free of charge for any failed patts or units until time W after purchase. In addition, the manufacturer guarantees the MTBF of the purchased equipment to be at least M. If the computed MTBF is less than M, the manufacturer will provide, at no cost to the buyer, (1) engineering analysis to determine the cause of failure to meet the guaranteed MTBF requirement, (2) Engineering Change Proposals, (3) modification of all existing units in accordance with approved engineering changes, and (4) consignment spares for buyer use until such time as it is shown that the MTBF is at least M.

2.2. Warranties for used products Warranties for second-hand products can involve features such cost limits, exclusions, and so on. Specifics of the warranty depend on the product and on the dealer's decision. The terms (e.g., duration, features) can vary from item to item and can depend on the condition of the item involved. They can also be influenced by the negotiation skills of the buyer. A taxonomy for onedimensional warranties on used items is given by Murthy and Chattopadhyay (1999). These policies can be divided into two groups, based on whether the policies offer a buy-back option or not. Policies without buy-back options imply that the seller has no obligation to take back an item sold. As a result, the warranty expires after the duration indicated in the warranty policy. Any failures within the warranty period are rectified according to the terms of the warranty policy. These policies can be further subdivided into two subgroups - G r o u p A: Non-

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renewing policies and Group B: Renewing policies. Each of these can be further subdivided into two sub-groups Group A1 (B1): Simple policies and Group A2 (B2): Combination policies. Under the buy-back option, the buyer is given a monetary refund (either a full refund or a fraction of the sale price) by returning the purchased item at any time within the warranty period. The warranty terminates when this occurs. We group such policies under Group C. All failures before the termination of the warranty are rectified according to the terms of the warranty. We describe a few warranty policies from Groups A and C. These are offered with most consumer durables and some industrial and commercial products. For a more comprehensive list of policies, see Chattopadhyay (1999).

2.2.1. Type A [non-renewing] policies Cost sharing warranty (CSW) policies: Under the CSW the buyer and the dealer share the repair cost. The basis for the sharing can vary as indicated below. The components of the product are divided into two disjointed sets, I (inclusion) and E (exclusion), with the components belonging to the set I covered under warranty. 2.2.1.1. Policy 7: Specified parts excluded (SPE) Under this policy, the dealer rectifies failures of components belonging to the set I at no cost to the buyer over the warranty period. The costs of rectifying failures of components belonging to the set E are borne by the buyer. (Note: Rectification of failures belonging to the set E can be carried out either by the dealer or by a third party).

Cost limit warranty ( C L W ) policies: Under the CLW policy, the dealer's obligations are determined by cost limits on either individual claims or total claims over the warranty period. There are many policies of this type. Two such are the following: 2.2.1.2. Policy 8: Limit on individual cost (LIC) Under this policy, all failures are rectified by the dealer. If the cost of a rectification is below the limit C~, then it is borne completely by the dealer and the buyer pays nothing. If the cost of a rectification exceeds CI, then the buyer pays the excess cost (the difference between the cost of rectification and CI). 2.2.1.3. Policy 9: Limits on individual and total cost (LITC) Under this policy, the cost to the dealer has an upper limit (CI) for each rectification and the warranty ceases when the total cost to the dealer exceeds CT or at time W whichever occurs first. The difference between the actual cost of rectification and the cost borne by the dealer is paid by the buyer.

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2.2.2. Type C [buy-back] policies 2.2.2.1. Policy 10: Money-back guarantee (MBG) Under the MBG policy, all failures within the warranty period [0,14/) are rectified at no cost to the buyer. If the number of failures over [0, W) exceed a specified value k (k > 1), then at the (k + 1)st failure, the buyer has the option of returning the item for a refund of 100% of the purchase price, and the warranty ceases when the buyer exercises this option. If the number of failures over [0, W) is either <_k or the buyer does not exercise the buy-back option when the (k + 1)st failure occurs, then the item is covered for all failures unti! time W from the time of purchase.

2.3. Extended warranties A warranty is ordinarily an integral part of a product sale and is offered by the manufacturer at no additional cost. The actual cost to the manufacturer is factored into the sale price. Extended warranties provide additional coverage over the base warranty and are obtained by the buyer by paying a premium. Extended warranties are optional warranties that are not tied to the sale process and can be either offered by the manufacturer or a third party. (For example, several credit card companies offer extended warranties for products bought using their credit cards and some large merchants offer extended warranties.)

3. Warranty cost analysis 3.1. Modeling warranty costs There are m a n y aspects to the analysis of the cost of a warranty. First, it is necessary to develop adequate cost models. These depend on the perspective (buyer or seller), the basis on which the costs are to be assessed, and the probabilistic structure of the random elements involved. An integrated framework to study warranty is given in Murthy and Blischke (1991a). In this section, we start with a simple system characterization that will enable us to build models for carrying out a cost analysis from both manufacturer and consumer perspectives.

3.1.1. System characterization A system characterization for the analysis warranty costs is given in Figure 1. The manufacturer produces products and sells them to consumers with a warranty policy included. Product performance is determined by the interaction between product characteristics (determined primarily by the manufacturer and influenced by design and manufacturing decisions) and product usage (determined by the consumer). When the buyer is not satisfied with product performance, a claim under warranty usually results. The cost of the warranty is the cost incurred by the manufacturer of servicing a claim under warranty. The

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I Manufacturer ]

]

Consumer

,L

Usage

Product Reliability

Warranty I

Product Performance

[ Satisfactory

[~

~

No Warranty Cost

No "1 Warranty Costs Fig. 1. Systemcharacterizationfor warranty cost analysis.

magnitude of this cost depends on the terms of the warranty policy and product reliability. The characterization of each of the elements in Figure 1 can involve many variables, depending on the degree of detail desired. A simple characterization is based on the following assumptions. 1. All consumers are alike in their usage. One can relax this assumption to divide consumers into two (high or low intensity usage) or more groups. 2. All new items are statistically similar. One can relax this assumption to include two types of items (conforming and non-conforming) to take into account quality variations in manufacturing. 3. The performance of the product is expressed in terms of a binary characterization, working or failed. The time to first failure is a random variable. Subsequent failures depend on the type of rectification action (repair or replacement; type of repair, and so on). 4. The cost to rectify each failure is the sum of administration, repair (or replacement), shipping, storage of spares, etc. We aggregate all of these into a single variable that, in general, is a random variable. 5. Whenever a failure occurs, it results in an immediate claim. Relaxing this assumption involves modeling the delay time between failure and claim. 6. All claims are valid. This can be relaxed by assuming that a fraction of the claims is invalid, either because the item was used in a mode not covered by the warranty or because it was a bogus claim.

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7. The time to rectify a failed item (either through repair or replacement) is sufficiently small in relation to the mean time between failures so that it can be approximated as being zero. 8. The manufacturer has the logistic support (spares and facilities) needed to carry out the rectification actions without any delays. Relaxing one or more of these assumptions results in more complex system characterizations.

3.1.2. Notation In the remainder of this chapter, the following symbols will be used: A W L U R F(t)

Age of used item at sale Warranty period (for one- and two-dimensional policies) Lifecycle of product Usage limit (for two-dimensional policy) Usage rate (for two-dimensional policy) random variable Failure distribution for first failure (for a new item) Failure rate associated with F(t) Failure distribution of a module (for the case of redundancy) Fm(0 Fù(t) Failure distribution for first failure (for a used item) Failure distribution for first failure (two-dimensional warranty) F(t, x) Distribution for the cost of each rectification under warranty G(c) Failure distribution of a non-conforming item I-I(O Density function for usage rate Number of failure (warranty claims) over the interval [0,t) N(O Renewal function associated with F(t) M(O Renewal function associated with FU,x ) M(«, x) X(t) Total usage by time t Sales rate s(O Total sales over the product lifecycle S Expected refund rate ~(0 Expected demand rate for spares p(O Expected repair rate 7(w) Remaining (Residual) life of item Product life cycle L Buyer's cost (sale price) per unit (one-dimensional warranty) Cb(~V) Cb(W, CO Buyer's cost (sale price) per unit (two-dimensional warranty) Manufacturing cost per unit for new items C, Manufacturer's cost per unit sold (one-dimensional Cm(t/V) warranty) Cm(W, CO Manufacturer's cost per unit sold (two-dimensional warranty) P(L, W) Manufacturer's profit Expected cost of each minimal repair Cr Buyer's cost per unit with no warranty Cù

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Cost o f / t h minimal repair for an item (random variable with distribution G(c)) Limit on total cost

3.1.3. Probabilistic models Each failure under warranty results in a claim. In order to analyze costs, it is necessary to model item failures under warranty. New and used item failures are modeled differently. Failures in the latter case depend on the item age at sale. In addition, first and subsequent failures must be treated separately, since the latter depend on the nature of rectification (repair versus replace, type of repair). Finally, modeling of one-dimensional policies is different from that for twodimensional policies. In general, an item consists of many components. One can either model at the component level or at the item level. Modeling at the component level is needed for certain types of policies for used items. Here we confine our attention to modeling at the system level. 3.1.3.1. Modeling time to first failure We first consider the first failure of a new item for one-dimensional policies. Let T denote the time to first failure. We model this by a failure distribution F(t) defined as

F(t) = P ( T <_ t) .

(1)

The corresponding density function, when it exists, is given by f i t ) = dF(t)/dt. A related function is the failure rate function r(t) given by

r(t) -- lf(t~)F(t)

(2)

For two-dimensional policies, two different approaches have been proposed for modeling the first failure. In the first approach, the time to first failure (7) and usage at failure (J0 are modeled by a two-dimensional distribution function

F(t,x) = P { T < t , X <_ x} .

(3)

In the second approach, usage is modeled as a function of time. Let X(t) denote the usage by time t. We assume that X(t) is given by

X(t) = Rt ,

(4)

where the usage rate R is a random variable. The failure rate r(t) is modeled as a function of t, R and X(t). In the simplest case (see Iskandar, 1993),

r(t) = Oo + 01t + 02R + 03X(t)

(5)

so that the model is reduced to a one-dimensional formulation. The failure distribution can be obtained from (5) using (2).

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For a used item, the time to first failure depends on the age A of the item at the time of sale. If A is known, then the failure distribution for the first failure under warranty is given by

B(t) =

F ( t + A) - F(A) 1 - F(A)

(6)

When A is unknown, the characterization of Fu(t) is more complex. 3.1.3.2. Modeling subsequent failures As mentioned previously, the distributions of times to subsequent failures depend on the rectification actions used. We first consider the case of new products sold with one-dimensional warranty policies. If the item is not repairable, then every failure results in the replacement of the failed item by a new one. Since the claims are made immediately and the time to replace is negligible, the subsequent failures over the warranty period occur according to a renewal process associated with the failure distribution F(t). The expected number of replacements over the warranty period W is given by

Z~[N(V/)] = ~ ( W ) ,

(7)

where M(t) is the renewal function (see, Ross, 1970) associated with F(t), determined as the solution of M(t) = F(t) +

./Òt

M(t - y)f(y)dy

.

(s)

If the item is repairable, then the manufacturer has the option of either repairing the failed item or replacing it with a new one. In the former case, the failure distribution of the repaired item depends on the type of repair. In the case of minimal repair, the failure rate of the item after a repair is the same as that just before failure. This model is appropriate when the item consists of a large number of components and failure is due to failure of one or a small number of components. Replacing the failed components by new ones does not materially affect the reliability of the system and hence the failure rate is unaffected. For more details of minimal repair, see Barlow and Hunter (1966). In this case, the number of failures over the warranty period occur according to a non-homogeneous Poisson process with intensity function r(t). The expected number of failures over the warranty period W is given by EIN(W)] =

jO"Wr(t)dt

.

(9)

As an example of two-dimensional policies, we consider non-repairable items and Warranty Policy 4. In this case, the expected number of failures M ( W , U) is given by the two-dimensional renewal function associated with F(t, x), namely

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M(t,x) = F(t,x) +

fo'fo xM ( t -

y,x - z)f(y,z)dzdy

.

(10)

A different approach is bettet suited for analysis of repairable items. We again assume minimal repair. In this case, conditional on R, failures over the warranty period occur according to a non-homogenous Poisson process with intensity function given by (5). This implies that failures over the warranty period occur according to doubly stochastic non-homogenous Poisson process since R is a random variable. Finally, in the case of non-repairable used items, the occurrence of subsequent failures over the warranty period depends on the type of item used in replacing failed items. Often, operable used items are used, so that it is necessary to model the failure distribution of such items. Let Fr(t) denote the failure distribution for such items. In this case, failures over the warranty period occur according to a modified renewal process, with the distribution of the first failure given by Fù(t) and that of subsequent by F,.(t). If the item is repairable and is subjected to minimal repair, then failures over the warranty period occur according to a nonhomogenous Poisson process with intensity function r(t), with A < t < A + W. In this case, the expected number of failures over the warranty period is given by

EIN(W)]

=

B

A+ W

r(t)dt

(11)

JA

3.1.4. Basis f o r cost calculation There are a number of approaches to the costing of warranty. Costs clearly are different for buyer and seller. In addition, the following are some of the methods for calculating costs that might be considered:

(1) Cost, to the seller, per item sold. This per unit cost may be calculated as the total cost of warranty, as determined by general principles of accounting, divided by number of items sold. (2) Cost per item to the buyer, averaging over all items purchased plus those obtained free or at reduced price under warranty. (3) Life cycle cost of ownership of an item with or without warranty, including purchase price, operating and maintenance cost, etc., and finally including cost of disposal. (4) Lifecycle cost of an item and its replacements, whether purchased at full price or replaced under warranty, over a fixed time horizon. (5) Cost per unit of time. These costs are random variables, since claims under warranty and the cost to rectify each claim are uncertain. The selection of an appropriate cost basis depends on the product, the context and perspective. The type of customer - individual, corporation, government - is important, as are many other factors. For a review of mathematical models in product warranty, see Murthy and Blischke (1991b).

Warranty and reliability

557

3.2. Warranty cost analysis for new products In this section we present the cost analysis for some of the policies discussed in Section 2.

3.2.1. Policy 1 We look first at the seller's cost per unit sold for non-repairable items sold under non-renewing F R W with warranty period W. The analysis of this warranty by Menke (1969) and Lowerre (1968) was one of the first theoretical analyses of warranty costs. The analyses presented were first-failure models, ignoring the possibility of multiple failures during the warranty period. Nonetheless, some useful first approximations were obtained. Blischke and Scheuer (1975, 1981) extended the model to account for multiple failures over the warranty period. We first consider the case where the item is non-repairable, so that any failures during the warranty period require replacement of the failed item by a new item. 3.2.1.1. Expected cost per unit to manufacturer (non-repairable product) Since failures under warranty are rectified instantaneously through replacement by new items, the number of failures over the period [0, t), N(t), is characterized by a renewal process with time between renewals distributed according to F(t). As a result, the warranty servicing cost to the manufacturer over the warranty period W is a random variable given by Cm(W) = Csll + N ( W ) ]

(12)

where we have included the manufacturing cost of the initial item sold to the buyer. The expected number of failures during warranty, E[N(W)] is given by (7). As a result, the expected warranty cost per unit to the manufacturer is given by

E[Cm(W)I = Cs[1 + M ( W ) ] .

(13)

EXAMPLE 1. A color television consists of several components, with the picture tube being the most important. The picture tube typically carries a warranty that is different from that on the remainder of the set. Suppose that the television is sold with an F R W policy with warranty period W for the picture tube. When a picture tube fails, it is replaced by a new tube, since it is not possible to repair one that has failed. Suppose that the failure time for a picture tube follows a Weibull distribution, i.e.,

F(t) = 1 - e (~«)~ with 2 = 0.2 per year and c~ = 1.5. This implies that the mean time to failure is # = 2-1 F(1 + 1/c0 = 4.51 years. Since c~ = 1.5, the failure rate is increasing with time.

558

D. N. P. Murthy and W. R. Blischke

Suppose that the manufacturer's cost of each item is Cs = $80. We consider three different values for W. It is not possible to obtain an analytical expression for the renewal function for a Weibull distribution. The values are determined either by the use of tables (Baxter et al., 1982) or by use of a renewal function solver (Blischke and Murthy, 1994). The results are given in Table l(a). If the shape parameter is ct = 1 (so that failures are exponentially distributed) with the same M T T F of 4.51 years (i.e., 2 = 1/4.51 =0.22173), the results are as shown in Table l(b). As can be seen, the results on warranty costs for the Weibull case are lower than for the exponential. The Weibull with increasing failure rate in fact has a lower failure rate (and hence higher reliability) for relatively small values. Thus the lower costs. 3.2.1.2. Expected life cycle cost for non-repairable products Here we are interested in the cost over a period L. The first failure after expiration of the warranty results in a new purchase by the buyer and this comes with a new identical warranty. Let Yi, i = 1, 2 , . . denote the time interval between successive repeat purchases, with the first purchase occurring at t = 0 . It is easily seen that the Y's are of the form Y = W + 7(W), where 7(14/) is the remaining life of the item in use at the expiration of the warranty. This is simply the excess age of the renewal process N(t) discussed earlier. Let FT(t) denote the distribution function for the excess age. This is given by (see Ross, 1970) FT(t ) = F ( W + t) - ~o W [1 - F ( W + t - x ) ] ~ ( x )

(14)

,

where M ( t ) is the renewal function given by (7). Since Y is a linear translation of 7(W), it is easily shown that the distribution function of Y is given by z.?(t)

= F(t)

- ~0 W [1

- F(t -.,c)]d~(.,c)

.

(15)

Table 1 Expected warranty costs for Example 1 W (years)

M(W)

E [Cm(W)] ($)

R(W)

(a) c~ = 1.5 0.5 1.0 2.0

0.0314 0.0878 0.2407

82.51 87.02 99.26

0.969 0.914 O.776

(b) c~ - 1.0 0.5 1.0 2.0

0.1109 0.2217 0.4435

88.87 99.74 115.48

0.895 0.801 0.642

Warranty and reliability

559

Ler Cb(L,W) denote t h e LCC to the buyer. Since the buyer purchases new items according to a renewal process with interval between purchases distributed according to Fy(t), we have E[Cb(L , W)] = Cb[1 +Mr(L)] ,

(16)

where M~t) is the renewal function associated with Fy(t). If L is large relative to W and E[Y], then one can use an asymptotic approximation (see Blischke and Murthy, 1996), which gives L Mr(L) ~ #[1 + M ( W ) ]

'

(17)

where # is the M T T F for the product and M(W) is obtained from (7). The LCC, Cm(L, W), to the manufacturer is simply the sum of the unit warranty costs for sales over the life cycle. As a result, we have

E[Cm(L, W)] = Cs[l ÷M(W)]E1 ÷My(L)]

(18)

Note that the expected profit to the manufacturer over the life cycle, EP(L, W), is given by EP(L, W) z {Cb -- Cs[1 +M(W)]}[1 +Mg(L)] .

(19)

EXAMPLE 2. Consider the picture tube of Example 1 and let the failure distribution be given by an exponential distribution F(t) = 1 - e -~t, with 2 = 0.5 per year, so that the M T T F is 2 years. In this case, F~(t)is also exponential and, as a result, the distribution of Y is simply the translated exponential Fr(t) = 1 - e -;~(t-W) The renewal function associated with this can be expressed analytically (see Cinlar, 1975). The result is Mr(L) =

- ~e i=1

x(L-iw)~~--~~IL ~j=l J!

J '

where [x] is the largest integer less than x. Suppose that the life cycle is 7.5 years and W = 1. In this case, we have Mr(7.5) = 7 -

v--, ( v - - , 7 i-1 [7.5 ) e-°5(75-i)~ ) , ~i=I

[" ,~j = l

)T

ilJ] f

= 2.222 "

This implies that the expected life cycle cost to the buyer is (from (16)) 3.2222Cb. If Cb = $100, then the expected life cycle cost is $322.20. Note that if 2 is changed to 0.4 (implying an M T T F of 2.5 years), then the expected life cycle cost

560

D. N. P. Murthy and W. R. Blisehke

to the buyer is $289.90. This is to be expected as the p r o d u c t has become more reliable. F o r the manufacturer, the expected life cycle cost (from (18) with 2 = 0.5 and Cm = $80) is $386.64. This changes to $360.86 when 2 = 0.4. This implies a reduction o f $25.28 over the p r o d u c t life cycle with this improvement in reliability. 3.2.1.3. Expected cost per unit to manufacturer, repairable p r o d u c t We first confine our attention to the case where all failures over the w a r r a n t y period are minimally repaired. As a result, the expected n u m b e r o f failures over the w a r r a n t y period is given by (9) and the expected w a r r a n t y cost to the manufacturer is given by E[Cm(m)] = Cs -F Cr

(20)

r(x)dx .

EXAMPLE 3. The item under consideration is a V C R with a Weibull failure distribution with shape parameter ~ and scale parameter 2. The V C R is sold under F R W with w a r r a n t y period W and costs $400 to manufacture. The m a n u f a c t u r e r rectifies all failures under w a r r a n t y using minimal repair at an average cost o f Cr = $50.00 per repair. Then f r o m (20), we have E[Cm(W)] ~- Cs + )~CrWp

Suppose that 2 = 0.25 per year, implying an M T T F o f 3.61 [3.55] years when c~ = 1.5 [2.0]. F r o m the above cost expression, the expected w a r r a n t y cost for W = l ..... 4 and c~ = 1.5 and 2.0 are as shown in Table 2. As can be seen the costs for ~ = 2 are greater than the corresponding costs for ~ = 1.5. 3.2.2. Policy 2 We confine our attention to the linear rebate function

q(t) =

(1 - t / W ) C b ,

0 < t < W

O,

otherwise .

(21)

Table 2 Expected warranty costs for Example 3 W(years)

1

2

3

4

(a) c~= 1.5 E [Cm(W)] ($)

412.50

435.36

469.52

500.00

(b) c~= 2.0 E[Cm(W)] ($)

412.50

450.00

512.50

700.00

561

Warranty and reliability

3.2.2.1. Expected cost per unit to manufacturer The cost per unit to the manufacturer is given by Cm(W) = Cs -- q(T)

(22)

where T is the lifetime of the item supplied. As a result, the expected cost per unit to the manufacturer is E[Cm(W)] = C, +

/0 ~ q ( t ) f ( t ) d t

(23)

.

Using (22) in (23) and carrying out the integration, we have #w] , E[Cm(W)I = Cs -~- Cb [F (W) - ~-~

(24)

where Bw =

~0 Wtf(t)dt

(25)

is the partial expectation of T. The expected profit per unit sale is given by 7"c(W) = Cb - E[Cm(W)] = Cb 1 - F ( W ) + ~

- C, .

(26)

3.2.2.2. Expected cost per unit to buyer The cost per unit to the buyer is Cb(W) = Cb -- q(T) ,

(27)

where T is the item failure time and q( ) is given by (21). As a result, the expected per unit cost to the buyer is E[Cb(W)]

= Cb[~-]-1

- F(W)l

.

(28)

EXAMPLE 4. Suppose that car battery failures occur according to an exponential distribution with parameter 2 = 0.5/year. The sale price Cb = $50 and the manufacturing cost is Cs = $30. For the exponential distribution, the partial expectation/~w is #w = ~1[1 - (1 + 2 W ) e ,~~v] Using this in (24) yields e -2W] E[Cm(W)] = C s -1- C b [i 1

k

D. N. P. Murthy and W. R. Blischke

562

Suppose that the product is sold with a warranty period of W = 1 year. Then the expected cost per unit to the manufacturer is Cs + 0.2131 Cb = $41.65. The expected warranty cost to the buyer (using (25) in (28)) is E[Cb(W)] = Cb -- {EICm(W)] -- Cs} • As a result, the expected per unit cost to the buyer is $38.35. This implies that the manufacturer's profit is $8.35 ( = 38.35 - 30.00 or 50.00 - 41.65) per item sold. If the manufacturer improves the reliability of the product so that 2 = 0.4, then the expected cost per unit to the manufacturer is Cs + 0.1758 Cm = $38.79. 3.2.2.3. Expected life cycle cost Let Cb(L, W) denote the life cycle cost to the buyer. Then, conditional o n X1, the failure time of the first item, we have (for L > 14/)

E[Cb(L, W]XI = x)] =

{

Cb+Cbx/W+E[Cb(L-x,W)]

if 0 _ < x < W ,

2Cb + E[Cb(L - x, W)]

if W < x < L,

Cb

if X > _ L (29)

On removing the conditioning, this becomes ElCh(L, W)] =

/0 ~ Cb m i n { x / W ,

1}f(x)dx +

/0 ~E[Cb(L -

x, W)f(x)dx (30)

This can be rewritten as (for details see Blischke and Murthy, 1994) ElCh(L, W)]

Cb{1 + F ( L ) - [ F ( W ) - # w / W ]

x [1 + M ( L -

W)]

«~~,~~,~,~»/~~~/0 ~-~,«~,«,~,~,~,} ,~~, The manufacturer's life cycle, Cm(L, V/) is the cost of supplying the items over the life cycle. Since failures occur according to a renewal process, we have E[Cm(L, W)] = Cs[1 -]-M(L)] .

(32)

The manufacturer's expected profit E[P(L, W)] is the difference between expected income (which is the expected cost to the buyer) and expected cost. As a result, E[»(L, W)] = < C b ( L , W)] - G[1 + M(~)] .

(33)

Warrantyand reliability

503

EXAMPLE 5. Consider the battery of Example 4. Since the failure distribution is exponential, it is possible to obtain an analytical expression for E[Cb(L, W)] given by (35). The result is

(

E[Cb(L,W)I=Cb l +e ;~w~ ( 2 L - 1 )T-W(1-e

)~w)

Let the warranty period be W = 1. The expected life cycle costs to the buyer and the manufacturer are given in Table 3 for 2 = 0.05 and 0.4 per year. Suppose that L = 10 years, Cb $50 and Cs = $30. Then for 2 = 0.5, the expected life cycle cost to the buyer and manufacturer are $237.73 and $180.00, respectively.

3.2.3. Policy 4. We carry out the cost analysis based on the second approach discussed in 3.1.3. The usage by time t is given by (4) with R a random variable with function k(u). We assume that failures are repaired minimally. As a result, over the warranty period, conditional on R = u, occur according to homogeneous Poisson process with intensity function given by

Section density failures a non-

r(tlu) = Oo+ 01t + 02u + 03ut .

(34)

3.2.3.1. Expected cost per unit to manufacturer Note that conditional on R = u, the warranty expires at time W if the usage rate u is less than ~ = U/W and at time Tu = U/u if the usage rate is u > ~. The expected numbers of failures over the warranty period for these two cases are

EEN(W, U)]uI =

/0~r(tlu)dt

and

EIN(W, U)lu ] =

/0~~r(tlu)dt .

(35)

respectively. On removing the conditioning, we obtain the expected number of failures during warranty and, using this, we find the expected warranty cost E[Cm(W,U)] to be

E[Cm(W, U)] =er { ~~ [foWr( t]u)dtl k(u)du + .~~

Ifor°r(tlu)dtlk(u)du}. (36)

Table 3 Expected warranty costs for Example 5 2

E[Cb(L,W)]

E[Cm(L,W)]

0.5 0.4

(0.8195 + 0.3935L)Cb (0.8461 + 0.3297L)Cb

(1 + 0.5L)Cm (1 + 0.4L)Cm

D. N. P. Murthy and W. R. Blischke

564

EXAMPLE 6. We consider an auto warranty for which the unit for usage U is 104 miles and for W the unit is years. Thus W = 1 and U = 2 corresponds to a time limit of 1 year and a usage limit of 20,000 miles, and the unit for R is 104 miles per year. Let k(u) = 0.2 for 0 5. This implies that R has a uniform distribution with a mean usage rate of 2.5 (or 25,000 miles per year). We consider three cases. In the first, the failure intensity is affected by age, usage rate and the total usage. In this case all the four parameters in (32) are nonzero. For the second case, we assume that the failure intensity is not affected by the usage rate, so that 02 = 0 and only age and usage influence the failure rate. Finally, for the third case, we assume a failure intensity that is not affected by either usage rate or total usage, so that 02 = 03 = 0, and only age influences the failure rate. We assume the following parameter values: Case (i): 00 = 0.003,

01 = 0 . 0 0 7 ,

02 = 0.003,

Case (ii): 00 = 0.003,

01 = 0.007,

02 = 0.000,

Case (iii): 00 = 0.003,

01 = 0.007,

02 = 0 . 0 0 0 ,

03 = 0.003.

03 = 0.003. 03 = 0.000.

The ratio of expected warranty cost to repair cost (obtained by solving (36) numerically) is shown for each case in Table 4 for U = 1.0, 1.5, and 2.0 and W = 0.50, 1.00, 1.50, and 2.00. As can be seen, for all three cases, the expected warranty costs increase with W and/or U increasing. 3.2.3.2. Expected life cycle cost A life cycle cost analysis can be carried out in a similar manner to that for Policy 1. The results can be found in Blischke and Murthy (1994).

Table 4 Expected warranty costs for Example 6 -1, Uand W--+

0.50

1.00

1.50

2.00

0.0024 0.0031 0.0035

0.0034 0.0048 0.0061

0.0040 0.0059 0.0070

0.0046 0.0069 0.0091

0.0014 0.0017 0.0018

0.0021 0.0027 0.0033

0.0026 0.0036 0.0043

0.0031 0.0043 0.0053

0.0017 0.0021 0.0023

0.0022 0.0029 0.0034

0.0025 0.0033 0.0040

0.0031 0.0036 0.0044

(a) Case (i) 1.0 1.5 2.0

(b) Case (ii) 1.0 1.5 2.0

(c) Case (iii) 1.0 1.5 2.0

Warranty and reliability

565

3.2.4. Other policies Blischke (H, Chapter 10) deals with one-dimensional free replacement and other rebate-related warranties and Patankar and Mitra (H, Chapter 11) deal with onedimensional pro-rata policies. One-dimensional combination policies were first studied by Nguyen and Murthy (1984). For further details, see Blischke (H, Chapter 12). Most of the warranty cost analysis is based on expected costs. Thomas (1989) does a comparative evaluation of the expected warranty cost with different failure distributions and same mean time to failure. Blischke and Vij (1997) explore this topic further through numerical studies. Sahin and Polatoglu (1998) deal with the probabilistic characterization of the warränty costs for the simple free replacement policy. A two-dimensional point process formulation for the study of two-dimensional free replacement policies was first proposed by Murthy et al. (1995a, b). The onedimensional approach was developed independently by Moskowitz and Chun (1994) and Iskandar (1993). Iskandar et al. (1994) deals with two-dimensional combination policies. Many different policies with different shapes for the warranty region have been proposed and studied - see, Iskandar (1993) and Singpurwalla and Wilson (1993). See Moskowitz and Chun (1996) and Wilson and Murthy (1996) for more on the cost analysis of two-dimensional warranties. Cumulative warranties were first studied by Guin (1984). Chapter 6 of W dealswith the cost analysis of several such policies. See also Zaio and Berke (1994). RIW policies are more complex, as they involve reliability development, testing, etc., and guarantees on M T T F and other reliability-related elements. Most models that have been developed are relatively simplistic in nature, for the most part ignoring stochastic aspects. This is a result of the complexity of the situation being modeled. On the other hand, in spite of the lack of modeling results, empirical evidence has shown that RIW has often proven to be cost effective. Additional analysis in this area is needed. A review of RIW models can be found in Chapter 7 of W. 3.3. Warranty cost analysis for used products We consider the case where A, the age of item at sale, is known and all failures over the warranty period are minimally repaired.

3.3.1. Policy 9 Ler TCj denote the cost of rectifying the firstj failures subsequent to the sale. It is given b J

TCj = Z

Cri

(j = 1 , 2 , . . ) .

(37)

i--1

where Cri is the cost of t h e / t h failure rectification. Since the total cost of claims to the dealer over the warranty period is limited to CT, we have for t h e / t h failure, the cost to the dealer is

Di = min{Cri, (CT - TC(i_I)}

(38)

D. N. P. Murthy and W. R. Blischke

566

and the cost to the buyer is

Bi = max{0, (TC, - Cx)}

(39)

Let TDj denote the cost to dealer associated with the first j failures. This cost is J

IDj = Z D i

(j = 1 , 2 , . . )

.

(40)

i-1 Similarly, let TBj denote the cost to buyer associated with the firstj failures. This cost is

TBj = ~ ~ B i

(j = 1 , 2 , . . )

.

(41)

i=1 with D~ and B~ given by (38) and (39), respectively. We take TDo and TBo to be 0. The w a r r a n t y can cease either when the c o m p o n e n t reaches age (A + W), with the n u m b e r o f failures being N(W; A), or earlier at t h e j t h failure if Tl) O _ 1) < Cx and TDj > Cx a n d j < N(W; A). This is shown in Figure 2. Let Z w be the cost o f rectifying all the failures over the period [0, W) and let V(z) be the distribution function for Zw. Then OO

V(z) = Z

P{Zw <_zIN(W;A ) = ]} × P{N(W;A) : j} .

(42)

j=0

Z w Cumulative cost for item 1

o

Ic12

Leve!~~os~!~g .......

t

L!m!!.°~.~°5~!~~st:

cT.

. . . . . . . . . . . . . . . . . . . . . .

a

]

1

W a r r a n t Y

Z w for item 2

\Cll

T ~3

7~ W Z W foritem3

C o s

t Time in warranty period [w] > Fig. 2. Limit on total cost warranty policy.

W

Warranty and reliability

567

Note that Zw, conditional on N(W; A) = j, is the sum o f j independent and identically distributed random variables with distribution G(z). Since N(W, A) is Poisson distributed we have

~~z~ ~[~~~~z~/~ ~

~~,~d, exp~

~/,/d,/~~'

~4~~

where GO~(z) is the j-fold convolution of G(z) with itself. The cost to the dealer is Z w if Z w < CT and CT if Z w > CT. In the former case, the warranty ceases at W and in the later case it ceases before W. Let Cd(W;A) denote this cost. The expected value of this is given by

E[Cd(W;A)] = ùf0cT zv(z)dz + CT~-(CT) ,

(44)

where v(z) is the density function ( = dV(z)/dz) associated with the distribution function V(z). The cost to the buyer over the period [0,W) is max {0, Z w - C T } . Let [Cb(W;A)] denote this cost. The expected value of this is given by

E[Cb(W;A)I =

(z - CT)V(z)dz .

(45)

T

When g(c) is an exponential density function and r(t) = 2 (i.e., failures occur according to stationary Poisson process), v(z) can be obtained analytically. (See Cox, 1962 for details.) For general r(t), it is not possible to obtain v(z) analytically and a simulation approach taust be used. EXAMPLE 7. Let the parameter values for the intensity function (given by (3)) be /~ = 2 and 2 ~- 0.443 and the parameter for the cost distribution (given by (6)) be p = 0.01. This implies that the expected cost of each repair is /~c = $100. Let CT = $450. The expected cost to the dealer, E[C«(W; A)], and the expected cost to the buyer, E[Cb(W; A)], for various combinations of A and W are shown in Table 5. (The values were obtained by simulation. For further details, see Chattopadhyay, 1998.) The sum of these two costs (i.e., cost to the dealer and buyer) is the expected warranty cost to the dealer under Policy 1 as the total cost is borne by the dealer. This is shown in Table 6.

3.3.2. Other policies Chattopadhyay (1999) is the first to deal with the cost analysis of warranties for used items. It studies many different types policies and also examines the case where the dealer has the option to upgrade an item (through overhaul, replacement of worn out components) prior to the sale.

D. N. P. Murthy and W. R. Blischke

568

Table 5 Expected warranty costs for Example 7 (Policy 9) W

A 1

2

5

6

7

[Cd(W;W)] 62.15 80.74 1 2 8 . 9 8 162.45 199,31 239.33 261.72 304.99

97.87 192.19 273.41 341.84

116.05 220.21 306.67 368.11

132.73 247.45 333.28 389.11

(b) Expected buyer's cost E [Cb(W;A)] 0.5 0.33 0.84 1.84 3.23 1.0 1.36 4.01 8.21 14.55 1.5 3.75 11.82 23.80 41.03 2.0 9.98 26.66 5 1 . 3 1 86.14

4.88 22.30 62.00 128.44

6.67 33.25 88.80 176.75

9.16 45.33 119.53 232.15

(a) Expected dealer's cost E 0.5 23.30 43.02 1.0 55.90 94.00 1.5 96.46 150.59 2.0 146.36 210.67

3

4

Table 6 Expected warranty cost for Example 7 (Policy 1) W

0.5 1.0 1.5 2.0

A 1

2

3

23.63 57.26 100.21 156.34

43.86 63.99 98.01 137.19 1 6 2 . 4 1 223.11 237.33 313.03

4

5

6

7

83.97 177 280.36 391.13

102.75 214.49 335.41 470.28

122.72 253.46 395.47 544.86

141.89 292.78 452.81 621.26

4. Engineering and management of reliability and warranty

4.1. Warranty and reliability improvement Most products consist of several components and the reliability of the product is a function of component reliabilities. If a critical component (one whose failure results in product failure) has low reliability, then a large number of failures (and claims) under warranty can result. This leads to high warranty costs. Two approaches to reducing the warranty cost are (i) building in redundancy for critical components, thereby improving the overall reliability of the product and (ii) improving the reliability through research and development.

4.1.1. Warranty and redundancy Typically, redundancy involves replication of the critical components. This possible only for components for which incorporation of such replication permissible by the functional design of the item. Building in redundancy results greater manufacturing cost per item and this is justified only if the reduction warranty costs exceeds this increase.

is is in in

Warranty and reliability

569

Various types of redundancies (hot, cold and warm standby) can be used. We discuss two of them for the case where redundancy involves a module comprising two identical components. Let F(t) denote the failure distribution of component and Fm(t) denote the failure distribution for the module. 4.1.1.1. Hot standby (Active redundancy) Here the two components in the module are in use so that module failure occurs when both of them fall. As a result, the failure time for the module is the bigger of the failure times for the two components in the module. This results in

Fm(t) = IF(t)] 2 .

(46)

4.1.1.2. Cold standby (Passive redundancy) Here the module involves a switch. When the first component of the module fails, the second is switched on. As a result, the time to failure for the module is the sum of the two component failure times. In general, the switch is imperfect. Ler q denote the probability that the switch functions properly when needed. As a result, the failure density function for the module is given by fm(t; q) = (1 - q)f(t) + q f ( t ) * f ( t ) ,

(47)

where * is the convolution operator and q is the probability that the switch functions properly when needed. (q = 1 corresponds to a perfect switch.) Redundancy in the context of warranty was first discussed by Murthy and Hussain (1994). Hussain (1997) deals with redundancy and warranty and examines the optimal redundancy decisions for hot, cold and warm standbys for both FRWs and PRWs.

4.1.2. Reliability growth Reliability growth has received a great deal of attention in the reliability literature. Nguyen (1984) was the first to study reliability development in the context of product warranty. Murthy and Nguyen (1988) deal with a model to determine the optimal development based on the reliability growth model proposed by Crow (1974). In real life, the outcome (reliability achieved at the end of the development period) is uncertain. Hussain (1997) developed more complex stochastic models for reliability growth that take this into account and derives optimal development strategy that achieves a trade-off between the development cost and the reduction in the expected warranty servicing cost. 4.2. Warranty and quality control 4.2.1. Basic concepts Because of variability in manufacturing, some of the items produced do not conform to design specifications. Items that conform to the design specifications are called "conforming" (of non-defective) and those that do not are called "non-

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conforming" (of defective) items. Non-conforming items are less reliable, and hence result in higher warranty costs. Here we look at modeling of these two types of items. Let F(t) and H ( t ) denote the failure distribution functions for conforming and non-conforming items, respectively. The failure rate for a non-conforming item is assumed to be higher than that for a conforming item over the interval [0, ec). This implies that H(t) > F(t) for all t. Modeling of the occurrence of non-conforming items depends on the type of manufacturing process used. The process to be used depends on the demand for the product and is determined by economic considerations. If the demand is high, then it is economical to use a continuous production process. If the demand is low to medium, then it is more economical to use a batch production process, where items are produced in lots (or batches). In either case, the state of the manufacturing process has a significant impact on the occurrence of non-conforming items. In the simplest characterization, the process state can be modeled as being in one of two possible states - (i) in-control and, (ii) out-of-control. When the process state is in-control, all the assignable causes are under control and, although non-conformance cannot be avoided entirely, the probability that an item produced is non-conforming is very small. The change from in control to out of control is due to one or more of the process parameters no longer being at required target values. This increases the probability that an item is non-conforming. Let Pi and Po denote the probability that an item produced is conforming when the process is in control and out of control, respectively. In general, Pi » Po. In the extreme cases, Pi--1, implying that all items produced are conforming when the state is in-control, and Po = 0, implying that all items produced are non-conforming when the process is out of control. 4.2.1.1. Continuous production Under continuous production, the manufacturing process begins in control and, after a r a n d o m length of time, it changes to out of control. When the process is in control, the probability that an item produced is conforming is Pi and that it is non-conforming is (1 -Pi). As a result, the failure distribution of an item can be modeled by a mixture of distributions, G(t) = p F ( t ) + (1 - p ) H ( t )

(48)

with p = Pi when the process is in control and p = Po when out of control. Once the process state changes from in-control to out-of-control, it remains in that state until it is brought back to in-control through some corrective action. 4.2.1.2. Batch production Here the items are produced in lots of size L. At the start of each lot production, the process state is checked to ensure that it is in control. If the process state is incontrol at the start of the production of an item, it can change to out of control

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with probability (1 - q), or continue to be in control with probability q. Once the state changes to out-of-control, it remains there until completion of the lot. As mentioned previously, an item produced with the state in-control [out-of-control], is conforming with probability Pi [Po]. Ler No denote the number of conforming items in a lot. Note that this is a random variable. We obtain the distribution of Nc using a conditional approach. Toward this end, let N denote the number of items produced before the process changes from in-control to out-of-control. This is a random variable that can assume integer values in the interval [0, L]. The probability distribution of N is given by

P{N = n} = { qL, qn(1 - q)'

n0_--
(49)

This follows since the process starts in-control and during the production of an item it can change to out-of-control with probability (1 - q) or remain in-control with probability q. Note that n = 0 corresponds to the process going out-ofcontrol during the production of the first item and, as a result, all items produced subsequently are produced with the process out-of-control. Similarly, n = L implies that the process state never changes during the production of the lot and, as a result, all items produced are with the process in-control. For 0 < n < L, the first n items are produced with the process in-control and the remaining (L-n) with the process out-of-control. The expected fraction of conforming items in a lot of size L, ~b(L), is given by (see, Djamaludin, 1993 for details) é(L) =

q(Pi -po)(1 _qL) (1 - q)L

~po.

(»o)

4.2.2. Control of reliability degradation A non-conforming item is less reliable than a conforming item. This has serious implicatious for the manufacturer as it leads to increased warranty servicing cost when items are sold with a warranty. Several approaches can be used to control reliability degradation during manufacturing. These can be broadly grouped into the following three categories:

1. Weeding: To detect and remove non-conforming items through inspection and testing.

2. Prevention: To reduce the occurrence of non-conforming items through control actions.

3. Process improvement: To reduce the occurrence of non-conforming items through changes in process design. The first two are "on-line" approaches and the third one is an "off-line" approach to controlling reliability degradation. In this section we briefly discuss some approaches in each of the three categories.

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4.2.2.1. Weeding out non-conforming items The aim of weeding is to essentially detect non-conforming items before they are released for sale. This can only be done by inspection and testing of each item. For static reliability, testing takes very little time, since a non-conforming item is detected immediately after it is put into operation. In contrast, the detection of non-conforming items for dynamic reliability involves testing for a positive time interval. Testing can be done either at the end of the final stage and/or at one or more of the intermediate stages of the manufacturing process. Detection of non-conformance at the earliest possible stage is desirable, as this allows for immediate corrective action. In some cases, the non-conforming item can be transformed into a conforming item by reworking it, and in other cases, the item must be scrapped. In either case, if a non-conformance is not detected at the earliest possible instant, the effort involved until it is detected is either wasted (if the item has to be scrapped) or the amount of rework required to make it conforming increases (if the non-conforming item can be fixed). Both result in extra cost. On the other hand, testing and inspection also costs money and it is necessary to achieve a suitable balance between these two costs. This implies that the location of inspection and testing stations in a multistage manufacturing process must be optimally selected. Another issue that must be addressed is the level ofinspection and testing effort necessary. One can either carry out 100% inspection or less than 100% inspection. Again, cost becomes an important factor in deciding on the level of inspection. The quality of inspection and testing is still another variable that taust be considered. If inspection and testing are perfect, then every non-conforming item tested is detected. With imperfect testing and inspection, not only may a non-conforming item not be detected, but a conforming item may be classified as non-conforming. As a result, the outgoing quality (the fraction of conforming items) depends on the level of testing and the quality of testing. It is 100% if testing is both 100% and perfect. Testing involves putting items on a test bed and operating them (either under normal or accelerated test conditions) for a certain length of time. This is also called "burn-in" and is discussed later in the section. 4.2.2.2. Prevention of the occurrence of non-conforming items When the process is in control, the occurrence of non-conforming items is due to uncontrollable factors. When the process is out of control, the occurrence of nonconforming items increases and the reason for the change in the process state is due to one or more of the controllable factors deviating significantly from their set values. In continuous production, the process begins in-control and changes to out-ofcontrol with the passage of time. If the change is detected immediately, then it can be brought back into control at once and the high occurrence of non-conforming items when the state is out-of-control would thereby be avoided. In a batch production, the process starts in-control and can go to out-of-control during the production of a lot. This affects the number of non-conforming items in the lot and the expected fraction of non-conforming items in a lot increases with

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the lot size L. This implies that the smaller the lot-size, the better the outgoing quality. On the other hand, the size of the lot has implications with regard to unit manufacturing cost, since each batch production results in a fixed set-up cost. This implies that it is necessary to determine the optimal lot size by a proper trade-off between this cost and the benefits derived through better outgoing quality. Djamaludin et al. (1994) deal with optimal lot sizing to control quality for products sold with warranty. For more details, see Djamaludin (1993). 4.2.2.3. Process improvement In the case of both continuous and batch manufacturing, the design of the manufacturing process has a significant impact on the probability that an item is conforming when the process is in-control. Ideally, one would like to have this probability one, so that no item produced is non-conforming. A manufacturing process is affected by several factors - some controllable and others, not. Taguchi (1981) proposed a method for determining optimal settings for the controllable factors, taking into account the influence of the uncontrollable factors. The method uses well-known concepts from design of experiments combined with the concept of "signal-to-noise" ratio from electrical communication engineering. Since the pioneering work of Taguchi, there has been considerable development in the design of optimal and robust manufacturing processes. 4.2.3. Burn-in

Burn-in involves testing all items for a period z. Those that fail during testing are scrapped. The rationale for this is that non-conforming items are more likely to fail than conforming items and hence are weeded out. For more on burn-in, see Leemis and Benke (1990). We first consider the use of burn-in in the context of weeding out non-conforming items and then discuss its use in pre-sale testing. 4.2.3.1. Weeding out non-conforming items From (48), the probability that a conforming [non-conforming] item will fail during testing for a period z is given by F(z) [H(z)]. As a result, the probability that an item that survives the test is conforming is given by (51)

pF(z) p l = p F ( ~ ) + (1 - p ) ~ ( ~ )

"

Since F ( z ) > Æ(z) , we have pl > p. The failure distribution of an item that survives the test is given by GI(t) = plFI(t) + (1 - p l ) H l ( t )

,

(52)

where F1 (t) and//1 (t) are given by (53)

Yl (t) = F ( t + z) - F ( z ) 1 -F(v)

'

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D. N. P. Murthy and W. R. Blischke

and H~ (t) = H ( t + ~) - H ( ~ ) 1 - H(,)

(54)

Murthy et al. (1993) deal with a model to obtain the optimal testing period to achieve a trade-off between the cost of testing and the reduction in the warranty cost. For more details, see Djamaludin (1993). 4.2.3.2. Pre-sale testing For products with bathtub failure rate, the failure rate is high in the initial period. As a result, if items are released without burn-in, a high fraction would fail in the early period, leading to high warranty costs and loss of customer goodwill. In this case, burn-in can be used to improve product reliability by consuming a part of the lifetime and thereby weeding out early failures. The approach is to test each item for a period ~ prior to its sale. Any item that fail within this period is minimally repaired. If the time to repair is small (in relation to z), so that it can be ignored, then the failure distribution of the item at the end of the test, FI (t), is given by (53), where F(t) is the failure distribution before testing. For burn-in to be effective, F1 (t) must be superior to F(t), so that the reliability of the item is improved. There are several ways of quantifying this improvement, e.g.,/71 (t) assumed to have a higher mean time to failure than F(t) or FI (t) < F(t) for all t, implying increased reliability for all t. Nguyen and Murthy (1982) were the first to look at optimal burn-in to reduce the expected warranty cost. See Mi (1997) for further results. 4.3. Warranty servicing 4.3.1. Dynamic sales

For products sold with warranty, the manufacturer is obligated to service all claims made under warranty. The actions taken by the manufacturer to accomplish this depend on the type and the terms of the warranty. Warranty servicing deals with study of such actions and related planning issues. Under Policy 2 (non-renewing PRW policy) the manufacturer is required to refund a fraction of the sale price on failure of an item in the warranty period. In order to carry this out, the manufacturer taust set aside a fraction of the sale price. This is called warranty reserving. For non-repairable items sold with Policy 1 (FRW policy), the manufacturer is required to supply a replacement item for failures under warranty. In this case, the number of spares needed is of interest and this is of importance in the context of production and inventory control. For repairable products sold with Policy 1, planning of repair facilities requires evaluation of the demand for repairs over the warranty period. This depends on the type of repair action and on anticipated sales over the product life cycle. Let L denote the product life cycle and s(t), 0 < t < L, denote sales per unit time over the life cycle. This includes both first and repeat purchases

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for the total consuming population. Total sales over the life cycle, S, is given by S :

~0L s ( t ) d t

(55)

.

It is assumed that the life cycle L exceeds W, the warranty period, and that items are put into use immediately after they are purchased. Since the manufacturer must provide a refund or replacement for items that fail before reaching age W, and since the last sale occurs at or before time L, the manufacturer has an obligation to service warranty claims over the interval [0, L + W). 4.3.1.1. Warranty reserves under Policy 2 Suppose that a product is sold with a non-renewing PRW policy with linear rebate function. The rebate over the interval [t, t + 6 0 is due to failure of items that are sold in the interval [t-0, t), where 0 = max{0, t - W} ,

(56)

and fail in the interval [t, t + 6 0 . Ler v(t) denote the expected refund rate (i.e., the amount refunded per unit time) at time t. Then, it is easily shown (for details, see Chapter 9 of W) that

v(O = cb

) dx I// s(x) k-~--]f(t-x ~'t 1

(57)

for 0 _< t _< (W + L). The expected total reserve needed to service the warranty over the product life cycle, ETR, is given by

B W+L ETR =

(58)

v(t)dt .

co

4.3.1.2. Demand for spares under Policy 1 When a non-repairable product is sold with a non-renewing FRW, the manufacturer is required to replace all items that fail within warranty period W. The demand for spares in the interval It, t + 60 is due to failure of items sold in the period [0, t) where 0 is given by (61). It can be shown (details the of derivation can be found in Chapter 9 of W) that the expected demand rate for spares at time t, p(t), is given by p(t) =

//

s(x)m(t - x)dx

(59)

,

where m ( t ) is the renewal density function associated with the failure distribution function F(t), given by ra(t) = f ( t ) +

m(t - x)f(x)dx

.

(60)

D. N. P. Murthy and W. R. Blischke

576

The expected total number of spares required to service the warranty over the product life cycle, ETS, is given by ETS = L +

B

L+W

p(t)d» .

(61)

J0

5.2.1.1. Demand for repairs for the F R W policy When a repairable product is sold with an F R W policy, the manufacturer has the option of repairing items that fail during the warranty period. We consider the case where the manufacturer always repairs failed items through minimal repair. For each item sold, failures over the warranty period occur according to a nonstationary Poisson process with intensity function r(t), where r(t) is the failure rate associated with the failure distribution function F(t). Let pr(t) denote the expected repair rate at time t. Then, using an approach similar to that used in determining the demand for spares, we have pf(t) =

s(x)r(t - x)dx

(62)

0 < t < L + W, with ~ given by (61). The total expected demand for repair over

the warranty period, EDR, is given by EDR = Cr

Ba0

L+W

pr(t)d« .

(63)

C o m m e n t s . The above topics were first discussed in Nguyen (1984). See also, Chapter 9 of W and Chapter 24 of H. 4.3.2. Replace vs repair

When a repairable item is returned to the manufacturer for repair under FRW, the manufacturer has the option of either repairing it or replacing it by a new one. The optimal strategy is one that minimises the expected cost of servicing the warranty over the warranty period. Nguyen and Murthy (1986, 1989) examine this topic and the optimal servicing strategies. See also Nguyen (1984). 4.3.3. Cost repair limit strategy

In general, the cost to repair a failed item is a random variable that can be characterized by a distribution function G(c). Analogous to the notion of a failure rate, one can define a repair cost rate given by {g(e)/[1 - G(c)]}, where g(c) is the derivative of G(c). Depending on the form of G(c), the repair cost rate can increase, decrease or remain constant with z. A decreasing repair cost rate is usually an appropriate characterization for the repair cost distribution (see, e.g., Mahon and Bailey, 1975). A simple model for this can be found in Murthy and Nguyen (1988).

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5. Reliability and warranty data sourees and analyses In order to assess and, especially importantly, predict warranty costs, it is necessary to have available valid and reliable data. There are many sources of relevant data in most applications, including test data, data on similar items, and so forth. Warranty claims data are often quite unreliable; people claim losses when they do not exist, and some fail to make a claim because it is not worth the effort, particularly near the end of the warranty period. As a result, most companies follow the practice of setting warranty policies based on extremely conservative approaches - we know that this will not fail for 10 years, so we will warranty it for 1 year. In today's competitive marketplace, this does not work. To do better, one needs data and a proper analysis of them. In this section, we address these issues.

5.1. Data sources and models for warranty analysis There are ordinarily many sources of data and other relevant information available that may be used in assessing the reliability of a product, even in the earliest stages of design and development. At the outset of the development of a new product, historical data on failure rates of similar products are ordinarily a part of the company's data base. Information on failures of patts and components, based on tests, supplier information, and operational information after product sales are often also available. All of this can be used in assessing and predicting product reliability. As is apparent from the many relationships expressed in the previous sections, this information can be used to predict warranty costs. The usefulness of the data depends on not only the reliability and validity of the data, but also on the validity of the models used in the analysis. This includes the probability models used to express the uncertainty in the data as well as the logic models used to connect the parts, components, etc., and the warranty cost models. The statistical estimation procedures used to estimate the model parameters and to assess the uncertainty in the estimates are also important in this process. If a Bayesian approach is used in the analysis, the choice of a prior distribution and the method of dealing with prior information also influences the results. (See Blischke and Murthy, 2000).

5.2. Estimation and prediction problems in warranty analysis 5.2.1. Estimation of system reliability Reliability analysis of systems and products may proceed along many lines, depending on the nature of the data and other information available. If data at the system level are available and a specific failure distribution can reasonably be assumed to adequately represent the data, maximum likelihood or other appropriate methods of estimation may be used to estimate the parameters of this distribution. This is discussed in detail in H and W and in most texts on theoretical statistics. Given estimators of the parameters of the distribution, system reliability may be estimated by expressing reliability as a function of the parameters and using

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this relationship to derive estimators of and confidence intervals for the reliability of the system. In some cases (e.g., the exponential distribution), exact confidence intervals can be obtained. In others, asymptotic results may be used. This requires derivation of the asymptotical distribution of the estimator, expressions for the parameters of this distribution, and estimations of these parameters. Under fairly general conditions, this asymptotic distribution is normal, and expressions for the mean and variance can be obtained in a straightforward manner. In this case, estimation of the asymptotic mean and variance is usually also straightforward and standard normal theory can be used to estimate and obtain a confidence interval for system reliability. For discussions of various approaches to this estimation problem, see Lawless (1982), Nelson (1982), Meeker and Escobar (1998), and R, Chapters 5 and 8. If lower level data are available (e.g., prior to full-scale system testing), the above approaches may be pursued to estimate the parameters of the distribution at this level or levels. Estimators of system reliability are then obtained as a function of these estimators. This is accomplished by first expressing system operation as a function of that of the components through an appropriate reliability logic model. As a result, system reliability is related to the reliability of the components, and procedures for dealing with functions of random variables such as those discussed above are applicable. Whether estimated directly or by use of logic models and estimators of component reliabilities, an estimator of system reliability may be used as a predictor of operational system reliability. In this application, two additional important considerations are: (1) If the reliability of an individual item is of interest, the variance of the estimator (i.e., the prediction) is substantially larger than in the case of estimation of average system reliability. This is analogous to the use of the sample mean Ä~ for estimating/~ or for predicting a single observation. In the former case, cr2/n is the appropriate variance, whereas «2(1 + 1/n) is the appropriate variance when Ä~ is used to predict an individual value. (2) In practice, variability is often found to be higher and reliability lower under operational conditions that they are under test conditions due to variability in the environment and/or usage intensity. Results should be interpreted conservatively. It should be noted that caution must be exercised in using asymptotical results in reliability analysis, particularly if the actual distribution of the variable in question (data, estimator, function of estimator, etc.) may be expected to be highly skewed. In this case, quite large sample sizes may be required to the asymptotical approximate to be adequate. 5.2.2. Estimation and prediction o f warranty costs

Once a warranty policy is selected, estimation/prediction of future warranty costs can proceed along basically the same lines as estimation/prediction of reliability, as discussed in the previous subsection. The approach requires explicit cost models

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for the warranty in question. In addition, this provides an opportunity to compare various warranty policies and terms (See, for example, W, Chapter 13), as well as to investigate warranty costs associated with different levels of reliability. Cost models are given in Section 3 above for the most common warranty policies, the F R W and PRW. In order to estimate costs for the F R W (manufacturer's cost per unit from (13) or life cycle cost from (17)), it is necessary to estimate the M T T F # and the renewal function M(W). Estimation of # is straightforward. M(W) may be estimated by estimation of the parameters of the assumed life distribution and then evaluation of the renewal function by direct calculation, numerical approximation, or table look-up. (See W for details.) For the PRW, manufacturer's average cost per unit is given in (24), and buyer's expected cost per unit in (28). To estimate these costs, we need to estimate the CDF F(W) and the partial expectation #w. There a r e a number of parametric and nonparametric approaches to estimation of the CDF. In cases where it is expressible in closed form, a simple approach is again to substitute parameter estimates into this expression. This is also the approach taken to estimation of #w. When closed forms do not exist, the usual approach is to use truncated infinite series expansions or asymptotic approximations and substitute parameter estimates in these expressions. Still another approach is computer simulation of the warranty process. Again, see W for details, and for examples involving the exponential and Weibull distributions. For other warranty policies, particularly renewing warranties, the cost models are much more complex, often requiring solution of an integral equation for evaluation, and the estimation problems are correspondingly more complex. Obtaining confidence interval estimates of warranty costs in a rauch more difficult problem. For the exponential distribution, usually the most tractable of life distributions, it is easy to obtain confidence intervals for the parameter 2 and for reliability, but not so easy to deal with cost estimation for the PRW. A confidence interval for #w and for warranty cost, which is given explicitly in Example 5 above, requires some tedious derivation, but it is doable. For the Weibull and most other life distributions, this approach is not feasible. Beyond the analytical problems in estimating and predicting warranty discussed briefty in the previous paragraphs, many data problems may be encountered. The procedures discussed above assume complete (simple random) samples, but this is not always feasible or practical, and many other types of data are often used. These include censored or other forms of incomplete data, which results from many tests, particularly of highly reliable components, where complete samples are either too costly or too time-consuming, or both, These complications lead to further difficulties in estimation. This topic has not been dealt with explicitly in estimation of warranty cost models, though conceptually the same approaches could be used as long as estimates of the (exact or asymptotic) variance of the estimator, based on an appropriate model for the data, can be obtained. Another approach to estimate future warranty costs is to use claims data. This is possible, of course, only after the product has been released into the marketplace and warranty claims are observed. There are many difficulties encountered

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in using claims data, including variable delays between failure time and the time at which a claim is made, bogus claims, and so forth, and many problems in statistical estimation in this context. In addition, prediction in the early stages of a product life cycle is difficult because of the lack of an adequate database. Models based on claims data are &ten empirical in nature and often use past data on similar products, particularly early on. For some approaches to the use of claims data, see Kalbfleisch and Lawless (H, Chapter 9), Chen et al. (H, Chapter 31) and Robinson and McDonald (1991). Additional discussion regarding data problems, including censoring, truncation, dishonest reporting, tampering, and haphazard reporting is given by Singpurwalla (1991). Methods of modeling and analysis of these phenomena are developed. Lawless (1998) reviews models and develops a framework for inference using claims data and other data of this type. Much can be done if a proper claims data base is devised and maintained, including prediction of future warranty costs, comparison of customer groups, comparison of warranty experience for different products, selection of a failure distribution, and estimation of operational reliability, failure distributions, and failure rates. Applications to automobile and refrigerator warranties are discussed, and many of the models for warranty data are applied in the context of these products. Related issues on estimation of reliability using field-performance data are discussed by Kalbfleisch and Lawless (1988). Sampling plans for the collection of field data on reliability and other performance measures and likelihood methods for analysis of data of this type are considered. Data requirements, including product and manufacturing characteristics, failure experience, and environmental data, are discussed in some detail and various methods ofestimation are compared. 6. Conclusions

In this chapter, we have examined the link between product warranty and reliability. We discussed briefly (i) the implications of reliability on warranty (in the context of warranty costs and warranty servicing) and (ii) the implications of warranty on reliability (in the context of design and manufacture). We focused on the statistical models and their analysis to capture the interaction between warranty and reliability and the analysis of warranty field data. From a manufacturer's viewpoint, Menezes and Quelch (1990) discuss warranty strategy from a mainly marketing viewpoint. Mitra and Patankar (H, Chapter 32) deal with a multi-criteria model for determining the warranty parameters. Warranty decisions must be made in a framework that takes into account the link between reliability and warranty and its impact on other technology and commercial oriented issues. Murthy and Blischke (2000) deal with this topic and Brennan (1994) also deal with some of these issues.

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References Barlow, R. E. and L. C. Hunter (1960). Optimum preventive maintenance policies. Oper. Res. 8, 90 100. Baxter, L. A., D. J. McConalogue, E. M. Scheuer and W. R. Blischke (1982). On the tabulation of the renewal function. Technometrics 24, 151-156. Blischke, W. R. and D. N. P. Murthy (1991). Product warranty management - I. A taxonomy for warranty policies. European J. Oper. Res. 62, 127 148. Blischke, W. R. and D. N. P Murthy (1994). Warranty Cost Analysis. Marcel Dekker, New York. Blischke, W. R. and D. N. P. Murthy (Eds.) (1996). Product Warranty Handbook. Marcel Dekker, New York. Blischke, W. R. and D. N. P. Murthy (2000). Reliability: Modeling, Prediction and Optimization. Wiley, New York. Blischke, W. R. and E. M. Scheuer (1975). Calculation of the cost of warranties as a function of estimated life distributions. Naval Res. Logist. Quart. 22, 681-696. Blischke, W. R. and E. M. Scheuer (t981). Applications of renewal theory in the analysis of the freereplacement warranty. Naval Res. Logist. Quart. 28, 193-205. Blischke, W. R. and S. D. Vij (1997). Quality and warranty: Sensitivity of warranty cost models to distributional assumptions. In Statistics ofQuality, pp. 361-386 (Eds. S. Ghosh, W. R. Schucany and W. B. Smith). Marcel Dekker, New York. Brennan, J. R. (1994). Warranties: Planning, Analysis and Implementation. McGraw-Hill, New York. Chattopadhyay, G. (1999). Modeling and analysis of warranty costs for second-hand products. Unpublished Ph.D. Thesis, The University of Queensland, Brisbane, Australia. Cinlar, E. (1975). Introduction to Stochastic Processes. Prentice Hall, Englewood Cliffs, NJ. Cox, D. R. (1962). Renewal Theory. Methuen, London. Crow, L. H. (1974). Reliability analysis for complex repairable systems. In Reliability and Biometry, pp. 397-410. (Eds. F. Proschan and R. J. Serling). SIAM, Philadelphia. Djamaludin, I. (1993). Quality control schemes for items sold with warranty. Unpublished Ph.D. Thesis, The University of Queensland, Brisbane, Australia. Djamaludin, I., D. N. P. Murthy and R. J. Wilson (1994). Quality control through lot sizing for items sold with warranty. Int. J. Prod. Eco. 33, 97 107. Gandara, A. and M. D. Rich (1977). Reliability improvement warranties for military procurement. Report No. R-2264-AF, Rand Corp., Santa Monica, CA. Guin, L. (1984). Cumulative warranties, conceptualization and analysis. Doctoral Dissertation, University of Southern California, Los Angeles. Hussain, A. Z. M. O. (1997). Warranty and product reliability. Unpublished Thesis, The University of Queensland, Australia. Iskandar, B. P. (1993). Modeling and analysis of two-dimensional warranty policies. Unpublished Thesis, The University of Queensland, Australia. Iskandar, B. P., R. J. Wilson and D. N. P. Murthy (1994). Two-dimensional combination warranty policies. RAIRO, Oper. Res. 28, 57-75. Kalbfleisch, J. D. and J. F. Lawless (1988). Estimation of reliability is field-performance studies. Technometrics 30, 365-378. Lawless, J. F. (1982). Statistical Models and Methodsfor Lifetime Data. Wiley, New York. Lawless, J. F. (1998). Statistical analysis of product warranty data. Int. Stat. Rer. 66, 41 60. Leemis, L. M. and M. Benke (1990). Burn-in models and methods: a review. IIE Trans. 22, 172-180. Lowerre, J. M. (1968). On warranties. J. Industrial Eng. 19, 359 360. Mahon, B. H. and R. J. M. Bailey (1975). A proposed improved replacement policy for army vehicles, Oper. Res. Quart. 26, 477-494. Meeker, W. Q. and L. A. Escobar (1998). Statistical Methods for Reliability Data. Wiley, New York. Menezes, M. A. J. and J. A. Quelch (1990). Leverage your warranty program. Sloan Man. Rer. 31, 69 80.

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Menke, W. W. (1969). Determination of warranty reserves. Management Sci. 15, B542-549. Mi, J. (1997). Warranty policies and burn-in. Naval Res. Logist. 44, 199-209. Moskowitz, H. and Y. H. Chun (1994). A Poisson regression modeI for two-attribute warranty policies. Naval Logist. Res. Quart. 41,355-376. Murthy, D. N. P. and W. R. Blischke (1991a). Product warranty management - II: an integrated framework for study. European J. Oper. Res. 62, 261-280. Murthy, D. N. P. and W. R. Blischke (1991b). Product warranty management III: a review of mathematical models. European J. Oper. Res. 62, 1-34. Murthy, D. N. P. and W. R. Blischke (2000). Strategic warranty management - a life cycle approach. IEEE Trans. Eng. Management 47, 40-54. Murthy, D. N. P. and G. Chattopadhyay (1999). Warranties for second hand products. In Flexible Automation and Intelligent Manufacturing (Eds. J. Ashayeri, W. G. Sullivan and M. M. Ahmad). Bagellhouse, New York. Murthy, D. N. P. and A. Z. M. O. Hussain (1994). Warranty and optimal redundancy design. Eng. Opt. 23, 301-314. Murthy, D. N. P., I. Djamaludin and R. J. Wilson (1995a). Product warranty and qnality control. Qual. Reliab. Eng. 9, 431M43. Murthy, D. N. P., B. Iskandar and R. J. Wilson (I995b). Two-dimensional; failure free warranties: two-dimensional point process models. Oper. Res. 43, 356 366. Mnrthy, D. N. P. and D. G. Nguyen (i987). Optimal development testing policies for products sold with warranty. Rel. Eng. 19, 113 123. Murthy, D. N. P. and D. G. Nguyen (1988). An optimal repair cost limit policy for servicing warranty. Math. Comp. Modeling 11, 595-599. Murthy, D. N. P., R. J. Wilson and I. Djmaludin (1993). Product warranty and qnality control. Qual. Rel. Eng. Int. 9, 431M43. Nelson, W. (1982). Applied Life Data Analysis. Wiley, New York. Nguyen, D. G. (1984). Studies in warranty policies and product reliability. Unpublished Ph.D. Thesis, The University of Queensland, Brisbane, Australia, 1998. Nguyen, D. G. and D. N. P. Murthy (1982). Optimal burn-in tirne to minimize costs for products sold under warranty, lIE Trans. 14, 16~174. Nguyen, D. G. and D. N. P. Murthy (1984). Cost analysis of warranty policies, Naval Res. Logist. Quart. 31, 525-541. Nguyen, D. G. and D. N. P. Murthy (1984). A general model for estimafing warranty costs for repairable products. IIE Trans. Rel. 16, 379 386. Nguyen, D. G. and D. N. P. Murthy (1986). An optimal policy for servicing warranty. J. Oper. Res. Soc. 37, 1081 1088. Nguyen, D. G. and D. N. P. Murthy (1989). Optimal replace-repair strategy for servicing products sold with warranty. Euro. J. Oper. Res. 39, 206-212. Robinson, J. A. and G. C. McDonald (1991). Issues related to field reliability and warranty data. In Data Quality Control: Theory and Pragmatics (Eds. G. E. Liepins and V. R. R. Uppuluri). Marcel Dekker, New York. Ross, S. M. (1970). Applied Probability Models. Holden Day, San Francisco. Sahin, I. and H. Potaloglu (1998). Quality, Warranty and Preventive Maintenance. Kluwver Publishers, Boston. Singpurwalla, N. D. (1991). Inference nnder planned manitenance, warranties, and other retrospective data. J. Stat. Plan. Inf 29, 171-185. Singpurwalla, N. D. and S. Wilson (1993). The warranty problem: its statistical and garne theoretic aspects. S I A M Rer. 35, 17~42. Taguchi (1981). On Line Quality Control During Produetion. Japanese Standard Association, Tokyo. Thomas, M. U. (1989). A prediction model for manufacturers warranty reserves. Man. Sci. 35, 15151519.

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Wilson, R. J. and D. N. P. Murthy (1996). Two-dimensional pro-rata and combination warranties. In Product Warranty Handboook (Eds. W.R. Blischke and D.N.P. Murthy), Marcel Dekker, New York. Zaino, N. A., Jr. and T. M. Berke (1994). Some renewal theory results with application to fleet warranties. Naval Res. Logist. Quart. 41, 465.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

"~ ]

Statistical Analysis of Reliability Warranty Data

Kazuyuki Suzuki, Md. Rezaul Karim and Lianhua Wang

Notation N, ns

rst F.t r« qt

number of products sold in month s, s = 0 , . . , S number of failures out of the Ns products sold in month s total reported number of claims for products of age t sold in m o n t h s total number of age t claims reported up to a specific month total number of claims reported in the month j , j = O,..., S the expected number of claims for a repairable product at age t, t = 0 , . . . , S or the probability of failure for a non-repairable one

qo + "" • + qt average cost of a claim in group k, k = 1 , . . , K expected number of claims of cost c(k) for product of age t lifetime variable of interest measured in actual operating time, e.g., X mileage censoring variable with covariate s, e.g., usage time during an interval s f(x),Æ(x) probability density function (pdf) of X and its corresponding survival function p d f of Y and its corresponding survival function g(Y), (~(Y) a vector of unknown parameters, taking values in the parameter 0 space ~2 observed quantities. Z = min(X, Y), 6 =_I[X <_ Y] (Z, a) indicator function of set [.] i[.] p* percentage of products followed up Qt

«(k) q,(k)

1. Introduetion

1.1. Importance of analysis of warranty data To achieve and improve "Customer Satisfaction" and data are valuable. Collecting accurate field data and assurance (QA) activities are critical for achieving high main purposes and uses of field reliability data are as 585

enhance it, field reliability translating it into quality quality and reliability. The follows:

586

K. Suzuki, Md. R. Karim and L. Wang

(1) The prediction of future claims; (2) The early warning/detection of bad designs, poor production processes, defective parts, poor materials, etc.; (3) Making decisions about discontinuing production or making design modifications; (4) Comparing the reliability of two or more competing products; (5) Grasping the relationships among the test data at the development stage, the inspection results of the production stage, and the field-performance; (6) Observing the targets of new product development, that is, were the targets achieved or not; (7) Constructing a database about the failure modes/mechanisms and their relation to both the environmental conditions and how the product is used; (8) Predicting product warranty costs, etc. A typical example of field reliability data is warranty claim data, which is collected economically and efficiently through service networks. For example, since 1993 in Japan every automobile manufacturer must offer warranties on almost every part of a new car for 5 years or 100,000 km. This means that the warranty is sufficient inducement for customers to report all failures within the warranty period. In Section 1, we describe various kinds of reliability warranty data and analytical problems associated with their analysis. Section 2 describes a general way of estimating the number of warranty claims. Section 3 deals with estimating the failure time distribution using follow-up information and a pseudo-likelihood approach. In Section 4, estimation of the failure time distribution using the usage time distribution is shown. In Section 5, the refinements of Section 4 are developed. Section 6 concludes the paper with some brief remarks.

1.2. Analytical problems of warranty data Manufacturers can easily obtain warranty data, but due to cost constraints and to the often disparate organizations of service departments or repair service networks, information from the warranty data is sometimes incomplete. In particular, (i) the warranty database may not provide the exact number of failures for a specific product sold in a particular month; (ii) the database may not supply the exact date the product was first used; (iii) monthly sales amounts may not be obtainable, and only the total sales amounts up to a point may be available; (iv) information may be only available for products which fail within their warranty period; (v) there may be a delay in entering the information into the database i.e., the reporting delay; (vi) sometime it is unknown how many products are still under warranty; (vii) failure analysis based on warranty data is important, but sometimes difficult to do because of missing data, and the possibility of more than one cause of failure, etc. This paper presents methods for analyzing such incomplete warranty data. Sometimes, sales information comes from the sales departments, as in Table 1, and the warranty records come from the maintenance department, as in Table 2. In this case, problem (i) occurs because the maintenance records may contain date of the maintenance activity but not the product's age. We call such data "Marginal Count Data". In other cases, the monthly sales amounts in Table 1 are not

Statistical

analysis

of reliability

warranty data

587

Table 1 Information from sales department - monthly sales amounts Sales month Sales amount

0

1

-.-

s

...

S

Total

No

N1

...

Ns

...

Ns

N

Table 2 Information from maintenance department Recorded month 0 Number of failures ro

case of refrigerator

1

...

s

...

S

Total

rl

•. •

rs

". •

rs

r

obtainable, but only the total sales amounts are available. We deal with these problems in Section 2. The method in that section can also be applied for (ii). The failure mechanism of a product is closely related to its actual operating time. For example, in the case of automobile, mileage until failure is important, but for a refrigerator, calendar time (days, months) is enough for the statistical analysis. Therefore, in general, products are categorized into two groups from the view point of the analysis. For (iv) above, the automobiles have a problem. If an automobile fails during its warranty period, its owner will ask the manufacturer to repair it under contract. On the other hand, if there is no failure, he will not report the mileage that the car has accumulated while on the warranty period. It is difficult to estimate its survival distribution without censoring data, because the usage time distributions of non-failure products are different from those of the failed products. Such records can be regarded as truncated data. Kalbfleisch and Lawless (1988) show that estimation based only on truncated data is quite uninformative when the truncation is heavy. Therefore we need supplemental information to estimate survival, such as the mileage of automobiles which did not fail as recorded in a follow-up survey, the usage time distribution at a given age, etc. An analysis (Suzuki, 1985a, b) using follow-up surveys is given in Section 3. Usage time distributions are obtainable from periodic maintenance records, especially in Japan, where, by law, every automobile must undergo preventive maintenance every two or three years. In Sections 4 and 5, the data structures of Tables 3 and 4 are analyzed using the usage time distributions which give the Table 3 Information from maintenance department - case of Recorded month

0

1

Failure mileages

zm

zll

---

•

'

automobile

s

--.

S

Zsl

• -

zsl

Zsr s

..,

2 •.,

Zoro Zlrl

ZSrs

Total count r

588

K. Suzuki, Md. R. Karim and L. Wang

Table 4 Desirable information from maintenance department caseof automobile Elapsed months from 0 the beginning of use

1

Failure number Failure miIeages

n0

nl

2"01

Zll

..

s

S

Total count

~s

nS

r

Zs1

r

ù. ù.

Zsns

ù.

ZOno Z*lnl

ZSns

distribution of the censoring random variable and it is shown that detailed information on the calendar time of the first use for each product is unnecessary. To develop feedback mechanisms for QA systems, we need to specify which component is most critical in the whole product. For this purpose, the competing risks problem in warranty data analysis is dealt with in Section 5. Section 6 contains concluding remarks. 1.3. Basic analytical tools f o r warranty data

Before doing a detailed analysis using recently developed methods, the following basic descriptions should be made to understand the over-all situation; (1) a cumulative number of failures versus calendar time plot, (2) a Pareto Diagram with a fixed observational period, (3) a cumulative failure rate versus calendar time (day, week, month) plot, (4) a cumulative failure rate versus usage time (day, week, month) plot, (5) a Weibull plot based on usage time, (6) a warranty cost versus time plot, (7) the number of repair parts and their cost value and etc. The Pareto Diagram in (2) shows the result for a fixed observational period. Therefore, it should be drawn with several different observational periods. In (3), the failure rate is estimated by h(t) =

@ of failures at time t B of products which are in use beyond time t

Based on h(t), the cumulative failure rate is estimated by

H(t)

~h(u),

u
u
Using the above idea, Kalbfleisch, Lawless and Robinson (1991) present methods for analyzing warranty data and discuss the properties of their estimates. They focus on the number of failures during a specific calendar interval. But if one is interested in the survival distribution of actual usage time, one encounters the following difficulties; during the warranty periods, the manufacturer can obtain data about the failed products only by the claims users make. Therefore, esti-

Statistical analysis of reliability warranty data

589

mating the (cumulative) failure rate is very difficult because the denominator of h(t) is unknown. This manuscript will be helpful for this problem. Descriptions, (1)-(7), should be done after stratifying by: (1) type of automobiles, (2) parts/products/subsystems, (3) production date, (4) production place (plant)/supplier, (5) dealer, (6) usage conditions, purpose for the use, (7) environmental conditions/area, (8) failure modes, (9) usage time/calendar time until failure and, etc. Cross classification by the above categories should be done. Without this stratification, the warranty analysis will not achieve its purpose. 1.4. Survey of warranty data analysis This section reviews the literature on the analysis of warranty claims data. Yun and Kalivoda (1977) present a simple probabilistic model to estimate product warranty return rates taking human factors into account. For analyzing incomplete warranty lifetime data for which information is only available for failed products, Suzuki (1985a, b) proposes a pseudo-likelihood approach using supplementary data. Kalbfleisch and Lawless (1988) developed this idea for the analysis of warranty data with missing covariate information. For this approach, see Section 3. Hu and Lawless (1996a) also apply this approach to the covariate analysis using estimating functions. More general types of pseudo-likelihood methods and their asymptotic properties when covariate information is missing are investigated by Hu and Lawless (1997). Suzuki (1987) discusses an analysis of warranty data based on the usage distribution and a non-homogeneous Poisson process. Kalbfleisch, Lawless and Robinson (1991) discuss the methodology for analyzing and predicting warranty claims using a Poisson model, where the Poisson parameter is a function of time in service. Lawless and Nadeau (1995) propose simple robust methods for the analysis of general recurrent events including regression. Assuming warranty claims follow a Poisson process, Hu and Lawless (1996b) suggest a technique for modeling claims as truncated data. Chukova and Dimitrov (1996) present some models for complex systems, where warranty periods are different for different components. Recently Blischke and Murthy (1996) provide an excellent summary of warranty data models from different disciplines, relating the different research areas. Lawless (1998) gives a comprehensive review of some methods for analyzing warranty data. He discusses the age-based analysis of warranty claims and costs, and the estimation of failure distributions or rates from warranty data. Lawless and Kalbfleisch (1992) also review some issues in the collection and analysis of warranty data. Sometimes manufacturers have warranty claims data only in aggregate form and they analyze based on these aggregate data. Trindade and Haugh (1980) discussed the complexities involved in statistically estimating the reliability of computer components from field data on systems having different total operating times for the systems at any specific reference time of analysis. They explained a renewal process method for estimating component reliability. Baxter (1994) describes a method of estimation from quasi-life tables where no observations of the lifetimes of individual components have been recorded; rather the numbers of

590

K. Suzuki, Md. R. Karim and L. Wang

components which fail between successive equally spaced time points are recorded. He shows how the theory of recurrent events can be used to construct a nonparametric estimator of a discretized version of the survival distribution. Tortorella (1996) studied a problem arising in the analysis of field reliability data generated by a repair process. In this process, data were collected in aggregate form and the component life duration data are not available. The author constructed a pooled discrete renewal process model to estimate the reliability of a component and used a maximum likelihood-like method to estimate the parameters. Karim et al. (2001a) discuss the methods for analyzing marginal count of failure data for both repairable and nonrepairable products. Also, Karim et al. (2001b) deal with change point problems by marginal count data. Much of the literature on warranty analysis considers failure models which are indexed by a single variable, such as age or mileage. There are situations where several characteristics are used together as criteria for judging the warranty eligibility of a failed product. For example, for automobiles, warranty coverage has • sometimes both age and mileage limits, and it is often important to develop methods based on both age and usage amounts. Moskowitz and Chun (1994) suggest a Poisson regression model to do this. They assumed that the number of failures under the two-dimensional warranty policies is distributed as a Poisson with parameters which can be expressed by a regression function of the age and usage amounts of a product. Lawless et al. (1995) considered the occurrence of warranty claims for automobiles when both age and mileage accumulation affect failure. They discussed methods to model the dependence of failures on age and mileage, and to estimate survival distributions and rates from warranty claims data using supplemental information about mileage accumulation. Singpurwalla and Wilson (1998) propose an approach for developing probabilistic models in a reliability setting indexed by two variables, time and a time-dependent quantity such as amount of use. They used these variables in an additive hazard model. Suzuki (1993) considers the lifetime estimation measured in mileage considering age as a concomitant variable. Given the concomitant variable, the random variable of interest is assumed to have a normal distribution. Independently of Suzuki (1993), Phillips and Sweeting (1996) deal with the analysis of exponentially distributed warranty data with an associated variable having a gamma distribution. From the manufacturers point of view, the estimation and prediction of the cost associated with the warranty policy is an essential aspect of managing the warranty program. There is a large literature on the analysis of warranty costs. Robinson and McDonald (1991) review the statistical literature on warranties relating to the cost of warranty, the advertising value of a warranty, the warranty as a product attribute, dealer relations, customer satisfaction, and reliability. They approach warranty modeling from the pricing and accounting perspective. Blischke and Scheuer (1975) analyzed pro-rata and free replacement warranty policies from both the buyer's and the seller's points of view. "Pro-rata" means that the manufacturers replace or repair a unit that fails during the warranty period at a charge to the customer that is prorated according to its age. "Free

Statistical analysis of reliability warranty data

591

replacement" means that it will be done, free of charge to the customer, during the period. They discussed the estimation of the lifetime distribution of the products and how to use these estimates for characterizing the distribution of the number of replacements and comparing costs. Blischke and Scheuer (1981) provide further application of renewal theory to the analysis of the free replacement warranty from the seller's points of view. Mamer (1982) estimated the short-run total cost and long-run average cost of products under warranty period. He showed that the expected warranty cost depends on the mean of the product's lifetime distribution and its failure rate. Mamer (1987) extended his previous research and proposed model for analyzing the trade-off between warranty and quality control, and to illustrate the sensitivity of warranty costs to environmental variables. Nguyen and Murthy (1984a, b) present a general model for repairable products for estimating both the warranty cost for a fixed lot size of sales and the number of units returned for repair in any time interval during their life cycle. Nguyen and Murthy (1988) later reviewed free warranty policies for nonrepairable products and derived the expected total cost to the consumer and the expected total profit to the manufacturer over the product life cycle. Matthews and Moore (1987) consider the problem of designing and pricing a product line of goods distinguished from each other only by different quality and warranty levels. Balcer and Sahin (1986) used renewal theory to estimate the replacement cost during the product life cycle under free replacement and pro-rata warranty policies. Frees (1986) discussed the performance of renewal function estimators, both parametric and nonparametric, of the expected cost of a warranty. Applying various time-varing failure correction processes such as replacement, minimal repair, and imperfect repair, Sahin and Polatogu (1998) investigate the jointly optimal repair/warranty-policy and repair-effort/maintenance-strategy configurations for repairable units. Vintr (1999) presents methods to minimize manufacturers cost and maximize the length of the warranty period from the manufacturers point of view. When claims data are reported to an organization or company, there are frequently substantial delays between the time a claim occurs and the time it is entered into the database used for prediction and analysis. References for the analysis of reporting delays include Kalbfleisch and Lawless (1991), Kalbfleisch, Lawless and Robinson (1991), Lawless (1994), Kalbfleisch and Lawless (1996) and Lawless (1998). Like expected warranty costs, forecasts of warranty claims, are also important to manufacturers. Articles by Robinson and McDonald (1971), Kalbfleisch, Lawless and Robinson (1991), Chen et al. (1996) and Lawless (1998) deal with methods for forecasting warranty claims. Generally field tracking studies provide better information about field reliability than warranty data, but such studies are more expensive. Amster et al. (1982) discussed several important aspects of planning and conducting field tracking studies.

592

K. Suzuki, Md. R. Karim and L. Wang

2. Estimation of the number of warranty elaims

2.1. Age-based claims anaIysis In this section we deal with methods for age-based claims analysis assuming that the expected number of claims per product at age t depends on the age of the product and is independent on other factors. Age is calculated in terms of the time since the product was sold or entered service. We use m o n t h as the unit for both age and calendar time without loss of generality. As shown in Table 5, {r«t,s = 0 , . . ,S; t = 0 , . . ,S - s} are the reported numbers of claims at age t for S-t those products sold in m o n t h s and r.t = }-~~s=0rst is the total number of age t claims reported up to month S. Here we assume that the r«t's and N / s are both known. In Section 2.3, we consider the situation where the Ns's are unknown. In Section 2.4 we deal with the case in which {r~t} are unknown and only the aggregated claims form time periods are available. F o r repairable products, ler qt be the expected number of claims for a product at age t. I f claims occur according to such a r a n d o m process, the expected value of rst is N«qt. Then the m o m e n t estimate of qt can be expressed as F. t

qt = R7

,

(l)

where Rt = vZ . ~' ss =-0 t N s is the total number of products sold up to month S - t. The cumulative estimate of (1) is t

O,=Z«,

•

(2)

u=0

These estimates are useful to manufacturers for various purposes mentioned in Section 1. The m o m e n t estimates (1) and (2) are the m a x i m u m likelihood estimates when r,t's are independent Poisson r a n d o m variables. Under the Poisson model, since Var{r.t} = Rtqt an estimate of the variance of Ôt is given by

Var{Ot}=~

q*

/__zLó~i

(3)

Table 5 General form of data for analysis based on age s

Ns

0

1

S-1

0 1

No NI

roo rlo

rm rn

ro,s-1

:

S S

1

Total

z

rl,s

S ros 1

:

Ns-1 Ns

rs-l,0 rso

rs-l,1

N

r.0

r.i

F.S-- I

F. S

Statistieal analysis of reliability warranty data

593

However, as pointed out Kalbfleisch and Lawless (1996) there is often extraPoisson variation in the claim frequencies and sometimes correlation also. They also discussed two approaches that allow extra-Poisson variation. For non-repairable products, we assume rso,...,rs,s-s have a multinomial distribution with parameters Ns and q o , . . . , qs-«, where s = 0 , . . , S. qt is defined as qt = Pr(t < X < t + 1), where X is a random variable representing the lifetime of an individual. The survival function /~(t) = 1 - ~ i =to q i is estimated by Kaplan-Meier (1958) as B(t) = 12I (1 - r'R~k)) ,

t=O,...,S,

(4)

k=0

v,S-k N s - ~t=o k-1 {re - ~ü=0r~-,,u},~ where R(o) = ~ s s= o N , and R(k)= ~s=o = 1 , . . , S, are the numbers of the risk set just before age k. Its properties are discussed in detail by Breslow and Crowley (1974).

2.2. Analysis of warranty costs Whenever a claim is made under warranty, the manufacturer incurs a cost to repair the product or to replace it with a new product. The warranty cost depends on the types of warranty policies and on the lifetime distribution of the products. Many approaches may be used to analyze warranty costs, depending on the types of warranty policies and the types of available data. We describe a simple approach, the age-based analysis o f w a r r a n t y costs, using the method of the analysis of warranty claims in Section 2.1. Kalbfleisch and Lawless (1996) and Lawless (1998) give a more thorough discussion. We assume that claims are stratified into k groups (k = 1 , 2 , . . ,K) based on the cost. Let c(k) be the average cost of a claim in group k and qt(k) be the expected number of claims of cost c(k) per product at age t. Under the assumptions of Section 2.1, the estimate of qt(k) can be written as 0t(k)

-

re(k)

,

(5)

Rt where r.t (k) = }-~~s=0 s-t rst(k) is the number of claims of cost c(k) for products ofage t reported up to month S. The cumulative estimate of warranty cost per product up to age t is K

dt = ~

c(k)Ot(k) ,

(6)

k=l

where ôt(k) = Y'~~ù=0 t q^ù (k) . The variance for dt can be estimated by applying the related method of the previous section. The method is applicable if the occurrences of claims of which costs are different are independent of each other. When they are not independent, a different method is needed.

K. Suzuki, M d . R. K a r i m and L. W a n g

594

2.3. Unknown monthly sales amount In Section 2.1 we dealt with the situation where the monthly sales amounts, Ns, s = O , . . . , S , are known. When the monthly sales amounts are not known, another method is needed. Suppose now that in Table 5, the Ns's are unknown s N s, and the rst's are known. but the total sales amount up to month S, N = ~s=0 We use a non-homogeneous Poisson process (NHPP) to mode1 claim counts (see also Ascher and Feingold, 1984). We assume that rst's are independently distributed as Poisson r a n d o m variables with parameters Nsqù where qt is the expected number of warranty claims at age t per product. This is a discretization of an age-based non-homogeneous Poisson process. The likelihood function based on rst is S

S t

L = H H e-N~q'(Nsqt)~~'

(7)

t--O s=O

s-t re, Ps = Ns~N, Ps = ~~=oPu and 0t = Nqt, the likelihood (7) Putting r.t = ~s=0 becomes S

S t

: II Il e-~'°'e~O,)~" t=O s=O

= IIe-~'«.' { I I H ( ~ ) kg=0

-u/

/ I,u=0s=0 \

= LI(0~0,.., ~s)L2(P0,.. ,PS) , where cq

(8)

OsPs-» Maximizing this, we get ês:

H

1-

,

s=O,...,S-1

,

(9)

l=s+ 1

S t

ô, = ~

~~~' ,

t = o, . . . , s

,

(lO)

s:0 Ps-t S l

l

S-l

where az = ~t=0 rlt and Dl = ~«=0 ~t=0 r« (Lawless and Kalbfleisch, 1992). In general, the estimate (9) gives a non-parametric M L E for Ps. This is derived by applying the reverse time hazard function gs = P r O ( - sIX <_ s) (Lagakos et al. 1988). Therefore, using fi;s = NÆs, we get for the multinomial model, =

-

,

~ = 0,..,s,

(11)

whereÆ(0) = ~s=0 s andÆ(k) = E s =s 0 k Ns ^ - 2 , =~-1 0 { r , - 2 ù =t 0 r ~ - , ù } , k = 1 , . . , s , are the estimated numbers of the risk set just before age k. The above approach is also applicable for the problem (ii) of Section 1.2.

Statistical analysis of reliability warranty data

595

2.4. MarginaI count data In the preceding sections r~t's are assumed to be known. But the information available is sometimes incomplete since many companies analyze their failure data by simply comparing sales for a total month with the total number of claims registered for that given month. This information is, thus, incomplete as it does not give the exact number of failures of the specific products that were sold in a particular month. For example, some electrical companies collect data on warranty claims for their products through world-wide networks. They cannot always report detailed records for each product because of limited of resources. Therefore, in the following, we investigate situations where manufacturers have only the monthly counts of warranty claims {rj}, not the detailed counts {r~t}, where rj = ~~=0 rs,j s, j = 0 , . . , S as shown in Table 6. We call {rj} the marginal count data and {re} the complete data. We treat both repairable products and nonrepairable products. 2.4.1. Repairable products For repairable products, we use a non-homogeneous Poisson process to model the failure counts. Assuming that the rst's have independent Poisson distributions with parameter Nsqt's, the complete data log-likelihood is given by log Lc(qtI{rst}) =

- ZN~qt + ~ s=0

t=O

r,t log(N~,qt)

.

(12)

t=O

However, we consider situations where only monthly marginal counts rfs are available. Under the above model, the rfs also have independent Poisson distributions with parameters ~Js=oN~qj_«, j = 0 , . . , S , and the log-likelihood function is logL= j=0

-~N~qs_~+rjlog s=0

N~qj_,

.

(13)

Table 6 General structure of monthly counted data Month

o

Sales

2

No N1 N2

S

Ns

1

Marginal total

Warrantyclaims 0

1

2

--.

S

tO0

rol

F02

ros

rlO

Fll

rl,s 1

r20

r2,s 2

rso

rO

F1

r2

FS

K. Suzuki, Md. R. Karim and L. Wang

596

2.4.1.1. Estimation of the parameters of the NHPP model Maximizing the log likelihood (13) with respect to qt, the unconstrained MLE is

[ ro/No t Ot = ), ( r t - y'~~~=lN~Ot_s)/No

fort=0, for 1 < t < S

(14)

This estimator has another expression, = N-lr , where

(15)

r = (rO,rl,...,rs)',

q = (qo,ql,...,es)',

and N is an ( S + 1) x ( S + l)

matrix, such that No

0

...

0

N1

No

...

0

ùNs

Ns-1

...

No

N=

ci is an unbiased estimator as well as a moment estimator. Since the covariance matrix of r is diag{Nq}, the covariance matrix of {i is given by covlci] = N-ldiag{Nq} (N-l) ' .

(16)

One undesirable characteristic of the unconstrained MLE, (14), is that it fails to provide valid estimätes of the parameters in certain situations. That is, Eq. (14) t ^ yield negative estimates of the parameter qt if the inequalities rt > ~~=1 Nsqt_s, do not hold for t = 1 , .., S. To avoid this, Karim et al. (200la) propose an estimator using the EM algorithm. The EM algorithm is a well-known algorithm for finding the M L E for incomplete data problems (Dempster et al., 1977). It consists of the E-step and the M-step. In this case these are: E-step: The conditional distribution of {r«j_s; 0 < s < j} given rj is a multinomial with sample size rj and probabilities Ns@-«/(Z~=0 N~qj_i),s = 0 , . . ,j, for j = 0 , . . , S. In general, the E-step finds the conditionally expected log likelihood of the complete data, given the observed data and the current fit of the parameters. For the linear exponential family, the E-step estimates the conditional expected values of the sufficient statistics of the complete data, given the observed data. For this model, at the (k + 1)th iteration, the E-step is r ~r (k)

ÆqL [r~«-'l~A - , ~ j /v«q)_~ . . (k), -

2-.~i=0lviq)

s = 0 , . . ,j .

(17)

i

M-step: The conditionally expected complete data log likelihood obtained in E-step is the same as (12) except for each rst replaced with Eq}kl[rst]r«+t]. Maximizing it with respect to qt, we have

Statistical analysis of reßabißty warranty data

0(k+l/ E,\-; e~~/{rs~lrs+t] t

=

S t ~,:0N~

t=O,...,S

.

597

(18)

2.4.2. Non-repairable products

For non-repairable products, rso,... ,rs,s s might be assumed to have a multinomial distribution with parameters Ns and q 0 , . . , qs s, for s = 0 , . . , S. qt can be interpreted as the probability that the lifetime of an individual is t. Marginal distributions from several heterogeneous multinomial distributions do not have a closed form in general, nor do the conditional distributions. Therefore if the EM algorithm is applied to estimate qt, a very computer intensive calculation will be required to obtain the conditional expectation of the complete data log-likelihood, given the observed data. For this problem, Karim et al. (200la) propose a Poisson approximation. This is highly accurate especially when many products are sold in each interval and a product is highly durable. If so, {rs«} are very small compared to {Ns}. Therefore the distribution of the r~t's can be approximated by a Poisson distribution with parameters N«qt,s = 0 , . . , S , t = 0 , . . , S - s. The likelihood functions based on the complete data and marginal data, and their estimating equations are the same as the NHPP model in Section 2.4.1. The computational complexity of an analysis which assumes multiple multinomial distributions is emphasized by Tortorella (1996) and Escobar and Meeker (1999). Escobar and Meeker (1999) also suggest Poisson approximations and Normal approximations. Karim et al. (200la) discuss the properties of the EM estimators derived in this section.

3. Estimation of the failure time distribution using follow-up information 3.1. Notation and assumptions

In the following sections, estimation of the failure time distribution is discussed. Let (X/, Y/), i = 1 , 2 , . . , N, represent independent, identically distributed pairs of random variables, where X/- is the variable of interest with pdf f ( x ) and survival function F(x), and Y~is some censoring variable with pdf 9(x) and survival function G(x). For example, 2(/might be the mileage to the first failure of product i, and Y/ might be the total mileage of product i dufing the warranty period. Also ler 0 represent a vector of unknown parameters, taking on values in the parameter space B. The observed quantities are (Zi, 6i), i = 1 , 2 , .. ,N : where Z/~_ min(X/, Y/), Bi =-I[X/< Y/], i = 1 , 2 , . . , N . Hefe/[.] means the indicator function of set [.]. That is, the random censoring model (e.g., Efron, 1967; Miller, 1981) is applied to the problem. The quantities Z/in the pairs (Zi, 6i) are unobserved for some is. Ler Di --= 1 if the/th product is followed up; otherwise, D i = 0 (i = 1 , 2 , . . . , N ) . Notice that Di is a known constant, not a random variable. In order to facilitate the description in this article, we shall make free use of terminology relevant to the automobile example which follows. Thus, we refer to

K. Suzuki, Md. R. Karim and L. Wang

598

X~ as mileage to the first failure of product i, and Yi as mileage in the warranty period of product i. Also, we define nu - ~ 6~ to be the number of automobiles that fail in the warranty period. The subscript u on nu means uncensored, and means ~ N 1 ( ~ will be used in this way throughout this article); nc = ~ ( 1 - 6~)Di to be the number of automobiles without failure in the warranty period but for which mileage was determined through follow-up. The subscript c on nc means censored; nl =- ~ ( 1 - 6i)(1 - D¢) to be the number of automobiles without failure that have not been followed up in the warranty period. The mileages for these automobiles have not been observed. The subscript 1 on nl means lost. Also we define N = n u + n c + n l to be the total number of automobiles and p* -- ( l / N ) ~ D ~ to be the percentage of automobiles followed up. Throughout Sections 3-5, we make the following assumptions: (1) X/ and Y/ (i = 1 , 2 , . . , N) are independent for all i; (2) The time scale of the rv's X and Y is assumed to be actual operating time (e.g., mileage, frequency, etc.); whereas the observational duration of the study is measured by calendar time (e.g., month, year, etc.); (3) The probability of the failure of a product depends only on its actual operating time; (4) All failures during the warranty period will be reported to the manufacturer. This is essential for obtaining the mileages of the failures. If there is no failure, the owner will not report the mileage in that period. Consequently, "no record of failure" means there has been no failure. Also, in this section we assume; (5) The percentage of follow-ups in the study, p*, is not equal to zero. Moreover, n c ¢ 0 and nu ~ O; (6) Individual automobiles to be followed up are selected randomly, and the correct mileages of followed-up automobiles are observed with probability 1, even if they have not failed.

3.2. Parametric approach for estimating the failure time distribution using follow-up information Under the assumptions in Section 3.1, the sampling distribution of the observed quantities (Zi, 6i), i = 1 , . . ,N, is given by N

~ - { f (Zi)G(Z~) } a,{g(Zi)p(Zi) } (1-a,)», {pr( 3i = O) } (~-a,)(1-D,) . i=l

Changing subscripts, the likelihood function becomes

L =

f(Zi)G

g(Zj)Æ

[Pr(X > Y)]"' ,

(19)

where Z i ( i = l , . . . , n u ) is the Zi conditioned on X / < Y / ( Ô i = l ) , and Zj (j = 1 , . . , nc) is the Zy conditioned on X~ > Y/ (c~~= 0) and Di = 1. Pr(X > Y) cannot be expressed in a simple closed form except for special cases (e.g., both X and Y are exponential; Miyagawa, 1982 investigated this special case assuming nc = 0). The distribution of the nl products that did not fail and for which no mileage frequencies are available is the same as that of2y. Ifwe try to estimate the

Statistical analysis of reliability warranty data

599

nl unobserved nonfailure mileages, we should redistribute these nl observations equally to the nc observed values of the Zj. Therefore, instead of [Pr(X > y)]nl, "c

1 n~/ùc

o(Zj)F(Zj)J

should be applied in maximizing L. Then we have pseudo-likelihood L** =

[~,,~,,~,~i,l~I~~«»~«,,, 1 l+ù,/~c

9

""

(20)

That is, every one of the observed non-failure data has an additional mass of nl/no along with its own observed mass of 1. If G(Z) and 9(Z) do not involve any parameter of interest, L** can be taken to be

L*= [Of(~.)] IO{Æ('j)}1+nl/nc ]

(21)

The proposed estimator is 0", the 0 in g2 which maximizes L*. 0* can be expected to have properties as good as those of the MLE, O, of 0 based on (19). This method is proposed by Suzuki (1985b). A similar approach is taken by Kalbfleish and Lawless (1988). They propose the pseudo-likelihood; L# =

Æ

P

.

(22)

This results from the fact that 1 + njnc converges to p*. Ler 0 # denote the value of 0 which maximizes (22). Even when the model for X is parametric and the model for Y is nonparametric, 0* and 0 # are valid. EXAMPLE 1. If X follows an exponential distribution Æ(x) = exp(-Zv), x > 0, we get L* =

2 exp(-2Zi

exp{-(1

+ù,/,c);,&}]

F r o m (8/62) log L* = 0, we obtain + (1 4- nl/nc)

)$ = nu

Zj j=l

Similarly, we obtain

}

/{ ..... i=1

for L #.

j=l

.

K. Suzuki, Md. R. Karimand L. Wang

600

EXAMPLE2. I f X follows a Weibull distribution Æ(x) = exp(-)xm), x > 0, we get L* =

2mZ~ -1 exp(-2Z, m)

exp -(1 + ni/nc)227

.

1_i:1 From (~/~m) log L* = 0 and (~/~2) log L* = 0, we have ~ u {(log Zi)Z: ~} + (1 + nl/no) ~ o { (log Zj)Z~ }

l / m + Z l o g Zi/nu = u

2=nu

2~Z;" + (1 + nl//nc)2cZ] n

/I~~ i~+ ,,+~,~~~,c~~:~l

,~~,

Here F u means ~i~1 and P c means ~ j =nol . The solution O* = (m*, 2")' of the preceding equations can be obtained by using the Newton-Raphson method. Similarly (1 + nl/nc) is replaced by p* for L #. The asymptotic properties of 0* and 0 # are as follows. For details and for the definition of J(0) and I(O,p*), refer to Suzuki (1985b). PROPERTY 1. Asymptotically, the solution 0* of (ô/~0) log L* = 0 coincides with the solution of 0* of (~/~0) log L1 = 0, where LI is the full likelihood that for each item it is known whether or not the item failed (i.e., p* = 1). Therefore, under regularity conditions (Zacks 1971, p. 194), 0* is a consistent estimator of 0. 0 # also has the above property. PROPERTY 2. Under the regularity conditions, v#N(O* - 0) ---+Nor (0, J(0) lI(O,p*)J(O)-l)

as N ---+oc .

Let Bp(O) represent the 100 × p percentile of the lifetime distribution of X, and assume that the partial derivatives (~/~O)Bp(O) exist. Then, PROPERTY 3. Bp(O*) becomes a consistent estimator of Bp(O), and V~(Bp(0*)- Bp(O)) --+ Not (0, Q(O)'J(O)-II(O, p*) × J(0)-IQ(0)) as N --+ ex} where Q(0) = (~/~ß)Bp(O) is the column vector of partial derivatives.

3.3. Non-parametric approach for estimating the failure time distribution using follow-up information In this section, we take a non-parametric approach. The generalized maximum likelihood estimator (Kiefer and Wolfowitz, 1956) of the survival function of fr(t)

Statistical analysis of reliability warranty data

601

of the random variable X is given and its statistical properties are described. We assume that X / a n d Y//are independent for all i, and that X~ is a discrete random variable taking values z 1 < ZK. Let z* denote the minimum of the uppermost support points of F(-) and G(.), and let K* be the maximum k that zk _< z*. To define an estimator of Æ(-), use the quantities (Zi*,d[, 6~), i = 1 , . . ,n* (n* <_ nu), instead of (Zi, öl). The Z[ are defined as the observed ordered distinct Zi's; that is, Z~ < ... < 2~, a[ is the multiplicity of Z/* (the number o f Z / s equal to Z*), and 6~ is the indicator associated with Zi*. If ties Zi = Zj for 6i ¢ 6j ,i ¢ j exist, we can break any such ties by treating the censored time as just slightly larger than the failure time. Then 6~ is either 0 or 1. <

•

•

•

PROPERTY 4. The generalized M L E of the survival function Æ(t) is given by B(t) = 1

for t < Z~,

[ I (1 - mi/Mi) a;

=

for Z~ < t < Z~, (24)

i:Z[ <_t

-- -

0

=Undefined

l" f @* = l

fort>Z~,

if6,].=0

fort>_Z2,

where n*

Mi =-- ~ mj, j=i

mi

=-- a[ (1 + nl/nc)d*

if 6 i = 1, if 6; = 0 .

If we apply the Kalbfleisch and Lawless (1988) approach, 1 + nl/nc in (24) changes to p*. This is a consistent estimator of Æ(.). B(t) in (24) coincides with the Kaplan Meier (1958) estimator when nl = 0. This property can be proved in several ways (Suzuki, 1985a): one is by Peterson's representation (1977, Theorem 2.1), which relates the subsurvival functions to the survival function/~(.). Moreover, Gill's (1981, Eqs. 6 and 7) result can be applied, which permits both subsurvival functions to have jumps in common. The idea of self-consistency, introduced by Efron (1967), is applicable to this problem. Using Efron's definition, we can derive the self-consistent estimator and prove that it coincides with F(.) in (24). Therefore the estimator is described simply via redistribution to the right (Efron, 1967). Table 7 shows an example for calculating B(t) where N = 9, nu = 4, nc = 2 and nl = 3. Out of 9 data, three are missing. 4. Estimation using the usage time distribution

4.1. Estimation with date-of-sale information In the case of automobiles, the products under consideration may have different ages measured in calendar time because of the variety of dates of sale. In other

602

K. Suzuki, Md. R. Karim and L. Wang

Table 7 An example of caiculating/~(t)

965 2124 2412 3113 3570 5090

1 0 1 0 1 1

1 2.5 1 2.5 1 1

?

0

nù

9.

0

mi =

?

0

g1 (1) 5.7 (~)

9 8 5.5 4.5 2 1

-

1

1

-

!9

(1 -~)(1 -5@5) -

1 11

(1-})(1 -1)(1-½) 1 B 5@5-t- ½ lq-5@5 + ~1+ T 1 (1 - } ) ( 1 - 1 ) ( 1 -½)(1 - } )

4, ne = 2 , nl = 3,N = 9 {1, ni __

6"=1 *

6i

1+g--1+3=2.5,

0

words, each product has its own observational period from the date of sale to the last date of observation measured in calendar time. For example, if observation ends in the Sth month, the observational period of the censored products sold in the sth month is S - s months. The censoring distributions then will be different from each other for the different observational periods. Please refer to the data structure in Table 4. That is, the distribution functions of Y~'s can be considered to have S distinct forms; there are Ns variables whose distribution functions are Gs(y) and whose s Ns = N. Here, s and survival functions are Gs(y), where s = 0, 1 , . . ,S and ~,=õ Ns denote the month of sale and the sales amount, respectively, as in Table 1 and Gs(y) represents the survival distribution of actual usage time for products sold in the sth month. For example, it is the distribution of the mileage accumulated by a car from the month of its first use to the month it was last observed whether it failed or not. Therefore, when the date-of-sale information is available, the likelihood function, the log likelihood function, and the corresponding MLE are given by:

L* = IIf(zk) ~~I I 1 k=l

f(zk)Gs(zk)

,

(25)

s=1 K*

S

log L* = Z d k l o g k=l

[

K*

f(zk) + Z ( N ~ - ns)log [1 - ~f(zk)Gs(zk) s=0

k=l

(26) and

f*(zk)

= dk

{s~ 0

(Ns~ K/s){~s (Zk) 1 - £y*_lf*(zj)Gs(z«)J

~-1 '

k = 1,..,K*

(27)

Statistical analysis o f reliability warranty data

603

where dk denotes the number of failures at Zk and ns is the number of failures in products sold in the sth month as shown in Table 4. z~l in the table corresponds to certain of the zk in (25). These are given by Hu et al. (1998). Since the M L E in (27) is not given in a closed form, they propose the following moment estimate and showed that there are almost no differences between them;

f(Zk) =

S

4

-

(28)

Z

4.2. Estimation without date-of-sale information For the data structure of Table 3, it is not known which of Go(y), G1 (y), ..., Gs(y) is the distribution function of Y~because there are no records on dates of sale for the failed products. Wang and Suzuki (2001) proved that f(zk) in (28) is the MLE when the censoring variables Y/s are distributed according to Ge(y) with probability N~/N. That is, when the Y,.'s have the distribution function s N G(y) - Z N G ~ ( y )

.

(29)

x=0

If the estimate derived under the above condition has almost the same good property as that of the M L E (27) when the exact distribution of Yi is known, then the information "recorded months" in Table 4 is not necessary. It is enough to store only the actual operating time until failure in the database. This is shown below. Under the above condition, the z~l (l = 0, 1 , . . ,rs; s = 0, 1 , . . ,S) in Table 3 can be considered as an observational set from Zi. Thus, the likelihood function can be written as L =

I I f(xi) 1-1 P r ( X / > Y~-) , Xi <--Yi

Xi>Yi

and K*

PRO(/> Ic//) =

K*

~f(zk)G(zk--) = 1 -- ~-~~f(z,)4(Zk--) k=l

k=l

S

K*

s=0

k=l

Therefore we get the likelihood function

L

H f ( z k ) d* 1 k=l

Zf(zl)G,(z,-) ~

and the log-likelihood function

N

l=1

(30)

K. Suzuki, Md. R. Karim and L. Wang

604

logL=Zdklogf(zk)+(N-r)log

Ns ~ Zf(zt)5,(zl_

1-

k=l

s=0

)

/=1

From the following equations 81ogL

dk

5f(zk)

f(zÆ)

(N - r) ~,:oN~G~(zk)/N s -

1 --

E s =s 0

Ns E / =x*I f(z~)G~(zz-)/N

= 0,

k=l,...,K*

,

we obtain the MLE o f f ( z k ) as /(z~)

=

d~

(31)

S

~~:oN, Gs(zk - ) This is the same as the moment estimator in Section 4.1. Furthermore, the MLE of F(x) can be written as

«k

(32)

Zk ~ N

The asymptotic variance of the estimator (32) is given by Avar(/~(Zk)) = ~

k

f(zi)

~ k

- E

k

Z

i=1 j = l

S

-

E,=o NsGs(z,-)G~(zj-)f(zi)f(zj) s NEs=0 sGs(zi-)ESs-oNsGs(zJ - )

(33)

For details, refer to Wang and Suzuki (2001). The estimator (31) is the same as the moment estimator (28) proposed by Hu et al. (1998) when the dates of sale are available. In this case, Hu et al. (1998) and Wang and Suzuki (2001) show that the moment estimator is almost the same as the corresponding M L E based on simulations. This means that the two MLEs derived from the two types of claim data, those with and those without information on dates of sale have almost the same precision. This indicates that the records of dates of sale are not so important for lifetime estimation when lifetime is measured in actual operating time. Therefore, the records of the actual operating times are sufficient for such a case.

5. R e f i n e m e n t

5.1. Extension to the case of competingfailure rnodes Next, we discuss the case of independent competing failure modes which occurs for many industrial products. Let Fu(x) be the distribution function of the failure

605

Statistical analysis of reliability warranty data

time variable Xù for failure mode u, u = a, b , . . . , U. Consider the competing risk model; Zi = min{X~i,Xbi, . . . ,Xvi, !4,}, i = 1 , . . ,N, {u ifZi=22ù, u = a , b , . . . , U (34) Õi = 0 if Z/ = Y,. , where the Y~'s are similar to the variable sequence defined in Section 4. For the case of two failure modes, based on claim data for which the dates of sale are known, the M L E F*, which has a non-closed form, is derived as follows log L* = Z

«~a{log fa(zk) + log Fb(Z~)}

s=0 K*

+ Z

dbb{1og fb(Zk) + log Æ~(Zk)}

k=l

+(Ns - n,)log

G(zk)Æb(z~)9,(zk)

+ const .

(35)

4k

f2(z~) =

~ [Nx

k i -, (Ns ns) ~j=oF£(z+)#,(zj)

k-1 db/s . ]

~s=O

F2 (x) = Z f ~

(36)

(Zk)

Zk <X

where daks and dbks are, respectively, the numbers of failures of failure modes "a" s and "b" at time zk out of the products sold in the sth month; dak ~s=0 da~, S K* dbk = ~~=odb~s and ns = ~k=l(d~k~ + db~). Similarly, fd(zk) and Fb*(zk) are obtained by interchanging the subscripts " a " and "b". 5.2. Estimation without date-of-sale information If there is no information on the covariates of Y/, assume the Y/s have the same s Ns~~ , Define the subdistribution functions as survival function G(y) = m 22s=0 ~ sLv). Fsub u(Z) = Pr(Zi < z, ö~ = u)

= =

i zPr(Y/_> x,X~ _> x, v =

a, b , . . . , U and v ¢ u)dFù(x)

70~G ( « - ) . fi~~/x-)dFu(x),

u = a, b , . . . , U .

(37)

v=a v¢u

From the above equations, we have Fu(x)=

.~0x G ( z -~d-F-sub-u ) -(z~)

)11(T&( ~":'~'z-"

u=a,b,...,U

.

(38)

606

K. Suzuki, Md. R. Karim and L. Wang

Let dùk be the number of failures of type u at time point zk, where u = a, b , . . . , U, zk-1 < zk, k = 1 , 2 , .. ,K*. It is known that the non-parametric MLEs of Fsub-u(Z) are 1 Bsub«,(Z)=~~'~duk, u=a,b,...,U. (39) Zk ~ Z

Let/?~(x) denote the non-parametric MLE ofFù(x). Then the following equations hold by the invariance property of the MLE.

P~(x)

:

fx

Jo

dFs~b u(z)

u ~o G(z-).I~:;;F~(z-

)

,

ù = a,b,...,

Calculating the Lebesgue-Stieltjes integral Bu(x) =/~ù(z«_i) for zk_l < _ x < z k , u = a , b , . . . , U ,

of k=

(40)

U .

(40) and noticing that 1 , 2 , . . , K * , we have the

MLE of Fu(x) dùk F~(x)=Z

~oF,, z s , zk<xI~ ù v( k-I)" ~ s = o N s G s ( z ~ - )

u=a,b,...,U

.

(41)

The properties of this estimator are discussed in Wang and Suzuki (2001).

6. Conclusion Although complete field performance data may not be possible to get, surrogates for some field performance data are collected when a product fails and a customer files a claim under the product's warranty. In this report we have pointed out why field performance data are important and we have given a survey of the literature pertaining to the analysis of warranty claim data. Testing whether two kinds of products have different mean lifetimes can be done using only the information of the failed items; however, without a supplemental data such as on items that do not fail (censored data) or on a useage time distribution at a given age, the estimation of a product's lifetime distribution and/ or other associated parameters is difficult. More work on this problem is needed. The development of the world-wide internet promises to make the collection of both primary field performance data and warranty claim data much easier and less expensive. For this reason, careful work needs to be done to identify which data of each type it is most important to collect.

Acknowledgements The authors thank Professor N. Balakrishnan who gave t h e m a chance to write this paper. They are also grateful to Dr. J. W. Halpern and Dr. W. Yamamoto for their careful reading of this manuscript and helpful suggestions.

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References Amster, S. J , G. G. Brush and B. Saperstein (1982). Planning and conducting fleld-tracking studies. Bell Syst. Techn. J. 61, 2333 2364. Ascher, H. and H. Feingold (1984). Repairable Systems Reliability. Marcel Dekker, New York. Balcer, Y. and I. Sahin (1986). Replacement costs under warranty: cost moments and time variability. Oper. Res. 34, 554~559. Baxter, L. A. (1994). Estimation from quasi life tables. Biometrika 81, 3, 567-577. Blischke, W. R. and D. N. P. Murthy (Eds.) (1996). Product Warranty Handbook. Marcel Dekker, New York. Blischke, W. R. and E. M. Scheuer (1975). Calculation of the cost of warranty policies as a function of estimated life distributions. Naval Res. Logist. Quart. 22, 681 696. Blischke, W. R. and E. M. Scheuer (1981). Applications of renewal theory in analysis of the free-replacement warranty. Naval Res. Logist. Quart. 28, 193-205. Breslow, N. E. and J. Crowley (1974). A large sample study of the life table and product limit estimates under random censorship. Ann. Stat. 2, 437-453. Chen, J., N. J. Lynn and N. D. Singpurwalla (1996). Forecasting warranty claims. In Product Warranty HandbooÆ, Chapter 31 (Eds. W. R. Blischke and D. N. P. Murthy). Marcel Dekker, New York. Chukova, S. and B. Dimitrov (1996). Warranty analysis for complex systems. In Product Warranty Handbook, Chapter 22 (Eds. W. R. Blischke and D. N. P. Murthy). Marcel Dekker, New York. Dempster, A. P., N. M. Laird and D. B. Rubin (1977). Maximum Iikelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Stat. Soc. B 39, 1-38. Efron, B. (1967). The two sample problem with censored data. In Proceedings of the Fifth Berkeley Symposium, Vol. 4, pp. 831 853. Escobar, L. A. and W. Q. Meeker (1999). Statistical prediction based on censored life data. Technometrics 41, 113-124. Frees, E. W. (1986). Warranty analysis and renewal function estimation. Naval Res. Logist. Quart. 33, 361 372. Gill, R. D. (1981). Testing with replacement and the product limit estimator. Ann. Stat. 9, 853-860. Hu, X. J. and J. F. Lawless (1996a). Estimation from truncated lifetime data with supplementary information on covariates and censoring times. Biometrika 83, 747-761. Hu, X. J. and J. F. Lawless (1996b). Estimation of rate and mean functions from truncated recurrent event data. J. Am. Star. Assoc. 91, 300-310. Hu, X. J. and J. F. Lawless (1997). Pseudolikelihood estimation in a class of problems with responsereiated missing covariates. Can. J. Stat. 25, 125-142. Hu, X. J., J. F. Lawless and K. Suzuki (1998). Nonparametric estimation of a lifetime distribution when censoring times are missing. Technometrics 40, 3-13. Kalbfleisch, J. D. and J. F. Lawless (1988). Estimation of reliability from field performance studies (with discussion). Technometries 30, 365-388. Kalbfleisch, J. D. and J. F. Lawless (1991). Regression models for right truncated data with applications to AIDS incubation times and reporting lags. Statistica Sinica 1, 19 32. Kalbfleisch, J. D. and J. F. Lawless (1996). Statistical analysis of warranty claims data. In Product Warranty Handbook, Chapter 9 (Eds. W. R. Blischke and D. N. P. Murthy). Marcel Dekker, New York. Kalbfleisch, J. D., J. F. Lawless and J. A. Robinson (1991), Methods for the analysis and prediction of warranty claims. Technometrics 33, 273-285. Kaplan, E. L. and P. Meier (1958). Nonparametric estimation from incomplete observations. J. Am. Stat. Assoc. 53, 457-481. Karim, M. R., W. Yamamoto and K. Suzuki (200la). Statistical analysis of marginal count failure data. Lifetime Data Anal. 7 (in print). Karim, M. R., W. Yamamoto and K. Suzuki (2001b). Change-point detection from marginal count failure data. J. Jpn. Soe. Qual. Control 31 (in print).

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Kiefer, J. and J. Wolfowitz (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Stat. 27, 887-906. Lagakos S. W., L. M. Barraj and V. De Gruttola (1988). Nonparametric analysis of truncated survival data, with application to AIDS. Biometrika 75, 515-523. Lawless, J. F. (1994). Adjustments for reporting delays and the prediction of occurred but not reported events. Can. J. Stat. 22(1), 15-31. Lawless, J. F. (1998). Statistical analysis of product warranty data. Int. Stat. Rer. 66(1), 41-60. Lawless, J. F. and J. D. Kalbfleisch (1992). Some issues in the collection and analysis of fleld reliability data. In Survival Analysis: State of the Art, pp. 141-152 (Eds. J. P. Klein and P. K. Goel). Kluwer Academic Publishers, Dordrecht. Lawless, J. F. and J. C. Nadeau (1995). Some simple robust methods for the analysis of recurrent events. Technometrics 37, 158-168. Lawless, J. F., X. J. Hu and J. Cao (1995). Methods for the estimation of failure distributions and rates from automobile warranty data. Lijetime Data Anal. 1, 227-240. Mamer, J. W. (1982). Cost analysis of pro-rata and free-replacement warranties. Naval Res. Logist. 29, 345-356. Mamer, J. W. (1987). Discounted and per unit costs of product warranty. Management Sci. 33(7), 916-930. Matthews, S. and J. Moore (1987). Monopoly provision of quality and warranties: An exploration in the theory of mnltidimensional screening. Econometrica 55, 441-467. Miller, R. G. (1981). Survival Analysis. Wiley, New York. Miyagawa, M. (1982). Statistical analysis of incomplete data in competing risks model. J. Japanese Soc. Qual. Contr. 12, 23-29 (in Japanese). Moskowitz, H. and Y. H. Chun (1994). A Poisson regression model for two-attribute warranty policies. Naval Res. Logist. 41, 355-376. Nguyen, D. G., and D. N. P. Murthy (1984a). A general model for estimating warranty costs for repairable products. HE Trans. 16, 379-386. Nguyen, D. G. and D. N. P. Murthy (i984b). Cost analysis of warranty policies. Naval Res. Logist. Quart. 31, 525-541. Nguyen, D. G. and D. N. P. Murthy (1988). Failure free warranty policies for non-repairable products: a review and some extensions. Oper. Res. 22(2), 205-220. Peterson, A. V. (1977). Expressing the Kaplan Meier estimator as a function of empiricai subsurvival functions. J. Am. Stat. Assoc. 72, 854-858. Phillips, M. J. and T. J. Sweeting (1996). Estimation for censored exponentiaI data when the censoring times are subject to error. J. Roy. Stat. Soc. B 58, 775-783. Robinson, J. A. and G. C. McDonald (1991). Issues related to field reliability and warranty data. In Data Quality Control: Theory and Pragmatics (Eds. G. E. Liepins and V. R. R. Uppuluri). Marcel Dekker, New York. Sahin, I. and H. Polatogu (1998). Quality, Warranty and Preventive Maintenace. Kluwer Academic Publishers, Boston. Singpurwalla, N. D. and S. P. Wilson (1998). Failure models indexed by two scales. Adv. Appl. Prob. 30, 1058 1072. Suzuki, K. (1985a). Nonparametric estimation of lifetime distribution from a record of failures and follow-ups. J. Am. Stat. Assoc. 80, 68-72. Suzuki, K. (1985b). Estimation of lifetime parameters from incomplete fleld data. Technometries 27, 263-271. Suzuki, K. (1987). Analysis of field failure data from a nonhomogeneous Poisson process. Rep. Star. Appl. Res. 43, 6-15. Suzuki, K. (1993). Estimation of usage lifetime distribution from calendar lifetime data. Rep. Star. Appl. Res. 40, 10-22. Tortorella, M. (1996). Lifetime data: models in reliability and survival analysis. In Life estimationfrom pooled discrete renewal counts, pp. 331-338 (Eds. N. P. Jewell et al.). Kluwer Academic Publishers, Dordrecht.

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Trindade, D. C. and L. D. Haugh (1980). Estimation of the reliability of computer components from field renewal data. Microelec. Reliab. 20, 205~18. Vintr, Z. (1999). Optimization of reliability requirements from manufacturer's point of view. In Proceedings Annual Reliability and Maintainability Symposium, pp. 183-189. Wang, L. and K. Suzuki (2001). Lifetime estimation based on warranty data without data-of-sale information cases where usage time distributions are known. J. Jpn. Soc. Qual. Contro131, 148 167. Yun, K. W. and F. E. Kalivoda (1977). A model for an estimation of the product warranty return rate. In Proceedings Annual Reliability and Maintainability Symposium, pp. 31 37. Zacks, S. (1971). The Theory ofStatistical Inference. Wiley, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook ofStatistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

~)~) z~z~

Prediction of Field Reliability of Units, Each under Differing Dynamic Stresses, from Accelerated Test Data

Wayne Nelson

The life of most products depends on the dynamic stresses they experience in service. This paper describes general statistical models and data analysis methods that use accelerated test data to predict the population reliability in service where each unit is under different stress profiles over time.

1. Introduetion

Purpose. A client sought a method for using accelerated test data from a stepstress test to predict field reliability of seals in brake cylinders. In service, seals in different cars experience different profiles of pressure and temperature over time. The client wanted an estimate of the life distribution of such seals in service, in particular, the fraction failing on warranty. This paper presents models and statistical methods for this problem, mentioned by Nelson (1990, p. 509). For the first time in this paper, a cumulative damage model is used both to analyze data from an accelerated test with time-varying stresses and to predict the field life of units each subjected to different time-varying stresses.

Overview. The organization of this paper is: • Section 2 presents a general accelerated life test model for constant-stress testing of units. • Section 3 describes a cumulative damage/exposure model for the life of units under varying stress in an accelerated test or in actual use. • Section 4 shows how to fit the model to accelerated life test data from a step-stress test. • Section 5 shows how to use the exposure model and the time-varying field stresses on different units to predict the life distribution of the population in service. • Section 6 outlines further work. 611

w. Nelson

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Background. To read this paper, the reader needs to be acquainted with accelerated life test models in Nelson (1990, Chapter 2) and with the basic cumulativeexposure model in Nelson (1990, Chapter 10).

2. Constant-stress model

Overview. This section presents a typical constant-stress model for accelerated life testing. In Section 3, this model is combined with a cumulative-exposure model to yield a model for life data from an accelerated step-stress test. The constant-stress model consists of (1) a life distribution and (2) an engineering relationship between typical life and the accelerating variables. The Weibull distribution and the Eyring relationship are used here for concreteness and simplicity. The results below readily extend to models with other distributions and relationships. Nelson (1990, Chapter 2) presents a number of such models. For clarity, assumptions are marked 3 . Distribution. For concreteness, we use the familiar ~ Weibull life distribution. Its cumulative distribution (cdf) for the population fraction failed by age t > 0 is

F(t) = 1 - e x p [ - ( t / ~ ) / ~ ] .

(1)

Here "time" t is any appropriate measure of usage, such as hours, cycles, miles, etc. The positive parameter ~ is called the characteristic life; it is the scale parameter, is always the 63.2th percentile, and has the same dimensions as time t. The positive parameter/~ is called the shape or slope parameter. Other distributions with a scale parameter can be used, for example, the lognormal distribution whose scale parameter is the distribution median.

Relationship. For concreteness, we use the ~ generalized Eyring relationship without an interaction term. It models the characteristic life c~ of seals as a function of constant pressure X and constant reciprocal temperature Ig = 1/T; namely, e(X, Y) = expl70 + 71X + y2Y] .

(2)

The model parameters 7o, 71, 72, and/~ have unknown values characteristic of the product; they usually are estimated from accelerated test data, but estimates may come from other sources. In (2), « is an Arrhenius function of absolute temperature T = 1/Y. Also, ln(X) could be used in place of tl, if physically appropriate; then e is an inverse power function of pressure X. For a lognormal life distribution, (2) is the equation for the distribution median. The combined model (1) and (2) is called the Weibull-Eyring model.

Variables. The above relationship for seals has two accelerating variables (also called "stresses") pressure X and reciprocal absolute temperature Y = 1/T.

Prediction of field reliability of units

613

Other applications may require more variables and a more complex relationship. For example, for the seals, it may be appropriate to include a thermal cycling variable if such cycling produces material fatigue. The ~ accelerating variables in the relationship taust be the stress variables in service. Also, the relationship may contain other engineering variables, such as product size and material and process variables. For example, Nelson (1990, p. 507) includes size in such a relationship. Neededmodels. The Eyring-Weibull model above (relationship and distribution) is

a simple, versatile one that is useful in many applications. Other applications will require other models. A major difficulty in applying the methods here to many products is a lack of a suitable physical relationship like (2). In some applications, even the appropriate stresses are not fully known. Engineers and physical scientists must first develop such physical models for their products in order to apply the methods here. The following methods are general and apply to most such models. Shape. In this model, the ~ shape parameter fl has a constant value. That is, fl does not depend on the accelerating variables X and Y. Dependence of fl on the accelerating variables cannot readily be incorporated into the cumulative-exposure theory below. However, ~ /3 can be a function of other non-accelerating engineering variables, such as product size, as described by Nelson (1990, p. 507). Equivalently, for a lognormal life distribution, ~r is constant. The lognormal distribution is often used to model fatigue life of metals; however, for metal fatigue, a is not constant but depends on the mechanical stress. Failure modes. Strictly speaking, ~ the model above is for a single failure mode.

Each product failure mode requires a different model with different parameter values and possibly different accelerating variables. The model parameters for a mode must be estimated from data on that mode. Then such models for the various modes are combined, using the series-system model, to obtain a model for the product with all failure modes operating, as described by Nelson (1990, Chapter 7). In practice, sometimes the simple model above is used as a first approximation to product life with all failure modes operating. For simplicity, the complication of competing modes is ignored here. 3. A cumulative-exposure model for varying stresses Purpose. This section briefly presents a cumulative-exposure model that appears in detail in Nelson (1990, Chapter 10). Such models are also called cumulativedamage models. This section first describes how the product stresses can vary over time in an accelerated test or in service. Next it presents the simple cumulativeexposure model used here. Then it derives the cumulative exposure for step-stress testing. Finally, it references other such models. Stress variation. The following general description of time-varying stress applies to step-stress testing and field experience. For example, test seal m is subjected to

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a known time varying pressure Xm(t) and reciprocal temperature Ym(t). Figure 1 depicts such a step-stress profile for one stress. Each test unit may have different known X and Y stress profiles (or functions) over time; this is indicated by the subscript m, where m = 1 , 2 , . . ,M, the total number of test units. Such stress functions can be constant, cyclic, steps, ramps, non-repeating, stochastic, or whatever. Singpurwalla (1995) reviews some stochastic and shock models for stress. In service, stress profiles are usually non-repeating, as depicted in Figure 2.

Exposure model. We use the ~ Weibull life distribution (1). Then if ~ the whole population of N units were each subjected to the same time-varying stress profiles [Xn(t), Yn(t)], the resulting population cdf would be Fn(t) = 1 - exp{-[e~(t)] ~} .

(3)

Here the "cumulative exposure" an(t) up to time t replaces (t/c~) in (1) and is modeled as the integral Stress X(t) X4 X3 X2

X l - -

0 0

tl

t2

t3

time t

Fig. 1. Step-stress profile.

x(t)

I? o

o

fimet

Fig. 2. Service stress profile and stabile stress distribution.

Prediction of field reliability of units

ex(t) =

~0t dt/~[Xn(t),Yù(t)]

615

(4)

.

Here c~[X~(t),Y~,(t)] is the "instantaneous" scale parameter as a time function of Xn(t) and Yn(t). This exposure is the cumulated time over the corresponding characteristic life. For example, for the Eyring relationship, the exposure integral is ~n(t) =

expl-70 - 71Xn(t) - 72Y,(t)]dt .

(5)

In practice, it may not be possible to evaluate the integral in a closed form. Then numerical quadrature is necessary. Nelson (1990, Chapter 10) applies this cumulative-exposure model to endurance of electrical insulating oil and cable insulation. Step stress. For a step-stress profile (Figure 1), the exposure follows. Suppose that, for a test unit, step i has stress levels (xi,yi) over the time interval (ti 1, ti), i = 1,2, 3 , . . , where to = 0. Ler 0~i denote the characteristic life (2) at the pair of stress levels (x;,y;); cq is also a function of the model parameters. Then the cumulative exposure for the test unit at time t in t h e / t h step is 4t) = E(t~ - o)/o~~1 + [(t~ - t~)/~~l +...

+ E(t- ti_,)/~~]

.

This is a piecewise linear function of t. Here the test unit/profile subscript m is omitted to simplify notation. Other models. The model (3) and (4) is a generalization of Miner's (1945) rule, used for metal fatigue under varying mechanical stress. This model corresponds to "linear damage" and has no sequence effect, as described by Nelson (1990, p. 503). Nelson (1990, Chapter 10) applied the model to endurance of electrical insulating oil and cable insulation. Singpurwalla (1995) surveys a variety of cumulative-exposure models. Bogdanoff and Kozin (1984) present their cumulativeexposure model for metal fatigue. Such models are not well developed and have rarely been confirmed in applications. Thus any such mode1 needs to be carefully assessed for its suitability in a particular application.

4. Model fitting Overview. This section briefiy outlines maximum likelihood (ML) fitting of the preceding model to accelerated life test data from a step-stress test. Needed background for this section is familiarity with M L methods. See Nelson (1990, Chapters 5 and 10) for details on M L fitting to data from step-stress, constantstress, ramp, and other stress patterns. This section first presents the sample likelihood for a step-stress test. Next it describes how to use the likelihood to

W. Nelson

616

obtain ML estimates of the mode1 parameters and corresponding confidence limits. Finally, it extends the methods to field data. LikeIihood. We taust have ~ a random sample of M units from the population. If sample unit m ran unfailed to time tm, its log-likelihood is the log of the snrvival probability 1 -Fm(tm) for Weibull-Eyring model in (1) and (2); namely, Lm(tm; 70, 71,72, il) = -[em(tm)l ~ •

(6)

If sample unit m failed at time tm, its likelihood for the Weibull Eyring model is the probability density fm(tm) = dFm(tm)/dtm = [dem(tm)/dtm]il[em(tm)] e

l exp{--[em(tm)]e}

Differentiating (4) yields dgm(tm)/dtm = 1/Œ[Xm(t),Ym(t)J. Thus the log-likelihood for unit m failed at time tm is

Lm(tm; 70, 71,72, fl) = - ln{°~[Xm(t),Ym(t)]} + i n ( i / ) + (il - 1)in[gin(im)]

-

[gm(lm)] fi •

(7)

The sample log-likelihood for all M sample units is sum of the unit log-likelihoods; namely,

L(tl,t2,

..,tm;70,71,72,fl)

= Z L m ( t m ; ~ 0 , 7 1 , 7 2 , il) •

(8)

m

M L estimates. The ML estimates 70, Y1,72, il are the parameter values that maximize the sample log-likelihood (8). They may be found by directly numerically maximizing (8) with respect to the parameters. Alternatively, they may be found, using calculus, as the parameter values that satisfy the simultaneous "likelihood equations" ÔL/Ôy o = O, ÖL~eT1 = O, ÖL~eT2 = O, ÖL/eß = 0 .

(9)

For most models and data, the ML estimates of the parameters are unique and have good statistical properties. Nelson (1990, Chapter 5) describes the calculation of ML estimates. Confidence limits. Nelson (1990, Chapters 5 and 8) describe in detail the calculation of ~ normal-approximation confidence limits for the model parameters and functions of them. This first requires the calculation of the sample Fisher information matrix of negative second partial derivatives of L with respect to the parameters. Difficult to calculate analytically, these derivatives can be numerically approximated with second differences obtained by numerically perturbing the sample likelihood about the maximum with respect to the model parameters. This sample information matrix is inverted to obtain an estimate of the covariance

Prediction offield reliability of units

617

matrix of the M L estimates of the model parameters. The covariance matrix yields standard errors of the estimates, which are then used to obtain the approximate confidence limits, as described by Nelson (1990). Nelson (1990, Chapter 5, Section 5.8) also describes the calculation of more accurate likelihood ratio confidence limits.

Field data. The theory above for accelerated test data also applies to fitting the model to field data. Then the field data taust include the observed stress functions Xn(t) and Yn(t) for each sample unit up to its currently observed failure or running time th. In practice, it is necessary to measure these sample stress functions.

5. Estimate population reliability Overview. This section shows how to estimate population reliability in service, in particular, the population cdf Fo(t). This function can then be used to estimate other quantities of interest, such as the population hazard function or mean life. In the application, the client was most interested in estimating the population fraction failing Fo(t t) by warranty age t ~. The section first discusses population stress profiles. Next it defines the true population Fo(t). Then it presents the cdf estimate and confidence limits for it. It discusses other sources of estimate uncertainty, prediction, and subpopulations. The section then presents simplified exposure calculations involving a stabile stress distribution for each unit in service. It concludes with estimation of other quantities of interest. Population stresses. To define Fo(t), we use the in-service stress functions [Xn(t),Yn(t)] and the corresponding cdf Fn(t) from (3) of each population unit n. Figure 2 depicts such an in-service stress function (say, pressure) for a seal in car; it shows brakings which vary with respect length and severity. Such stress functions could reflect seasonal variation in temperature, geographical location, and other factors. In practice, we may use just a sample of units and their stress functions, or we may somehow estimate all the population stress functions.

Population cdf The true population cdf for all N population units is Fo(t) = [Fl(t)--F2(t) + ' " 4 - F N ( t ) I / N .

(10)

This average of the N cdf functions is equivalent to saying that ~ the N units are randomly assigned to the N stress profiles. That is, the seals are randomly assigned to the cars in the population, where each car is operated differently.

Cdf estimate. For the stress-unit pair n, we use its known stress functions [Xn(t),Yn(t)] and (3) to evaluate Fn(t), which contains the unknown true parameter values 7o, 71, 72, /L We replace them with their estimates 7;,7~,7~,/3" from the accelerated test and denote the resulting cdf estimate by F~*(t) = Fn (t; 7;, 7~, ~~,/~*)Substitute these unit cdf estimates into (10) to get the estimate

618

W. Nelson

Fö(t ) = [FI*(«) +F~(t) + . . . +F~~r(t)]/N .

(11)

Evaluated at the warranty age t', (11) provides an estimate of the population fraction failing on warranty. In practice, if we have ~ a random sample of just N' stress profiles, we evaluate the corresponding N' cdf estimates F ~ ( t ) , F ] ( t ) , . . , F~, (t). Then the alternate estimate of the population cdf is Fö(t ) = [Fi*(t) +F](t) + . . . +F~v,(t)]/N' .

(12)

Confidence limits. The estimate Fó*(t) in (11) or (12) is usually based on a small number of sample units from an accelerated test, and the estimate usually involves extrapolating from high test stress conditions to low service stress conditions. Consequently, F ö (t) may have rauch statistical uncertainty, and confidence limits are important so one can judge its accuracy. Such confidence limits are easy to calculate, if the Fö(t ) employs (11) with all N known stress profiles, and if the ML estimates 7~, 71, 7~,/~* are based on a small number of sample units compared to N. Then the normal-approximation confidence limits for Fö(t ) = F0(t; 7~, 7~, 72,/~ ) are obtained using the usual propagation of error technique (Taylor series) with respect to 7;, 71, 7*2~ /?* and their covariance matrix, as described by Nelson (1990, Chapter 5, Sections 5.6 and 5.7). The needed derivatives of Fó*(t) with respect to 7~, 7~, 7~,/3* can be approximated with a perturbation calculation. If Fõ(t) is (12), it employs a small number N' of sample profiles, and the calculation of confidence limits is more complicated. Further uncertainty. The confidence limits above reflect only the random sampling uncertainty due to the small number of units used to estimate the model parameters from accelerated test data. In practice, the following additional sources of uncertainty in the estimate (11) or (12) may be important. • The estimate employs a small number N / of sample stress profiles rather than all N. • The stress profiles are known only approximately, perhaps due to measurement error. • The model distribution and relationship are in error to some degree. The uncertainties contributed to the estimate (11) or (12) by these sources are difficult to quantify.

Prediction. The preceding confidence limits are appropriate only if the number M of specimens in the accelerated test is small, say, less than 10% of the population size. If field data from more than 10% of the population were used to estimate the model, the confidence limits above would be incorrect. Then one has a onesample prediction problem and needs corresponding prediction limits; Nelson (1982, p. 257) gives an example of one-sample prediction limits. Subpopulations. The methods above can be applied to any portion of the population, for example, units in a certain production period, units in a certain

Prediction offield reliability of units

619

geographic area, etc. Comparison of such subpopulations can lead to engineering and management insights and product improvements. If the population is partitioned into non-overlapping subpopulations, the estimates from the subpopulations can be combined to yield an estimate for the entire population, using statistical methods for stratified populations.

Simplified profiles. In some applications, the stress profile IXn(t), Y~(t)] of unit n in service may be simplified as follows. Imagine that the entire time fange (perhaps to infinity) is partitioned into many equal time intervals so short that the stress levels of Xn(t) and Yn(t) are essentially constant over each interval. For example, for the seals, the intervals could be one second long. Then there is a joint distribution of stress levels with density, say, gù (x,y). Moreover, suppose, over any period of time suitably smaller than the spread in the life distribution Fn(t), that the joint stress distribution On(x,y) is essentially ~ stabile. Then the cumulativeexposure integral (4) simplifies to

en(t) = t H [1/~(x,y)]gù(x,y)dx dy .

(13)

Here the double integral runs over the joint range of the stresses x and y. Let

c~n-1/{~[1/o~(x,y)]gn(x,y)dx d y } .

(14)

(This is Miner's rule when the population model is lognormal and an is the lognormal median.) For the Weibull model, then ~n(t) = t / ~ ~

.

(15)

Substituting this cumulative exposure into (3), we find that the population cdf

Fn(t), when all units are under the same pair of stress profiles [Xn(t),Yn(t)], is Weibull with characteristic life c~nfrom (14) and the same shape parameter value B. When the population units are under different stress profiles, the population cdf is the mixture (10) of these Weibull unit cdfs.

Stress distribution. The stabile stress distribution gn(x,y) for unit n in service can be estimated nonparametrically. Also, it can be modeled with a parametric distribution, and the distribution parameters are estimated from observed stress data from the unit in service. Then the population can be regarded as a mixture of such distributions, each with different parameter values. The statistical uncertainty in these parameter estimates contribute to the uncertainty in the estimate F ö (t) and should be taken into account in calculating its confidence limits. Discrete stress distribution. In some applications, the distribution of stress is approximated with a ~ discrete distribution with K stress levels. For example, for the seals application, unit n has a stabile stress distribution that is approximated

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with K pairs of stress levels (xl,yl), ( x 2 , y 2 ) , . . , (XK,YK) applied corresponding fractions of the time 9 1 , 9 2 , . . , 9K and with characteristic lives el, e 2 , • •, C¢K.Then the characteristic life for that unit is ~~ = 1/[(g~/cq) + (g2/c~2) + ' "

+ (gK/C¢K)l •

(16)

Thus unit n has a Weibull life distribution with this ~n and the same/3. Other estimates. The preceding methods yield estimates and confidence limits for the model parameters and Fo(t). Those estimates can be used to obtain an estimate and confidence limits for most any quantity of interest. Examples include the population mean and hazard function, which is offen ofinterest in engineering work. Lognormal. The preceding results for the Weibull distribution extend readily to the lognormal distribution. Corresponding formulas for the lognormal distribution are given here. The lognormal cumulative distribution function (cdf) for the population fraction failed by age t > 0 is F(t) = ~ { [ l n ( t / m ) J / a }

.

(1')

Here ~{ } is the standard normal cdf, m is the population median, and a is the log standard deviation. The median m(x,y) is ~ modeled as a function of the accelerating variables x and y, and ~ a is constant. The corresponding cumulative exposure for unit n subjected to the time-varying stresses [X,(t),Y,(t)l is en(t) =

/0 '

dt/m[Xn(t),Yn(t)]

.

(4')

The corresponding cdf for unit n is Fn(t) = 4~{ln[eù(t)l/a ) .

(3')

Suppose we approximate the stabile stress distribution of unit n with K pairs of stress levels (xl,yi), ( x 2 , y 2 ) , . . , (xK,y~;) applied corresponding fractions of the time 9 1 , 9 2 , . . , 9K and with median lives rnl, m 2 , . . , mK. Then the median life for unit n is m n = 1 / [ ( g l / m l ) + (92/m2) + ' " + (gK/mK)] • (16') Miner's rule for metal fatigue usually is given in this form. The corresponding cdf of unit n is F» (t) = q){ln[t/mù]/a}

.

(3")

6. Further work

Overview. This section outlines needed further work on reliability prediction of such populations. Such work includes a computer program, better models, and good test plans.

Prediction of field reliability of units

621

Computer program. The preceding theory needs to be implemented in a simple computer program suitable for engineers. The program would provide estimates and confidence limits for the population cdf and other quantities of interest, for example, the mean life. The program would include the Weibull, lognormal, and other suitable distributions and a variety of engineering relationships. Developing such a program would require a team of engineers, statisticians, and programmers who are analytically inclined. Such a program has long been needed and would be a major undertaking. Hopefully this paper will stimulate its development. It is widely needed. Better models. The statistical theory above is quite general. It applies to many

existing models. But the methods are only as good as the physical models. For the seals, the models above are reasonable and potentially useful, but other distributions, relationships, and cumulative-exposure models may be bettet. Those who wish to use the methods here need suitable models. They may need to search their physical literature for them or may have to invent and assess them. Nelson (1990, 2002) describes residuals and other means of statistically assessing proposed models fitted to data. Test plans. The client used step-stress testing, because it quickly yields failures and estimates. Bettet accelerated test plans can yield more accurate estimates of quantities of interest, such as the population fraction failing on warranty. Such plans depend on the model, the quantity being estimated, test constraints, costs, etc. Plans could be evaluated using analytic theory or simulation, as described by Nelson (1990, Chapter 6, Sections 5 and 6). Simulation would require the computer program for model fitting mentioned above.

Acknowledgments The author wishes to acknowledge Mr. Harry Rudy of G M Delco Moraine N D H Dir., who posed the problem and supported the initial documentation of the methods here. The author is grateful to Prof. N. "Bala" Balakrishnan for his invitation and encouragement to expand and contribute this material to this book.

References Bogdanoff,J. L. and F. Kozin (1984). ProbabilisticModels ofCumulative Damage. Wiley,New York. Miner, M. A. (1945). Cumulative damage in fatigue. J. Appl. Meeh. 12, A159 A164. Nelson, W. (1982). Applied Life Data Analysis. Wiley, New York. Nelson, W. (1990).Accelerated Testing: Statistical Models, Test Plans, and Data Analyses. Wiley,New York, Order Dept. (800)879-4539. Nelson, W. (2002). Definition and Analysis of Residuals from Accelerated Tests with Varying Stress. (in preparation). Singpurwalla, N. D. (1995). Smwivalin dynamic environments. Stat. Sei. 10, 86-103.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statisti«s, Vol. 20 © 2001 Elsevier Science B.V. All rights reserve&

t") 2

z~ ùJ

Step-Stress Accelerated Life Test

E. Gouno and N. Balakrishnan

This work is an overview of step-stress accelerated lifetime tests. In Section 2, we review different methodologies and terminologies for step-stress testing. The notion of acceleration model is presented in Section 3. The cumulative exposure model and other methods to derive the lifetime distribution under step-stress pattern are presented in Section 4. Sections 5 7 are devoted to the problem of inference; specifically, m a x i m u m likelihood, non-parametric and Bayesian approaches are described. Section 8 addresses the question of optimal plan.

1. Introduetion

Accelerated life test is a popular experimental strategy to obtain information on life distribution of highly reliable product. The main idea in it is to submit material to higher-than-usual environmental conditions inducing failures in a shorter time. D a t a thus obtained need to be extrapolated to estimate lifetime distribution under normal conditions. The environmental conditions (stress) which are considered here to be controlled can be a single stress or a combination of stresses. Classical stress are temperature, voltage, currents, pressure, cycling rate or load. The tests can be conducted using constant stress, step-stress or linearly varying stress. We focus here on step-stress which corresponds to the situation when stress changes at pre-specified times or after pre-specified number of failures. The main advantage of this test strategy is that, while making the stress increase or decrease, reasonable numbers of failures are ensured to occur. It can further reduce test time and the variability of the failure times. However, a step-stress does not mimic actual conditions; most products tun at constant stress. The step-stress test aims to assess the lifetime under constant used-condition and the model must take into account the effect of accumulation of successive stress exposure. This fact leads to a sophisticated and more complex model than in the case of constant stress.

623

E. Gouno and N. BaNkrishnan

624

2. Step-stress testing: Definitions A simple step-stress test, also called partially accelerated life test, is a test in which we have only one change o f stress. W h e n several stress change-times are considered, we have a multiple step-stress test. A simple step-stress is, therefore, a p a r t i c u l a r case o f a m u l t i p l e step-stress test. W e n o w define the time-step stress test a n d thefailure-step stress test as follows. I n the former, stress is c h a n g e d at a pre-specified time a n d in the latter, it is c h a n g e d after a fixed n u m b e r o f failures have occurred. These schemes o f test can be c o m b i n e d with classical censoring mechanisms. I f items are n o t run until failure, then we have a step-stress test with type I censoring; i.e., the test ends at a pre-specified time at the last level o f stress. W i t h a time-step stress, at each level o f stress, the n u m b e r o f failures is r a n d o m . I f the test ends when a pre-specified n u m b e r o f failures have occurred at the last level, we have a step-stress test with type II censoring. In this case, the end time o f the test is r a n d o m a n d is the c o m m o n censoring time. Ler us n o w c o n s i d e r a general step-stress test with m levels o f stress ss,. • •, sm. Let ~ 1 , • •, Zm be the times o f stress change. L e t n be the n u m b e r o f specimens u n d e r test. E a c h specimen is t u n at a c o n s t a n t stress level sl until time 'cl. I f a n y specimen does n o t fail, then it is subjected to a higher stress level s2 until time 'c2, a n d so on. The stress on a specimen is thus increased step by step until it fails. The o r d e r e d collection (Ss, 'Cl,S2, "C2,-.. »Sm, "Cm)is called a time stress pattern. I n a lifetest, each specimen o r g r o u p o f specimens can be s u b m i t t e d to different stress patterns. EXAMPLE 1. To illustrate a d a t a f r o m step-stress test, let us consider an e x a m p l e f r o m N e l s o n (1980). The test in this case concerns cable insulation. The stress here is voltage (kilovolts). T h e test was run to estimate life at a certain design stress. F o u r g r o u p s o f specimens were first held for 10 min each at 5, 10, 15 a n d 20 kV. T h e n each g r o u p went into step 5-11 successively at higher level o f stress (see table) for different times exposure. W e have four stress patterns.

Step Stress (kV) Times exposure (min)

1

2

3

4

5

10

15

20

10 10 10 10

10 10 10 10

10 10 10 10

10 10 10 10

5 26

6

7

28.5 31

8

9

33.4 36

10

11

38.5 41

15 15 15 15 15 15 15 60 60 60 60 60 60 60 240 240 240 240 240 240 240 960 960 960 960 960 960 960

I n the s i t u a t i o n c o n s i d e r e d above, times in the o r d e r e d collection are prespecified. I f a failure-step stress test was considered, times will be r a n d o m a n d will c o r r e s p o n d to times to failure o f a certain n u m b e r (pre-specified) o f failures at each step. This m e t h o d , however, is n o t very c o m m o n . W e will, therefore, focus o u r discussion on time-step stress testing a n d inferential m e t h o d s for it.

625

Step-stress accelerated life test 45 40 35

Step 5

ml

a0

20

I V-

10

r

l

0

50

1O0

150

times (min)

Fig. 1. Representation of step-stress test. 3. Acceleration models

The main problem with step-stress testing (and with accelerated life tests, in general) is to assess the relationship between lifetime in test and lifetime in used (normal) conditions. Many approaches are possible to model this relationship. Basically, a relationship can be set on the lifetime. This functional relationship is called the accelerationfunction or time transformationfunction. Let us denote O(s) for the stress function. Some examples are as follows. Let the time to failure under stress be a nonnegative random variable X; let Y be a nonnegative random variable corresponding to the time to failure under used-conditions. The acceleration function is the function qo0:[0,+oc ) --+ [0, +oc) such that Y -- po(X). If ~o0 is a Cl-diffeomorphism, the relationship can be written in terms of cumulative distribution functions. Let Fx and Fr denote the cumulative distribution functions of X and Y, respectively. Then, Fr(y) =

Fx ((Po1(y)) •

(1)

This is the general acceleratedfailure time model; see, for example, Bagdonavicius (1990). It serves as a basis to build the lifetime model under step-stress with the cumulative exposure model. F r o m (1), we readily obtain a relationship between the probability density functions of X and Y as 5

fr (y) = fx (~oÓ (y) ) ~y I~oö1(y)] .

(2)

if 2:: and 2r are the failure rates (hazard functions) associated with X and Y, respectively, we have the relationship

B~Y(Y) =

f x ( ~ o o l ( y ) ) ~y (pöl (y) ; 1 -- Fx((,oÖ 1(y)) =

qOõl(y) /~x ((pO1 (y)) .

(3)

626

E. Gouno and N. Balakrishnan

We have thus arrived at a model which generalizes the so-called Cox model (see Cox, 1972), but it is not, in general, a proportional hazards model, except for certain choices of distributions. A classical assumption on (Po is a linear function given by (Po(X) = O(s)X,

O(s) > 1 .

In this case, O(s) is the so-called acceleration factor. relationship in (3) reduces to

(4) In such a case, the

1 tT~S)

which is a proportional hazards model (see Lawless, 1982) since

;«(x) _ 0(s) ,t~(y) The ratio of the failure rates under two different stresses in this case is a constant. Another possible form for
O(s, t)dt .

This model has been studied by Bagdonavicius (1990), and it is called as model o f additive accumulation o f d a m a g e s (AAD). For step-stress testing with stress being constant at each step, this model becomes i-I (Po(X) = Z O(Sj)('Cj -- 27j_1) -}- O(Si)(X -- "Ci-I) • j=I

Another possible time transformation is a power function Y = A X °(~) . With such an acceleration function, a Weibull(~, fl) distribution is transformed to a Weibull distribution with parameters (Ac~e, il~O). Some classical choices for the stress function O(s) are: • the Power or Inverse Power L a w (see Nelson, 1990), when stress is tension, given by 0(s) = CsP ; (6) • the Arrhenius L a w (see Jensen, 1985; Nachlas, 1986), used for life tests with temperature, given by (7) where K is the Boltzmann constant;

Step-stress accelerated life test

627

• the Eyring L a w (see Schmoyer, 1986), used for life tests with temperature combined with other stress variables, can be expressed in the form O(s) = CsA exp{Bs} .

(8)

This model can be viewed as a combination of the inverse power and the Arrhenius models. In some cases, assumptions on the lifetime cumulative distribution function and the acceleration function can lead to relationships between distribution parameters and stress. For example, i f X is exponential with mean life O x and if the Eyring model in (8) with A = 0 is assumed as stress function with a linear acceleration function, then according to (5), we have Ox = C exp{Bs}Oy ,

where Oy is the mean life under used-condition. Now, let us denote a = log(COr) and b = B; then we have log O x = a + bs, which is a classical relation stated by m a n y authors; see, for example, Miller and Nelson (1983), Bai et al. (1989), and Xiong (1998). Another c o m m o n assumption involves a Weibull distribution with parameters (c~, fi), in which the scale parameter « depends on stress according to an inverse power law c~(s) = eo/Cs p (see Nelson, 1980), where «0 is the scale parameter under used-condition. This approach implies a linear time transformation of the form (4) with stress function of the form (6). Note that, in this case, we have a proportional hazards model, with the ratio of the hazard rates being constant equal to (CsP) ~.

4. Lifetime distribution under step-stress pattern A m o n g the classical parametric models used in the analysis of lifetime, the most frequently used ones in the context of step-stress tests are exponential, Weibull and log-normal. One m a y refer to Johnson et al. (1994) for a comprehensive discussion on these distributions. 4.1. The cumulative exposure model

Introduced by Nelson (1980), the cumulative exposure model is a classical assumption for the statistical analysis of step-stress test data. The basic idea in this model is to suppose that the remaining life of specimens depends only on the current cumulative fraction failed and current stress - regardless of how the fraction accumulated. Specifically, let F~- be the cumulative distribution function of the time to failure under stress si. The cumulative distribution function of the time to failure under a step-stress pattern, F0, is obtained by considering that the lifetime ti-1 under si-1 has an equivalent time ui under si such that E - l ( t i - 1 ) = Fi(ui). Then, the model is built as follows.

628

E. Gouno and N. Balakrishnan

We assume that the population cumulative fraction of specimens failing under stress sl, in Step 1, is

Fo(t)=F~(t),

O
.

In Step 2, we write F 2 ( U l ) = FI('Cl) t o obtain Ul that is the time-to-failure which would have p r o d u c e d the same p o p u l a t i o n cumulative fraction failing under s» We then state the population cumulative fraction of specimens failing in Step 2 by time t as

Fo(t) = F 2 ( t - "Cl ~- U l ) ,

"cI ~ t ~ "c2 .

Similarly, in Step 3, the unit has survived Step 2 and we consider an equivalent time u2 under s3 such that F3(u2) = F2(z2 - "el ÷ Ul) , where ~2 - "el -~- Ul is an equivalent time under s2. Then, we have Fo(t) = F 3 ( t -

z2 + u2),

z2 <~ t < z3 .

In general, in Step i, we have Fo(t) = Fi(t - "ci-1 + u i - 1 ) ,

"ci_ 1 ~ t <~ 77i ,

where ui-1 is the solution of Fi(ui_l) = Fi-l(zi-1 - zi-2 + ui-2). Finally, the cumulative exposure model can therefore be written as

[" F1 (t), ] F2(t - "q + Fo(t) = ~ F3(t - z2 +

o < t < ~,, b/l),

"C1 < t < z2,

b/2),

T 2 ~ t ~ Œ3,

I

[, Æn(t),

"Cm 1 <-- t < +oc ,

where u0 = z0 = 0 and ui is the solution of F~+l(U/)---Fi(ri- zi-i +ui-1), for i = 1 , . . , m - 1. The main advantage of this model is that, in its general form, there are no assumptions on the acceleration function. However, parametric analysis requires assumptions. F o r example, Nelson (1980) has considered the case when F~ is a Weibull distribution, the time t r a n s f o r m a t i o n is linear, and the stress function is of the power-law form. Specifically, in Step i, the cumulative distribution function is taken to be

Fi(t)

1 - exp [ - { C 4 t } ~ J

.

(9)

Here, the scale p a r a m e t e r is e(si) = 1 / 6 ' 4 . In this case, for t E [zi-i, ri],

Fo(t) = 1 - exp [ - { C 4 ( t - ' / i - - 1 -

Ui--1)}fi]

Step-stressacceleratedlife test

629

with ui-1 = (si-1/si) ~ (~i-1- ~i-2- ui 2). Then, step by step the cumulative exposure model transforms successively times to failure according to a linear acceleration function with an acceleration factor of the form (s~_l/s~)B.

4.2. Other methods The time to failure model under a step-stress pattern can be deduced from assumptions on the failure rate shape. For example, suppose that the failure rate associated to the time to failure is constant (Z~) at each level of stress x~, and that it is increasing (2i 1 ~ 2i, i = 2 , . . , m). Then, m

Z(t) = Z 2:1I[~, ,~,)(t) .

(10)

j=l

This assumption allows us to find the cumulative distribution function of failure time of a test unit under step-stress using the relationship

F(t)=l-exp{-

fotfl(u)du}

(11)

as

m(Vi-1 Fo(t)=l-i~=lexpt-[j~_12JAj+2i(t-~i_l )]1 II[~, ,~i)(y),

(12)

where Aj ='cj- "Cj-1. Eq. (10) is, of course, equivalent to assuming that the cumulative distribution function F,. at each level of stress si is exponential with parameter 4» The expression in (12) could have also been obtained using the cumulative exposure model. Yet another possible assumption on the failure rate shape can be a piecewise Weibull. In each interval, the failure rate is assumed to be of the form

~(t) = õ¢~t~_l In this case, instead of (10), we obtain m

B(g) = ~

6

- - tO-1 ~[[zi-l,vi)(t) .

B0i

Then using (11), the cumulative distribution function is obtained as

Fo(t) =

~/I~

1- Zexp i=1

1,

-

Lj=I

~:Oj-1+1 ~,~«~7}. Oi j ][[zi-l,vi)(Y)"

(13)

Inferential issues for this case have been discussed by Khamis and Higgins (1998). Optimal time to change stress for simple step-stress, when c~ is known, has been discussed by Bai et al. (1989).

630

E. Gouno and N. Balakrishnan

Note that this model is not a cumulative exposure model with a Weibull distribution. It has been proposed by Khamis and Higgins (1998) in order to avoid the mathematical intractability of the latter. The assumption of a cumulative exposure model with a Weibull distribution leads to the following expression for the cumulative distribution function 1 -- exp -- ( Z 1-exp

{_ (t-~, ~,xP/ \~2 + ; ) j~,

1-exp No(t ) . . . .

'

+~T- + ~)

0
(14)

\ aa

ùCi_1 ~ t ~ Z'i~

Fm(t),

Zm l < t _ < + O C .

5. Inference

Different estimation methods can be applied in the context of step-stress accelerated life test. In most situations, some plotting procedures are possible and least-squares estimation can be performed. Though somewhat involved, maximum likelihood estimation method is used frequently because of "optimal" properties. Under this method, inference is straightforward, and the asymptotic theory then will give approximate confidence limits for the parameters and the percentiles. The major drawback, however, is the computational complexity. The estimators are rarely in closed form and extensive iterative methods must be used in order to determine the MLEs. Depending on the assumed model, the computations can become quite cumbersome.

5.1. Simple step-stress 5.1.1. Exponential lifetime In simple step-stress, two stress levels Sl and s2 (Sl < s2) are used. Assuming that the logarithm of the mean life is a linear function of stress of the form a + bs, maximum likelihood method can be applied to obtain estimates of a and b. Miller and Nelson (1983) investigated the situation wherein units are observed continuously until all test units are run to fall for time-step stress and for failure stepstress. They also presented optimal test plans. The case with censoring has been addressed by Bai et al. (1989). Xiong (1998) has derived the MLEs of a and b and

Step-stress accelerated life test

631

also provided a test of hypothesis to assess if stress has an effect on lifetime with T y p e II censored data.

5.1.2. Weibull lifetime The case of simple step-stress with a Weibull lifetime model is a particular case of the p r o b l e m treated by Nelson (1980); see below. Type I censoring was assumed in this case. We investigate here the case of T y p e II censored Weibull data. Let r be the pre-specified n u m b e r of observed lifetimes. Let ni be the n u m b e r of failures observed at step i (i = 1,2), and r = nl + n2. I f nl = r, we have a classical constant stress. X,ö is the failure time of test unit i at stress sj, for i = 1 , . . , nj and j = 1,2. The assumptions of cumulative exposure model and Weibull distributed life with p a r a m e t e r s (e;, fi) at any constant stress si imply that the cumulative distribution function of a test unit under simple step-stress test is of the f o r m (14) with m = 2. Thus, the probability density function of a specimen is

F(t) =

B t/~-I exp ( - - ( 7 ~ ) } , «1 (t-~~ + exp B \ ~2

0_
+

71

~ < t < cc . -

'

Let us n o w assume that cq = C~. Then the log-likelihood, expressed as a function of the p a r a m e t e r s C, p and fl, is

L(C,p, fi) = r l o g fi - nlfi(log C + p l o g s1) + ( f i - 1)

j=l

,,=,

log Xl,j - n21og C + ~ l o g

1

,=~

j=l

--

n2(log C + p l o g s2)

/X2«

T, "~

~-7-2 +71) \

2

(16)

In order to obtain the M L E s of the p a r a m e t e r s C, p and fi, we need to solve the likelihood equations

~

L(c,p,/~)

= 0,

~pL( C, p, fi) = O,

BL(C,p,8) = 0 • The numerical p r o b l e m of solving these three equations can be s o m e w h a t simplified as follows. The first equation leads to the expression

632

E. Gouno and N. Balakrishnan «

_ u ( p , fl)

(17)

n~ +n2 where

~,~ +

+«)

+,ù r~(~~~ ~~1)~

Substituting (17) into the two other equations, we obtain

7/1

B + plog

//2

Z log Xx2,j - r

j=l

z

/

V(p, fl) _ O,

j=l

~j~l/'log s2, log sl "~ / / ' x 2 j - 77 ~ ) -,1plog,~-,21og~2+/B-1/= k~-~~2,~-~/+-Z-~)//---g-+~,

fl [~--~~(Xl,j ) fllOg Sl ~_~-~ ßX2,j -- Œ "C"~fl-1/'X2,j -- "C, 77 Cfl [.j--1\ 4 / j=l k 4 -I-~ll) kTIOgS2-[-~ll 1OgS1) x2,n2-77+77q)fl-1 (~ x2'n2-771°gs2 +«log 77 sl)l = 0 , +(.-r)(4

where j=I

\~j\~j

+~log[X2,j-"c

z

/`x2,j-"c

"c5 ~

+ ~,,-r~~o~C~'~-~ + ~) (~~'7 ~+ ~) ~ These two equations can be solved using Newton's method in order to find the MLEs o f p and c« Once they have been determined, the MLE of C can be easily computed from (17). 5.2. Multiple step-stress 5.2.1. Exponential lifetime

When stress is temperature, it is common to consider an Arrhenius model as stress function and an exponential life distribution, with the time transformation being again linear; see, for example, Tobias and Trindale (1986) and Gouno (1999). An approach based on a generalized Arrhenius can be found in Nachlas (1986). The method discussed by Gouno (1999) deals with grouped data. Numbers of failures and numbers of withdrawals from the test (then censored) are observed at each step stress. Graphical and maximum likelihood estimators for the failure rate and

Step-stress accelerated life test

633

for the acceleration mode1 parameters are then derived. The failure is supposed to be constant at each level of temperature leading to an expression of the form (10). Assuming a linear time transformation and a stress function of the form (7), we arrive at a proportional hazards model

2i=exp

-~-

-

20 ,

where Ea and )~0 are unknown parameters that need to be estimated, ~ is the temperature at step i, To is the temperature in used-condition, and K is the Boltzmann's constant (K = 8.617 x 10 -» eV/°K). The MLEs of Ea and 2o are obtained and compared with the graphical estimates. These graphical estimates of Ea and 2o are obtained, respectively, from the slope and the intercept of the closest

straightlinetothesetofpoints(-~(~-~),log2i),where2iistheMLEof2i. 5.2.2. Weibull lifetime When an acceleration mode1 is specified, inference can be carried out using the transformed observed times and hypothesis on lifetime cumulative distribution function; see Tobias and Trindale (1986) and G o u n o (1999). M a x i m u m likelihood and graphical analysis can be conducted in order to obtain estimators of O(s) parameters. Nelson (1980) has given solutions with a linear acceleration function, a stress function O(s) = Csp, and a lifetime distribution supposed to be Weibull with a constant shape parameter. His work also includes the case of Type I censored data.

6. Non-parametric approach When no assumptions are made about the form of the underlying distribution, a non-parametric estimator of F0 can be proposed; see, for example, Shaked and Singpurwalla (1983). The method proposed by Shaked and Singpurwalla (1983) advocates the assumption on the form of the time-transformation and the parameters of this need to be estimated later. To do so, a test must be conducted wherein items are submitted to at least 2 stress levels. Ler us suppose that the test lasts a time z. In the case of a linear time-transformation with an inverse powerlaw stress function, (9) holds and an acceleration factor between two stresses can then be defined by

Oi-l'i

\ S i /I

Let F / b e the empirical function under stress si defined by B(t) =

number of observations < t ni

E. Gouno and N. Balakrishnan

634

where ni is the number of items in step i. Ler ~-1 be the right continuous inverse of ~. defined as ~.-l(u)=sup{t:/6z(t) < u } ,

uE[0,1] .

Then an estimator Ôz-l,i of Oi-l,i can be proposed as

ôi-l,i

u ^--1

f0 ~ - l ( t ) dt U ^--1 f; F/ (t)dt

»

where u = min(F/_1 (z), Fz(z)). An estimate o f p is naturally given by log Ôi-l,i B - log(si_l/si) Then, the observed times to failure can be transformed as

I(~~/~o)~X

(S1/So)a'Cl @ (s2/so)ô:(X - -el)

Y=

i-1

~

~j=I(Sj/So) ('Cj

-

-

& "eS_l) -~ (Si/So) (X - "c/_i)

if0_<X
if ~i-1 _<X _< ~i

(18) The empirical distribution function F0 based on the transformed lifetime is readily obtained. If several stress patterns are available, the estimate of F0 is computed by superimposing the data. Shaked and Singpurwalla (1983) have studied asymptotic properties of such an estimator.

7. Bayesian analysis A Bayesian analysis of partially accelerated life testing has been proposed by DeGroot and Goel (1979)• They considered ~, the total lifetime of the test item i (i = 1 , . . , n), defined by the relation Yi = {Xi

ri- O(s)(Xi - ~i)

i f X / < zi, i f X / > ri •

Before time % items are run under standard environment, but this can be extended to any stress level. O(s)(Xi - zi) is interpreted as an equivalent lifetime under used-conditions. This model is a particular case of model (18). We have a linear time transformation function and O(s) is the acceleration factor. DeGroot and Goel (1979) then assumed that X has an exponential distribution with parameter 2 and derived the estimators of 2 and O(s). Note that their

Step-stress accelerated life test

635

approach does not allow one to assess the form of the stress function, and it just provides a point estimator. For loss functions of the form L()~, 2) = )~k2l(2 -- 2) 2 ,

(19)

where - 2 < k < 0 and - c o < l < +oc, D e G r o o t and Goel (1979) obtained Bayes estimators. They first assumed that 2 = 2o is known, worked with the parameter B = l/a, and considered a prior distribution of fi to be a gamma distribution with parameters r and b20. The posterior distribution of fi turns out to be again a gamma distribution with parameters rl and bi20, where rl = m + r and bi = b + ~i~(Y~ - ~i), with A being the set of indices corresponding to lifetimes of items that failed after being switched to the higher stress-level. For r + l > 0, the Bayes estimator/) with respect to the loss function L is B = "yk(rl -t- l) ~1 , sl Oo

where the function 7k is defined (for , / > 0) as

{ [

1 k+l+

( ~)1/2] 1-

1+~1

for-2
When 2 is unknown, one needs to build re(il, 2), a conjugate family o f j o i n t prior distribution for (fi, 2). This distribution can be specified as rc(fi, 2) = rc(fil2)~z(2), where rc(fil2) is the conditional prior distribution of fi given 2 (a gamma distribution with parameters r and b120) and re(2) is the prior distribution of 2 (a gamma distribution with parameters r0 and b0). Then, the posterior distribution of 2 is a gamma distribution with parameters r2 and b2, where l'2=rO-}-n--m

and

b2=bo-FEzi4-ZY j . iEA jCe(

If r0 + l > 2, the Bayes estimator )~ of 2 with respect to the loss function (19) turns out to be L s2 / For r + l > 0 and r0 > l + 2, the Bayes estimator is

B = ~k((rl + z)(r= - I-

~ [ (r, + z)~2 ]

r [ ~ r 2 - -1 2)_/L(r~ - z

1)all

D e G r o o t and Goel (1979) have mentioned that other loss functions and other lifetime distributions could be used. They did not give any directions, however, for the choice of the prior distributions and of the loss function. The case of multiple step has also not been investigated.

E. Gouno and N. Balakrishnan

636

It seems natural to assume a form for the failure rate under step-stress testing such as 20__<21_<'''_<2m • I f we assume 2i to be constant in each step, then (10) holds. This formula will allow one to compute the probability density function of failure time of a test unit under step-stress using the formula

(20)

f ( x ) = 2(x) exp -

{/0 ~2(t)dt } .

We then obtain

f(xl)O=Z2iexpi=1

,~jAj-[-2i(x-.ci_1)

]I[~'i i "te)(y)

(21)

1,

where Ai = "ci - Ti 1. The reliability (survival) function is given by

R ( Y ) = £ e x p { - [ ~ ~ J A[_j=l J@'~i(Y-Œi-1)J}

~[zi-l'zi)(y)

(22)

Let us assume that at each readout time Zg which are the stress-change time, a number kg of times to failure xi,j ~ / = 1 , . . , ki) are observed and a number ci of units are withdrawn. Then the number ng of pieces still in the ith interval follows the relation

t'lizni_ 1 - k i _ 1 --Ci_l~

i=2,

..,m

.

We have nl = n, to = 0, and tm is the end of the test. Note here that the numbers cg are not random. Some specimens are arbitrarily withdrawn from the test at every time % Let us set, for convenience in notation, m

m-1

m

cm = n - Z kj - Z cj j=l j=l

sothat

m

n = E kJ + Z cJ " j=l j=l

The number of units in test in the time interval [Zg-1,zi) is m

ni = Z ( k :

j--i

i-1

+ cj) = n - ~ ( k j

+ cj)

j=l

In m a n y practical applications, exact failure times are not available. To work with times to failure, one can draw randomly kg failure time YU (l = 1 , . . , kg) between ~i-1 and zi using a uniform distribution, for example (middles of intervals could have been used equally). To obtain Bayes estimator of 2, we need to consider a prior distribution ~z(2) on this parameter. Then the posterior distribution rc(2[x) is f(x12)=(2), where

Step-stress accelerated life test

mFk, ] S(xl~.) o~ H/IIs(~<','l'~)/[R(T'i)]ci i=1

L I=I

637

(23)

1

After some algebraic manipulations, using (21) and (22), we obtain roT/

/(xI,L) oc II,'t.~'

exp{-2iTTTi}

(24)

i=1

with TTTi = ~lIÈ1 (xi>1- 77i-1) 4- (#ili - k i ) A i denoting the total time on test in the /th time interval. Let us assume that the prior distribution of 2i is a gamma distribution with parameters (ai, bi). These distributions form a conjugate family in this problem and the posterior distribution of ,~i turns out to be again a gamma distribution with parameters (ki + ai, TTTi + bi). Then, if we assume a quadratic loss function, a Bayes estimator 2 of 2 will be the posterior expectation given by )~i--

kiq-ai

TTTi+bi'

i = 1, ..,rn .

(25)

Van Dorp et al. (1995) proposed a Bayesian analysis of a problem of the same kind but with only numbers of failures ki (i = 1 , . . , m) as observations. In this case, the likelihood takes on the form m

H )'/k'(l - Äi)"'-k'

(26)

i=I

Setting a transformation of the failure rate ui = exp{-c2i}, they considered a multivariate ordered Dirichlet distribution on u = (Ul,...,Um). Expressing the likelihood in terms of u, they then obtained the posterior distribution of u as a mixture of generalized Dirichlet distributions. Relying on this result, Dietrich and Mazzuchi (1996) extended the work to multi-stress situation and investigated the use of combination of design of experiment techniques with multiple stress testing.

8. Optimal test plan A question naturally arising in setting a step-stress test is how to select the best stress levels to estimate the desired distribution. This is the problem of optimal design. The question includes the problem of how long items are exposed to each levels. Chernoff (1962) developed a technique to obtain optimal accelerated life designs. He considered five possible models with a constant stress. In the case of simple step-stress tests, Miller and Nelson (1983) presented optimal design under the assumption of an exponential distribution. Their optimization criterion was to minimize the asymptotic (large-sample) variance of the maximum likelihood estimator of the mean at a specified design stress. This criterion leads to optimizing three quantities:

638

E. Gouno and N. Balakrishnan

1. the level of the first stress; 2. the level of the second stress; 3. the time of the change stress. The optimal time at low stress first for simple time step-stress tests is ZL = OL log [24 + 1)/4] , where ~ = (st - S D ) / ( S H -- SL) and OL is the mean of the time to failure under low stress expressed by OL = exp(y0 + y~s). For simple failure step-stress, the optimal test switches from SL to sn when the fraction of failures under SE reaches (1 + ~)/(2~ + 1). The expression of the optimum asymptotic variance is Varlog (ÔD) = (24 + 1)2/n • For an optimal simple step-stress plan, the accuracy of the estimator depends on the sample size n. If one wishes ÔD to be within a factor co > 1 of the time OD with (high) probability fi, i.e., Pr(OD/CO < ôD < (D(~D) = fl , then the approximate sample size achieving this will be n ~ [NB(2~ + 1)/in(co)] 2 , where Nô is the standard (1 + fl)/2 fractile. Bai et al. (1989) extended this work to the case in which a prescribed censoring time is involved. In partially accelerated life testing, the cost of the experiment will depend on the stress change times zi (i = 1 , . . ,n). DeGroot and Goel (1979) have given optimal designs for the estimation of the unknown parameters choosing the n points zi such that the total risk is a minimum. 9. Conelusion

In this paper, we have presented an overview of step-stress accelerated lifetime tests. We have described the cumulative exposure model as well as other methods of deriving the lifetime distribution under step-stress model. We have then discussed inferential issues with step-stress testing and specifically described the maximum likelihood, non-parametric, and Bayesian approaches. Finally, we have given some details on the construction of optimal plans. References

Bagdonavicius, V. B. (1990). Acceleratedlife models when the stress is not constant. Kybernetika 26, 289-295. Bagdonavicius, V. B. and M. S. Nikulin (1997). Transfer functionals and semiparametricregression models. Biometrika 84, 365-378.

Step-stress accelerated life test

639

Bai, D. S., M. S. Kim and S. H. Lee (1989). Optimum simple step-stress accelerated life tests with censoring. IEEE Trans. Reliab. 38. Chernoff, H. (1962). Optimum accelerated life design for estimation. Technometrics 4, 381~408. Cox, D. R. (1972). Regression models and life-tables. J. Roy. Stat. Soc. Ser. B 34, 187-220. DeGroot, M. H. and P. K. Goel (1979). Bayesian estimation and optimal designs in partially accelerated life testing. Naval Res. Logist. Quart. 26, 223 235. Dietrich, D. L. and A. M. Mazzuchi (1996). An alternative method of analysing multi-stress multi-level life and accelerated life tests. In Proceedings of Annual Reliability and Maintainability Symposium. Gouno, E. (1999). An inference method for temperature step-stress accelerated life test. Quality and Reliab. Eng. Int. Jensen, F. (1985). Activation energies and the Arrhenius equation. Quality and Reliab. Eng. Int. 1, 13-17. Johnson, N. L., S. Kotz and N. Balakrishnan (1994). Continuous Univariate Distributions vol. 1, 2nd edn. Wiley, New York. Khamis, I. H. (1997). Optimum M-step, step-stress test with k stress variables. IEEE Trans. Reliab. 47. Khamis, I. H. and J. J. Higgins (1996). An alternative to the Weibull cumulative exposure model. Proceedings of the American Statistical Association, Section on Quality and Productivity. Khamis, I. H. and J. J. Higgins (1998). A new mode1 for step-stress testing. IEEE Trans. Reliab. 47. Lawless, J. F. (1982). Statistical Models and Methodsfor Lifetime Data. Wiley, New York. Miller, R. and W. Nelson (1983). Optimum simple step-stress plans for accelerated life testing. IEEE Trans. Reliab. 32. Nachlas, J. A. (1986). A general model for age acceleration during thermal cycling. Quality and Reliab. Eng. Int. 2, 3-6. Nelson, W. (1980). Accelerated life testing - step-stress model and data analysis. IEEE Trans. Reliab. 29, 103-108. Nelson, W. (1990). Accelerated Testing. Wiley, New York. Schmoyer, R. L. (1986). An exact distribution-free analysis for aecelerated life testing at several levels of a single stress. Technometrics 28, 165 175. Shaked, M. and N. D. Singpurwalla (1983), Inference for step-stress accelerated life tests. J. Stat. Planning and Inferenee 7, 295-306. Singpurwalla, N. D. (1995). Survival in dynamic environments. Stat. Sci. 10, 86 103. Tang, L. C., Y. S. Sun, T. N. Goh and H. L. Ong (1996). Analysis of step-stress accelerated-life test data: a new approach, IEEE Trans. Reliab. 45. Tobias, P. A. and D. Trindade (1986). Applied Reliability. Van Nostrand Reinhold, New York. Van Dorp, J. R., T. A. Mazzuchi, G. E. Fornell and L. R. Pollock (1995). A Bayes approach to step-stress accelerated life testing. IEEE Trans. Reliab. 38. Xiong, C. (1998). Inferences on a simple step-stress model with type-II censored exponential data. IEEE Trans. Reliab. 47.

N. Balakrishnan and C. R. Rao, eds., Handbook ofStatistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

~A

A..,~

Estimation of Correlation under Destructive Testing

Richard Johnson and Wenqing Lu

1. Introduction

Any reliability analysis of structures made of dimension lumber must overcome the problem that it is very difficult, if not impossible, to pair full sized specimens of dimension lumber so that their individual strengths are highly correlated. More particularly, the problem we address arises when, for instance, a roof of a house is loaded by wind or snow. Some of the structural members of the roof system will simultaneously undergo two modes of stress. The integrity of the roof system depends on both the bending and tensile strength of certain members. More generally, any reliability-based design approach to evaluating roofs, floors, walls or even complete buildings made of wood needs to use the joint distributions of two or three strength properties (see Suddarth et al., 1978). The idea of proof loading has been widely used in engineering (see Johnson, 1980). The scope of application was extended by Galligan et al. (1981) to provide a design of experiments for estimating the correlation of two strength properties, measured on an individual specimen. Ordinarily we would need to estimate the correlation by observing the two strength properties on an individual specimen. Unfortunately, when specimens are tested by loading to failure in any one strength mode the specimen is destroyed. Somehow, information on both strength properties must be obtained from a single specimen. Galligan et al. (1981) proposed the first practical experiments to obtain information about the correlation between two strength properties that follow a bivariate normal distribution. Their design consists of two steps:

Step 1. Load a specimen in strength mode 1, not to failure, but to a specified proof load. If the unit fails, record the mode 1 strength x. Otherwise remove the load and proceed to step 2. Step 2. Load the specimen to failure in strength mode 2. Record the mode 2 strength y. Data collected from these initial experiments provided the first ever estimates of correlation.

641

642

R. Johnson and W. Lu

Because of the concern that the proof load would damage survivors and thus change the joint distribution, the proof loads in the first experiments (Galligan et al., 1981) were purposely selected to be in the lower tail of the distribution. Although these designs provided the first estimates of correlation, they were not particularly efficient and needed to be improved upon. Most of the work to-date concerns estimation of the correlation coefficient in bivariate normal distributions. We first review the normal theory results and then give new multiple proof load designs in Section 3. Next we give some new results concerning the bivariate exponential and bivariate Weibull in Section 4. Finally, in Section 5, we present some results concerning a non-parametric approach. 2. Current designs for estimating correlation

The first approach to selecting a good proof load in the design was made by Evans et al. (1984). They estimated optimal proof loads for the bivariate normal model, for various values of correlation, by performing a simulation study. Their choice of the proof loads were those that yielded the smallest estimated variance for the maximum likelihood estimate/5 of correlation. In some applications, the marginal distributions are essentially known, so the case of known marginal distribution was considered as well as the five-parameter unknown case. De Amorim and Johnson (1986) took a more systematic approach that maximized the Fisher information about p. To ger the Fisher information, we first define T~=

{Xi ifXi _< Lx, Yi ifX~>Lx,

and {~ C/=

ifXi ifX~

<_Lx, > Lx ,

where Lx is the proof load. We set 0 = (/fi, a 2, p, a 2, #2)'. Then, the log-likelihood function of one observation (Tl = t, CI = c) can be expressed as

-I[c

{1

2

= 2] ~ l n «2 -+

(t-/~2)2 ~ä~

ln[1 - O(ax)]

}

,

(1)

where ax = (2x - a 2 1 p ( t - #2))/~/1 - p2 and )ox = (Lx - #1)/«1. For the known marginal case, the Fisher information concerning p, or 133, is given by

1 Fco(z2 -

h3 = (1 - p2) 3

p2x)2h(ax)~b(ax)Ó(z2)dz2

,

Estimation of correlation under destructive testing

643

where qb(.) is the standard normal density and h(.) = ~b(.)/[1 - q~(.)] is the standard normal hazard rate. It is enough to consider standardized variables and then determine the standardized p r o o f load that maximizes the Fisher information for p. We call 2x the standard p r o o f load. The optimal p r o o f load for X is then: L x = o-1.~x @/A 1 .

(2)

Figure 1 illustrates the Fisher information as a function of standardized p r o o f load 2 -- 2» The information is a unimodal curve with the m a x i m u m at a positive B~opt for p = 0.1,0.3, 0.5 and 0.9• Also, if the 2 selected is within 0.1 unit of 2opt, there is only a small loss in the information about p. On the other hand if 2 = - 1

rho = 0.1

rho = 0 . 3

d d e'-

00

r-

ù0_ ci

.o_ d

E 0 r'-

o4

0

0 0

-1.0

0.0

1.0

-1.0

0.0

Lambda

Lambda

rho = 0 . 5

rho = 0 . 9

1.0

t,D

(5

E: 0

(:5 1(5 C'M

L{1

c5

ei

-1.0

0.0

Lambda

1.0

-1.0

0.0

1.0

Lambda

Fig. 1. Fisher i n f o r m a t i o n as a function of ), w h e n marginals are known.

644

R. Johnson and W. Lu

is chosen as the proof load, the loss of information is great and it will be very difficult to estimate p. This Fisher information calculation shows why Galligan et al. (1981) had difficulty getting good estimates of correlation with the low proof loads used in the first experiments. A straightforward numerical integration essentially reproduces some entries in Table 2 of De Amorin and Johnson (1986). These new values are given in Table 1. Note that the optimal proof loads are in the upper tail of the distribution of strength mode 1. Further, as p increases, -~opt decreases from 0.608 to 0.221 and the corresponding Fisher information Iopt increases from 0.4111 to 5.1832. According to the information Iopt, a higher correlation can be estimated more efficiently than a low correlation. Table 1 Optimal proof loads and Fisher information for a single proof-load design marginal distributions known p

0.1

0.3

0.5

0.7

0.9

Iopt Bopt

0.4111 0.608

0.4668 0.577

0.6273 0.508

1.1383 0.401

5.1832 0.221

We now treat the case where all five parameters are unknown. The first-order partial derivatives of the log-likelihood (1) are: 8ll _ I[« = 1] ~1 + I[« = 2] 0"1 l2~'~h ~( a- x- P) ~ll ~=I[c=

z2 - 1 ~ 2xh(ax) 1]-~a12 + I [ c = 2 1 2 ù 2 ~ ,

el_~l = I[c = 2] (z2 - p2x)h(ax) ~D

(1 - p2) 3/2

(3) '

1

oz~ i{c-- 21(z2

2

'

p»(ax)

The Fisher information can be obtained from these first-order partial derivatives or from the second-order partial derivatives as in De Amorim and Johnson (1986). I=E

[(~ll) (~/1"~'] [ ~2[1] ~Õ \ ~ C B J = E - ~ - Õ ~ j

(4)

The results in De Amorin and Johnson (1986) were a surprise. We had not expected to see much change in the optimal proof loads when all five parameters are unknown. However, with five parameters unknown, the information is

Estimation of correlation under destructive testing

645

especially small for small p and even experiments with large sample sizes are essentially non-informative. The optimal proof loads are summarized in Table 2, along with the associated element /33 of the inverse of the Fisher information matrix, for the proof-load design when all five parameters are unknown. That is, the entry i33 is the (3, 3)th element of inverse of the information matrix corresponding to p. For all of the cases presented, ,~opt is negative. The optimal loads are very different from these in the known marginal case. Table 2 Optimal proof loads and /-33, from the inverse of the Fisher information matrix. Single proof-load design (#1, a 2, P, a~, #2) unknown p

0.1

0.3

0.5

0.7

i33

87227.6 -1.140

936.255 -1.087

87.3909 -0.981

1 1 . 6 5 4 3 0.767947 -0.812 -0.516

Bopt

0.9

De Amorin and Johnson (1986) established the asymptotic normality of the maximum likelihood estimate ~. In particular, where the parameter space is specified as in Theorem 1 V~(~_p)

D»N(0,/33)

.

This justifies, at least for large samples, our approach of selecting the design which minimizes 133. We recalculate the Fisher information using De Amorin and Johnson's formulas but evaluate the integrals by a new adaptive integral routine due to Lau (1995, p. 299). It uses Simpson's rule with a Richardson Correction. All of our new calculations agree with De Amorin and Johnson (1986) up to 10 -4. The large /33 values, especially for small p, establish that the single proof-load design is inadequate for estimating p when the parameters of the marginal distribution are unknown. The original proof load design is asymmetric. This is necessary in some applications. For a full-sized lumber specimen, the compressive strength could only be measured on two machines in the US. However, a related strength may be obtained by cutting the specimen into small sections and loading these in compression on one of numerous smaller machines available. Here, the full sized specimen compressive strength must be the mode 1 strength. The two strength modes cannot be interchanged. When the proof loading could be done in either strength mode, De Amorin and Johnson (1984) suggested a symmetric proof load scheine that leads to much greater improvements. Half of the sample is first proof loaded in mode 1 and the other half is first proof loaded in mode 2. The Fisher information for the symmetric design is

Isp

=~EIp+

Ip~] ,

(5)

646

R. Johnson and W. Lu

where I o is the Fisher information matrix for the single proof-load case and Ipr is the transpose of Ip along the second (SW-NE) diagonal. The optimal proof loads and the corresponding Fisher information, for the symmetric design, are summarized in Table 3. There is significant improvement over the basic single proof-load design. Most importantly, there is a sharp reduction in the value of i33, especially for small p's. Note, however, that the optimal proof loads are now positive, and so lie in the right-hand tail of the distribution. Table 3 OptimaI proof loads and 133, from the inverse of the Fisher information matrix. Symmetric single proof-load design (#1, al2, P, a2a, #2) unknown p

0.1

0.3

0.5

0.7

0.9

/33

2.9620 0.803

2.4874 0.729

1.7816 0.628

0.9560 0.496

0.2064 0.292

~'opt

De Amorin and Johnson (1986) also recommend a hybrid design, where some specimens are specifically loaded to failure in mode 2, for applications where the symmetric design cannot be implemented. 3. A new double proof load design

In cases where the strength modes are interchangeable so that a specimen can be proof loaded in either order, strength mode 1 or mode 2, it is possible to consider a three-stage design. Johnson and Lu (2000) propose the procedure: Design step 1. Load the unit in strength mode 1 up to an established maximum load L» If the unit fails, record its mode 1 strength x. If it does not fail, remove the load and proceed to Design Step 2. Design step 2. Load the unit in strength mode 2 up to an established maximum load Le. If the unit fails, record the mode 2 strength y. If it does not fail, remove the load and proceed to Design Step 3. Design step 3. Load the unit to failure in strength mode 1. Record the mode 1 strength x. The observations in this experiment consist of the breaking strength T~ of the specimen and the stage Ci at which the specimen breaks. The stage carries the information on the strength mode. To set notation, we define X~. ifX,.<_Lx, T~ = Y~ if X / > Lx and Y~_< Ly, X,. i f X g > L x a n d g > L y and Ci

{ {

1 ifX~ _ Lx and ~ < Ly, 3 if X/ > Lx and Y/> Ly .

647

Estimation o f correlation under destructive testing

Then, the log-likelihood function, 12, of one observation (T = t, C = c), but for two proof loads, can be expressed as 12(O;t,c) = - - ~ l n ( 2 ~ ) - I [ c

= 1] [~lncr2 q (t--20#1)2_1 .2

{1 2 (t -- #2) 2 - I [ c = 2] ~ l n % q 202

ln[1 - e(a,)]/

{1 2 (t -- #1 )2 - I [ c = 3] ~ l n % -~ 20 2

lnll - ~(ay)] } ,

(6)

where 2x - p t 72**2 2y

2x - L- x --

#1

,

ay -

O t-~~ "~Y - - -«l ,

and

Ly - #2 02

3.1. Fisher information for a double proof-load design

Two proof loads Lx and Ly will be selected to jointly maximize the Fisher information about p. To obtain the Fisher information matrix, we first calculate the first-order derivatives of the log-likelihood 12 with respect to 0 = (#1, ~2, p, az, #2)'. Define the standardized variables zi =.(t - #i)/«i, for i = 1,2, and recall that h(a) = (B(a)/[1 - ~(a)], is the hazard rate for a standard normal. The five first-order derivatives are 51q~12- I [ c = 1 ] ~7 + I [ c = 21 ol ~12 - - [[C

~ßa21

11z 2 - 1 q - I I C 2«2

= 21

lxf~7-p + i [ c = 31( ~

olph(aY)lx/~7-p 2 ,

2xh(ax) 2«2 V/1 _ p2

+ IEc = 3]~o ~~ ~ - 1

(7)

el2 _ I[c = 2] (z2 - p2x)h(ax) + I[c = 3] (Zl - p2y)h(ay) ~P (1 -- p2) 3/2 (1 -- p2) 3/2 ' ~12 _ I[c = 2] 1 ~ ~a 2 ~ßa22

-1

N

el2 _ Il« = 23 (z2

ph(ax)

pz2h(ax)

B

+

I[c

~ +I[« = 3]

=

3]

2yh(ay)

2«~lB77-p2'

h(ay)

648

R. Johnson and W. Lu

Let I(e) = (I[c = 1], I[c = 2], I[c = 3])' be the vector of indicators and

then, we can write (7) as

~J22 = Mt(c)

(8)

00

The Fisher information matrix \~]

I = E

(9)

then has elements

Iik = E(milmklI[C = 1] + mi2mk2IIC = 2] + mi3mk3I[C = 3]) ,

(10)

where mik is the (i, k)th element of M. To simplify (10), consider the expectation of any integrable function 9('), which we obtain by conditioning on C. E(g(Zl)I[C

~-

1])=

F"

g(Zl)~(z1)dz1 ,

(11)

O(3

E(o(z~)I[c = 21) = P [ c = ~ e [ c

a]E[o(z~)lc = 2] 2] J

f I

]-oo g(z2)c)(z2)[1 - O(ax)]dz2 ,

(12)

where f(z2]C = 2) is the conditional density of Z2 given C = 2. Similarly,

E(g(Z1)I[C = 3])

g(zl)O(zl)[1 - ~b(ay)]dzl .

(13)

Estimation of correlation under destructive testing

649

Applying relations (11)-(13) to each term in Iik in (10), we obtain Iik =

mi2mk2~)(z2)[1-- ~(ax)ldz2

milmkl~(Zl)dZl + O0

+

O0

mi3m~30(zl)[1 -- ~(ay)]dzl .

(14)

Notice that each Iik is the sum of 3 integrals. The first integral is either 0, because m i l = 0 for i > 3, or can be done analytically. The other two integrals can only be evaluated numerically. 3.2. Optimal selection o f double proof loads

We numerically evaluate the Fisher information for the double proof load design. Let us first consider the known marginal case. Figure 2 shows the contour plots of Fisher information as 21 and 22 vary over the fange between - 1 and 2.0. Again, p = 0.1,0.3,0.5 and 0.9 are used to represent the whole range of p. A unique maximum occurs at a pair of values with ~1 < ,~» Table 4 summarizes the optimal proof loads }q,opt and -~2,opt and the corresponding Fisher information for p = 0 . 1 , 0 . 3 , . . , 0 . 9 . The pair (21,opt, ,~2,opt) changes in a somewhat complex pattern as p increases. In each case, from Table 4, the optimal first proof load )~l,opt is 1ower than the respective single optimal proof load )~opt in Table 1. Actually, )~l,opt is negative when p = 0.9. The optimal second proof load 22,opt is rauch higher than the single optimal proof load. Is there rauch gain of the optimal double proof-load design over the optimal single proof-load design? Table 5 presents the increase and relative increase in the maximum Fisher information numbers Iopt obtained by employing the two-stage proof-loading scheine. When the correlation p is less than or equal to 0.5, the increase is less than 10%. By contrast, when p is greater than 0.5, the increase is greater than 10% and can be as large as 57%. This latter situation is the case most commonly encountered in the wood industry where two strength properties on the same specimen are typically highly correlated. We hext consider the case where (/11, a~, t3, «22, ~2] , 2~ are unknown. The optimal proof loads and 133 from the inverse of the matrix of Fisher information, for the double proof-load design, are summarized in Table 6. Comparing an optimal 133 in Table 6 with the corresponding information in Table 3 for the single but symmetric proof load design, we see that the double proof-load design is not as efficient as the symmetric single proof-load design, especially for the small/3's. However, when/3 increases to 0.9, the difference decreases to a moderate value. 3.3. A symmetric version o f the double proof load design

To make the double proof load design comparable to the symmetric single proof load design, we can symmetrize the double proof load design. In particular, when all five parameters are unknown, half of the specimens can be subjected to a double proof-load design starting with strength mode 1; and the other half of the

650

R. Johnson and W. Lu

rho=0.3

rho=0.1 q

q

Ol

•-:

../..

0

Lt')

c5 ._1

0

¢5

°

"~.1 -1.0

0

0.0

1.0

0~.2 -1.0

2.0

i 0.2 0./1 0.0

1.0

Lambdal

Lambdal

rho=0.5

rho=0.9

2.0

q Ol

0 Ol LQ

t~'-

04 "0

E

.

c5

._1 0c5

.-I

o

O

o.

I-

0

-I .0

0.0

1.0

2.0

-1.0

Lambdal

0.0 1.0 Lambdal

2.0

Fig. 2. Fisher information as a fuuction of 21 and 22 when the marginals of the bivariate normal distribution are known.

Table 4 Optimal double proof loads and Fisher information for a double proof-load design marginal distributions known ,o

0.1

0.3

0.5

0.7

0.9

Iopt Bl,opt Æ2~opt

0.4296 0.509 1.245

0.4940 0.464 1.356

0.6837 0.339 1.531

1.3568 0.006 1.425

8.1459 -0.377 0.835

651

Estimation of correlation under destructive testing

Table 5 Comparison of Fisher information numbers from the single and double proof-load designs marginal distributions knowna p

0.1

0.3

0.5

0.7

0.9

Increase Rel. inc.

0.0185 0.05

0.0273 0.06

0.0564 0.09

0.2185 0.19

2.9627 0.57

a Increase = Iopt,double- /opt,single; rel. inc, = increase/Iopt,single. Table 6 Optimal proof load and 133 from the inverse of the Fisher information matrix. Double proof-load design - (#1, er2, P, «22,#2) unknown p

0.1

0.3

0.5

0.7

0.9

i33 Bl,opt

11.6893 -0.934 1.198

7.4972 - 1.668 0.740

3.9009 - 1.618 0.666

1.5587 - 1.454 0.596

0.2548 - 1.076 0.448

B2,opt

specimens are subjected to a d o u b l e p r o o f - l o a d starting with strength m o d e 2. This design will generate m o r e i n f o r m a t i o n t h a n the s y m m e t r i c single p r o o f - l o a d design. The question is h o w m u c h m o r e ? T h e F i s h e r i n f o r m a t i o n o f the s y m m e t r i c d o u b l e p r o o f l o a d design follows the same relation (5) as does the single p r o o f - l o a d design. But n o w I o is the F i s h e r i n f o r m a t i o n u n d e r the d o u b l e p r o o f - l o a d i n g design. T h e n u m e r i c a l results c o n c e r n i n g the o p t i m a l ¤33, f r o m the inverse o f the F i s h e r i n f o r m a t i o n m a t r i x , are p r e s e n t e d in T a b l e 7. C o m p a r i n g a n y entry in T a b l e 7 with the c o r r e s p o n d i n g e n t r y in T a b l e 3, we see t h a t there is only a small gain in 133 when p is small. H o w e v e r , the gain in 133 can be s u b s t a n t i a l when p is large. N o t e t h a t w h e n p is 0.9, the r e d u c t i o n in 133 is 34%. W h e n strength p r o p e r t i e s have a high correlation, it is clearly w o r t h w h i l e to s y m m e t r i z e the double proof-loading procedure. Table 7 Optimal standardized double proof loads and /33 from the inverse of the Fisher information matrix. Symmetric double proof-load design - (#1, er2, P, er2, #2) unknown p

0.1

0.3

0.5

0.7

0.9

i33

2.8949 0.762 1.443

2.4023 0.672 1.545

1.6787 0.519 1.705

0.8254 0.208 1.569

o. 1372 -0.198 0.950

Bi,opt B2,opt

3.4. Asymptotic results

The b i v a r i a t e n o r m a l d i s t r i b u t i o n r e m a i n s a r e g u l a r case even u n d e r multiple p r o o f loading. T h e m a x i m u m l i k e l i h o o d estimate ~ satisfies v / n ( ~ - p) is as-

652

R. Johnson and W. Lu

ymptotically normal with variance ¤33. Our approach has been to determine proof loads to minimize this asymptotic variance. We now present the asymptotic results for the maximum likelihood estimate under the double proof-loading design. THEOREM 1. Let O ---- {[/ii] _<M , M -1 <_ a 2 -< M, Ipl^< 1 - - M - l , i

= 1,2}, where

M > 1. Let Oo be the true value of parameter 0 and 0 be the maximum likeIihood estimate of 0 under double proof-load design. Under a bivariate normal modeI, 1. if Oo C O, Ô ~a's" Oo as n ----+ oo. 2. if Oo ~ interior of O, v~(Ô - Oo) D» N(0, i00_l)"

PROOF. Compare the log-likelihood function (6) under the double proof-load design and the log-likelihood function (1) under the single proof-load design. The third additional term in (6) is just the symmetric version of the second term with subscripts 1 and 2 interchanged. The log-likelihood function and its second order derivatives for the single proof-load case are bounded by some integrable functions (see De Amorim and Johnson, 1986). Because of the symmetry of the parameter space O and the terms in the log-likelihood function, the log-likelihood function and its second order derivatives for the double proof-load case are then also bounded by integrable functions. The asymptotic results follow according to the same argument as in the single proof-load case treated in De Amorim and Johnson (1986).

4. An example using the bivariate Weibull distribution Not surprisingly, the normal distribution does not adequately model many strength properties. Overall (see Johnson et al., 1999), the Weibull distribution seems to be a better model for the various strength properties of lumber. We illustrate our general approach by considering single proof-load designs for estimating p in the bivariate Weibull distribution having survival function F(x,y)=P[X>x,Y>yl=ex

p

-

(Ul)~+(

-

,

0<3_<1

.

The likelihood can be shown to depend on the observations only through

O1j

'

~,02 j

Therefore, even for the case of unknown marginal distributions, we can reduce the calculations to the standard case with fll = f12 = 01 = 02 = 1. This reduces the problem to the bivariate exponential distribution (see Lu and Bhattacharyya, 1990). The optimal proof load )[opt then corresponds to an optimal proof load Lx = 01 (,~opt)l/B1 on the original scale.

Estimation of correlation under destructive testing

653

For the bivariate exponential, 2[F(6 + 1)]2

Corr(X, Y) = v(2a + 1)

1 .

(15)

For instance, when 6 = 0.22, we have Corr(X, Y ) = 0.88. Using the same numerical integration procedure as used for the normal distribution, our computer calculation gives 2opt = 0.502 or Lx = 01 (0.502) 1/fil on the original scale. A plot of the asymptotic variance of a, 133, versus 2 is presented in Figure 3. These numerical calculations establish an interesting extension of the proof load technique for estimating the correlation in the bivariate Weibull distribution. We will continue to pursue this investigation. 5. Nonparametric estimation of correlation and conditional survival

A nonparametric approach can also be taken to estimating correlation and the conditional survival function. We restrict discussion to the single proof load scheine. Johnson and Wu (1991) show that a dense set of proof loads {Li} is required in order to identify the underlying joint distribution. From a practical standpoint several different proof loads are required to obtain reasonable estimates of correlation. Let (X, Y) be a bivariate random variable with support in [0, ec) x [0, oc). We study inferences under the following finite proof load scheme. Fix m increasing proof loads Lm = { L 1 , L 2 , . . , L m } , where Lm is contained in the interior of supp(X). We place ni units under the proof load scheme with proof load Li, i = 1 , 2 , . . , m. The total sample size is n = ~im=l n i.

rho = 0 . 2 2

T,-

o > ,r-

e-

V'--

i

0.2

,

'

,

d

0.6

i

b

1.0

Lambda Fig. 3. The asymptoticvariance of the mle 6 as a function of 2 when the marginals of the bivariate Weibull distribution are unknown.

R. Johnsonand W. Lu

654

T h a t is, for each L~, we consider a r a n d o m sample (X~j, Y~j),j = 1 , 2 , . . ,nj which we transform to

Zij

=

{ Xij if Xij ~ Li Bj if Xij > Li,

if X/j _< Li,

{1

and (~ij ~-

0 ifXij > Li

(16)

for i = 1 , 2 , . . , m . T o obtain estimates of mixed moments, Johnson and Wu (1991) first verify that, if E[X«Y~I[X > LI, Y > L2]I exists for some non-negative e, fl with c ~ + f i > 0,

E[X~Y~I[X > L1, Y ) L2]] = L~L~2Æ(L1,L2) + L~

/7

fly-lF(Ll,y)dy

2

+ LB2fLT o:x«-lÆ(x,L2)dx

+

(17)

~ßx~-~W 1F(x,y)dx dy 1

2

Setting ~ = 0, L1 = Li and L2 = 0, we obtain the conditional mean for a unit that survives p r o o f load Li.

E[Ye[X > L,] -

1 E[Ye]I[X > Fx(L,)

Lg]] --

1 f0 °° Fx(L,) p/-7"(L~,y)dy (18)

for fi > 0. This form for the conditional expectation suggests the intuitive estimator 1

?t i

(19)

E[YalÆ > Li] - n k - L Z YifjI[Xij > Li] , i X ( i ) j=l

where ~'x(Li) is any estimator of ff'X(Li). For simplicity, we take F'x(Li) to be the empirical estimator ni

Bx(Li) = n i 1 ~ I [ X ~ 7 > Li] . j=l

We write f'i for P[X > Li]. THEOREM 2.

i = 1,..,m.

Suppose Then

that ~-1 ¢ ~

and that

ni/n ~ 2i(0 < 2i < 1) .for

x/n(Ê[Y~]X > Li] - E[Y~tX > Li]) __+DN(O, a 2) , where the asymptotic variance is (see Johnson and Wu, 1991)

(20)

Estimation of correlation under destructive testing

= Æi-l(1

-Æi-1)2E2[yB[Li 1 < X < Li] +

Bi-I (/z~i-1 -/~i)

-1- ,~i

--

2

-

-

655

V a r ( r e I [ x > L~-I]) B~i-1 (fr'i-1 -- fr, i)2 Var(Y~I[X > Li]) ,~/(L-1 - &)2

1

1 --

Æi-1

E[Y~I[Li_I < X < Li]IE[Y~I[X > Li-1]]

ù

Bi--1 (Æi--1 -- Æi) ~

1-Æi

. ~ . . IIL~ 1 < X < Li]IE[YfiI[X > Li]] ,

2 2i(/~~._~ ~ i ) J E [ Y

(21)

The estimator in this last theorem can be used to develop approximations to the mixed moments and the correlation coefficient. For a reasonable number of fixed proof loads m, we can numerically approximate the expectation m-1

E[X~Y fl] = ~ E[X~Y~I[Li < X <_Li+I]] + E[X~YflI[Lm < X]]

(22)

i=0

over the grid of proof loads Lm = { L 1 , . . ,Len) by m-1

ELm [X~Y~] = ~-~~L;E[Y~I[Li < X < Li+ll] + L~mE[Y~I[Lm < X]] .

(23)

i--O The approximation may be quite good if the grid of proof loads Lm is fine enough. Finally, Johnson and Wu (1991) and Wu (1993) suggest the estimator en--]

EL,,, ~X~Y ~] = ~-~ L~E[Yfl[[Li < X ~ Li+I] ] Av L~E[YflI[Lm < X]l ,

(24)

i=0

where we write

e[Y~I[Li < X < Li+I]] = e[r~I[Li < x]] - e[Y~I[Li+I < X]]

(25)

and, as in (19), we use the intuitive estimator Bi

1 ~j=l #I[X~j > Li] • Ê[reI[L, < x]] = n-7

(26)

This estimator ean be shown to be consistent as the proof loads become dense and the minimum ni --+ oo. The survival function F(Li, y), evaluated at x = proof load Li, can be estimated by the empirical cumulative distribution function (e.d.f.)

B,~(Li,y) = ~~ßl I[X~7 > Li, Y//j > y] F/i

(27)

R. Johnson and W. Lu

656

The conditional distribution of specimens that survive a proof load is also of interest. In the context of lumber testing, the stronger specimens survive a proof load. If the proof load is low enough not to cause damage, the strength distribution of survivors would be stochastically larger than the original strength distributions for either mode 1 or mode 2 strength. There are two choices for estimating the denominator, P[X > Lil, of the conditional probability. We can use only the observations loaded to proof load Li or all of the observations loaded to at least that strength. Campbell and Foldes (1980) considered a path-dependent estimator of F'x,y(x,y) = P[X > x, Y > y]. Instead, we estimate F(y[X > x) using the empirical c.d.f, as the denominator

~E

kn~¢i,y) Fx,n,(Li)

F<(y[X > Li) = ~

(28)

Alternatively, we could estimate F(yl X > x) using a Kaplan-Meier type estimator

F-~KM n (ylX > Li)

kn,(Li,y)

(29)

where ~'xKM(Li) is the Kaplan-Meier estimator of Fx(Li) and F'n~(Li,y), Fx,ni(Li) are the values of the empirical c.d.f.'s^corresponding to P(Li,y), Px(Li) respectively. Notice that Pm(Li,y) and Fx,ni(Li) are estimated from the data (Zij, ~ij),J = 1 , 2 , . . , nj, generated exclusively under the p r o o f load Li. Further, we let: Nr be the number of survivors at strength Lr-1 and Dr the number of failures in the strength interval (Lr-l,Lrl for r = 1 , 2 , . . , m. Then, we have m nj

Nr = Z Z { I [ Z j «

> Lr-, 6jq = 11 + ( 1 - fij«)}

j--r q=l

m nj. : ZZ{I[Lr_, j - r q-1 m

> L r I],

nj

n't nj

j--r q=l

j=r q=l

and

k5~ (L,) = (-I :vr- Dr ~~

> ~j']}

(30)

nj

= ~~.[[Xj.q j=r q-1 m

<Xjq ~Lj]@Z[Xjq

Nr

Estimation o f correlation under destructive testing

We are interested in the asymptotic distributions

657

of the estimators

-~E

Bnr~(y[X > Li) and Fùi(ylX > Li). For simplicity, denote B ( y ) = Æ(Li,y),

for everyy_> 0, i = 1 , 2 , . . , m

.

(31)

THEOREM 3. L e t m

g/ =

Y/r~

~nr- ----+ ~~r(0

< ~~_< 1)

r=l

as minl<~<m

nr ---+ oo and 2[k, m] : ~~=k m 2r" Then, f o r each f i x e d i, 1 Li) - F/(0) ~ ( y ) ]J

D> N(0, a KM,i(Y)), 2 as minl
(32) where

«t,(y) = [~(y)]= 1-F,.e) '

LF~(O) J

/~iFi (y)

~

F»_,(0)F~(0)

= ff'k-~,/71]

(33)

The proof is given in Johnson and Wu (2000). C O R O L L A R Y 4.

B

[~f(ylX Li) >

- -F/OF) Fi(O) 1j » >N(O, a 2,i( )Y) ,

as minl<_r<_m n r --> O0 ,

(34) where

2

I~'(y) ] 2 [ 1 - F / ( y ) 1

o-~,0,) = L~(O) j

~.(y)

-E'(0)]

~

.

(35)

PROOF. Taking m = 1, the result follows immediately from Theorem 3. Somewhat surprisingly, a calculation of the asymptotic efficiency suggests that ~KA/ Bn~ (Yl X > Li) is a more efficient estimator than Fn (y]X > Li). References De Amorin, S. D. and R. A. Johnson, (1986). Experimental designs for estimating the correlation between two destructively tested variables 3. Am. Star. Assoc. 81, 807-812. Evans, J. W., R. A. Johnson and D. W. Green (1984). Estimating the correlation between variables under destructive testing, or how to break the same board twice. Technometrics 26, 285~90. Galligan, W. L., R. A. J o h n s o n and J. R. Taylor (1981) Examination of the concomitant properties of lumber. In Proceedings o f the Metal Plate Wood Truss Conference, 1979, Forest Products Laboratory, Madison, WI.

658

R. Johnson and W. Lu

Johnson, R. A. (1980). Current statistical methods for estimating lumber properties by proofloading. Forest Products J. 30, 14-22. Johnson, R. A., J. W. Evans and D. W. Green (1999). Some bivariate distributions for modeling the strength properties of lumber. United Stares Department of Agriculture - Forest Product LaboratorF. Johnson, R. A. and W. Lu (2000). A new multiple proof loads approach for estimating correlations. In Recent Advances in Reliability Theory Methodology, Practice and [nference, pp. 245-257 (Eds. N. Limnios and M. Nikulin). Birk häuser, Boston. Johnson, R. A. and K. T. Wu (1991). Inferences about correlation structure based on proof loading experiments. Stat. Prob. Lett. 12, 517 525. Johnson, R. A. and K. T. Wu (2000). In Nonparametric Estimation of Conditional Survival under Destructive Testing (Ed. M. Puri). Asymptotics in Statistics Papers in Itonor of George Gregory Roussas, pp. 243 258 (Ed. M. Puri). International Science Publishers, The Netherlands. Lau, H. T. (1995). A Numerical Library in C for Scientists and Engineers, pp. 299-303. CRC Press, FL. Lu, J. C. and G. K. Bhattacharyya (1990). Some new constructions of bivariate Weibull models. Ann. Inst. Stat. Math. 42(3), 543 559. Suddarth, S. K., F. E. Woeste and W. L. Galligan (1978). Probabilistic Engineering Applied to Wood Members in Bending/Tension. Research Paper FPL 302, US Department of Agriculture, Forest Service, Forest Products, Madison, WI. Wu, K. T. (1993). An Experimental Design for Nonparametric Estimation of Correlation Under Destructive Testing, Unpublished Ph.D. dissertation, Department of Statistics, University of Wisconsin-Madison.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statisti«s, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

~) «

z~ ù.,,

System-Based Component Test Plans for Reliability Demonstration: A Review and Survey of the State-of-the-Art

Jayant Rajgopal and Mainak Mazumdar

1. Introduction

An important step in the design, development and eventual deployment of complex and highly reliable systems is a dedicated program of testing and analysis. The purpose of the testing program is to demonstrate that the system will perform at a level of reliability that is acceptable with respect to the mission for which it was designed. A customary requirement is to demonstrate (at a specified level of confidence) that the system reliability exceeds some pre-specified level. The reliability parameter is typically expressed as a mean time to failure or as the probability of no failures over some specified mission time (such as a warranty period or a deployment interval). Reliability tests for the system can be conducted at various levels of assembly, e.g., component, subsystem, or system, and can be done under different conditions or in different environments. The simplest way to evaluate system reliability is to assemble the complete system and then test it as required. However, this is often undesirable or eren infeasible; system-level tests can be prohibitively expensive due to the time and effort required to first assemble the system prior to testing, and because system-level tests are inherently more expensive than component-level tests. Using component tests to evaluate system reliability has several advantages, such as: 1. They are usually less expensive, more timely, and result in a shorter overall test schedule. 2. They can be done by different organizational entities and proceed at different locations and times. 3. They are offen more informative in that many more aspects of performance can be instrumented at the component level than would be possible in a system test.

659

660

J. Rajgopal and M. Mazumdar

4. Component tests are consistent with principles of Total Quality Management and Design for Manufacture; the entire system does not have to be assembled until it is guaranteed to be reliable. In addition to the economic advantages given above, system-based component tests are particularly useful in the following situations: Systems mixing old and new components: Many systems are designed by combining new components and/of new subsystems together with others that have been successfully used before in earlier designs. This situation is common when the new system is the "next generation" of an existing product, or is one of a family of products with functional similarities. When this situation arises, component testing to demonstrate a system reliability requirement is particularly effective. For a subsystem that has been nsed previously, little or no additional testing may be necessary and additional test effort is required only for the new components, which should now be tested from an overall system perspective. Iterative or evolving design stages: Many systems follow an evolutionary pattern where design improvement is continual as technology advances. At certain points in time, the design is temporarily "frozen" and the product using this design is released into the market place. However, in actuality, the design process is continuous. The personal computer market is one example of a continually changing design. For such systems, component test plans would be more practical to evaluate system reliability. When some particular subsystem changes to the extent that it renders the previous mode1 obsolete, reliability tests can be conducted on it. The corresponding data can then be combined with the most recent test results of other subsystems in order to evaluate the reliability of the new system. Complex multi-function systems: If a new system is intended to perform many complex tasks, it is likely that diverse design teams will have been assembled as part of the overall design project. These design teams are likely to be in different locations and working according to different (albeit coordinated) schedules. In these situations, component test plans represent a more practical approach to system reliability evaluation. The tests can be initiated sooner and at the most appropriate location. The data can then be pooled without having to first assemble the system into its final configuration. There is also another, rather fundamental justification for component-level testing. One of the topics of research that has received a great deal of attention in reliability theory is the development of models that can express system reliability as a function of the reliabilities of its constituent components. Assuming that one accepts the validity of these models, it then stands to reason that results from component tests could be used to make a valid inference on system reliability. This fact has not been lost on other investigators. A snbstantial body of literature exists (Mann et al., 1974) on obtaining interval estimates of system reliability based on component test data. However, there is another related issue that has

System-based component test plansfor reliability demonstration

661

received relatively less attention. This is the design of the component tests whose results will be used (along with the model relating system and component reliabilities) to draw inferences on whether the system is reliable or not. In other words, how does one design component tests that are efficient from an economic perspective, while simultaneously being statistically sound and mathematically tractable? The objective of this paper is to present the various issues involved in the design of minimum cost component test plans, to review the research literature to date that has addressed these, and to summarize the open issues in this area. In general, the resulting problems represent a challenging and unique blend of statistics and mathematical programming. Finally, before proceeding further it should be emphasized that component testing does not necessarily preclude all system testing. There will be instances where system testing is unavoidable. Examples of such situations would be when component failures are not independent, when interfaces between components that make up a system are unreliable, or even more simply, when a system designer is uncomfortable with not doing any system testing. However, even in such situations it might be possible to reduce the amount of system-level testing by combining it with the results of some component testing, and in fact, some recent work has started to examine this approach. 1.1. Problem formulation Before providing a general formulation of the component-test design problem, it is worth commenting on a matter of some importance. The most general and desirable approach would be one that could be applied to systems of all configurations, and a non-parametric one where no assumptions are made on the component failure time distributions. However, the resulting optimization problems are intractable and provide no insight whatsoever into the problem structure. Thus, orte is restricted to examining the design problem for specific types of system configurations and distributions. Furthermore, while other formulations may be equally plausible, the formulation used in virtually all the research in this area takes the viewpoint that there is a cost associated with testing each component type that comprises the system, and that in general, this cost may vary from one component to another. The fundamental question then arises as to how the test program should be designed so that (a) adequate protection is guaranteed against unwarranted acceptance or rejection of the system (i.e., against Type I and Type II errors), and (b) overall test costs are minimized by optimally allocating test effort among different components. The test design problem may be viewed as the experimental design converse of using component test data to estimate system reliability. The question posed in this statistical design problem is the following: " H o w should the total available resources for testing be allocated among the components comprising the system so that the inference about the system reliability can be obtained in the most effective

J. Rajgopal and M. Mazumdar

662

manner?" When the component testing costs are unknown or are not of much significance, this approach also yields the minimum sample sizes for each component that will provide protection at the system level. A general formulation for the component-test design problem may now be developed. Suppose that associated with each component j of the system is a parameter set 0j which determines its reliability (for example, with exponentially distributed component lifetimes the single parameter 2j representing the failure rate for that component measures its reliability, so 0j - ~v). The exact values of this parameter set are unknown. The system reliability Rs may then be expressed as a function of 01, 02, etc., i.e., Rs = f(_0) where _0= [01, 0 2 , . . , 0n] is a vector of parameters, and the exact form of the function f depends on the system configuration (series, parallel, serial connection of parallel systems, etc.). Suppose further, that associated with each component j there is an index Cj that measures the cost of testing componentj. Once again, the specific form of Cj will depend on the format followed by the test plan. Finally, define the two sets

S1 = {OIRs = f ( 0 ) _> R1} ,

(1.1)

So = {OIRs = f(_0) _< R0} •

(1.2)

Here R0 is an upper bound on the unacceptable level of reliability, while RI(>R0) is a lower bound on the acceptable level of system reliability. It may be noted that $1 is the set of all combinations of (the unknown) values for the parameters which lead to a system with a definitely acceptable reliability, while So is the set of values which leads to a system with a definitely unacceptable reliability. The optimization problem of minimizing total test costs subject to constraints on Type I and Type II error probabilities may then be stated as follows:

Problem P: Minimize Z = Z

Cj

J

st Minimum_0~s~{Prob("Accept the system")} _> 1 - c~

(1.3)

Maximum0cs0 {Prob("Accept the system")} _
System-based component test plans for reliability demonstration

663

an acceptable system. Hence, for a given t, the probability of system acceptance should be at least (1 - c0 for all O_E S1, or equivalently, the minimum probability of acceptance across all _0 E $1 should be at least (1 - «). This is the first constraint. Along similar lines, the second constraint imposes a restriction on the maximum probability of accepting an unacceptable system. Thus, given t, the LHS of these two constraints lead to two optimization subproblems (in _0) over the sets Sa and So, respectively. If the optimum values of these two subproblems are, respectively, _>(1 - c0 and
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estimates for the component failure rates, while in others one has virtually none. In many cases one typically has some limited knowledge about component reliabilities (e.g., as upper bounds on component failure rates). In such cases the optimization procedure should seek to exploit such prior knowledge to arrive at a plan that will be less costly, and to avoid situations that may be intuitively unappealing to a practitioner. In summary, the exact structure of the optimization problem (Problem P) will depend on the system configuration, the distributions of the component failure times, the specific format of the test plan, censoring procedures (if any) that are used, a priori information on failure rates, and the nature of the test data. Thus, the optimization problem for a series system of components with exponentially distributed failure times could be quite different from the situation of component life tests for a parallel system with components that have a gamma distributed failure time. The optimization problems could in general, be quite complex and the choice of the algorithm for solving a particular problem will depend on its structure. An interesting feature of Problem P is that of the two optimization subproblems at the inner stage (minimization over Sa and maximization over So) one problem typically turns out to be fairly easy, while the other is usually very hard to solve. That is, a test plan that satisfies one of producer's and consumer's risks is relatively easy to find, but simultaneously satisfying both is usually very challenging. Finally, it should also be kept in mind that although the above discussion is in terms of acceptance or rejection of a system using the concept of an operating characteristic (OC) curve under the traditional decision theoretic framework, a decision to accept or reject the system need not necessarily follow after the completion of the component tests since additional non-statistical considerations may be required for a decision. 2. Review of researeh literature

We begin with a discussion of two seminal papers from which rauch of the research on system-based component test plans has arisen. The original paper to address the topic of system-based component testing is due to Gal (1974). In this work, Gal considered an arbitrary coherent system consisting of n different component types with independent failures. In his scheme, each component typej is tested for tj time units, and the system is "accepted" i f n o failure occurs for each of the component types during these prescribed testing periods. He described a general procedure for obtaining the optimum test times that minimize the total testing cost C(t) = cltl + c2t2 + . . . -t- c~tn, subject to Pr{"accept system"

when Rs <_ R0} _
(2.1)

Here cj denotes the cost of testing component j per unit time, R s denotes the system reliability for a unit time period (e.g., a suitably scaled version of the mission time), Ro is an upper bound on the unacceptable level of reliability, and B is a suitably low pre-assigned probability value. He also gave some specific

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examples for series systems, parallel systems, and a serial connection of redundant units under the assumption that the component lifetimes are exponentially distributed. While Gal's work was the first to formalize the problem it had one inherent weakness. The acceptance rule that he used was very demanding and it was quite conceivable that this could result in a good system being unnecessarily rejected; in the next paper on this subject Mazumdar (1977) addressed this issue. He considered the case of a series system with exponentially distributed component lifetimes. Specifically, he assumed that (i) the times to failure for the components are independently distributed exponential random variables where the jth component has a mean time to failure of 1/2j, and (il) without loss of generality, the mission time is one time unit long. The system reliability over its mission time is given by

Rs = I-[ exp(-2j) = exp j=l

. j=l

(2.2)

/

Using (2.2), it may be seen that the sets $1 and So defined by relations (1.1) and (1.2) reduce to

Sl={)~E~nI~-~2j<_--lnRl,j

2j_>OVj} ,

(2.3)

So={2E~n,Z2j>_-lnRo,j

2j_>OVj} .

(2.4)

Mazumdar considered a formulation identical to that considered by Ga1, but with two important differences. First, he added another probability constraint in keeping with the standard statistical practice for determining the sample size. This constraint is: Pr {"accept system"

when Rs >_R1} >_ 1 - c~ ,

(2.5)

where RI(>R0) is a lower bound on the "definitely acceptable" level of system reliability, and :~ is some suitably low probability. Second, since in general it might be impossible to satisfy both probability constraints (2.1) and (2.5) using the stringent criterion proposed by Gal for system acceptance, he proposed an acceptance rule that was not as demanding as Gal's. Specifically, Mazumdar assumed that component testing took place with replacement. Let J(j denote the number of failures of componentj that occur when it is tested for tj time units. He then proposed the following acceptance criterion which he referred to as the sum rule: "accept the system if ~ j X j < m," where m is an integer valued decision variable. Since ~ j X j follows a Poisson distribution with mean A = ~j2jtj it follows that the probability of acceptance is equal to Fm(A), where Fm(A) is the distribution function of a Poisson random variable with mean A, i.e., Fm(A) = Pr{X < m}.

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Then for given m, Problem P of Section 2.1 reduces to

Program P(m): Minimize

Z cjtj

(2.6)

J /

st

MinimumA~s,

MaximumAcso

\

Fm(~-~Ajtj~ >_ l -:~

(2.7)

\j //•

(2.8) Solving this problem he showed that the optimum component test times are the same for each component irrespective of the testing costs. In his original work, Mazumdar did not explicitly prove that a feasible m was guaranteed to exist but speculated that this was true and that the smallest feasible m was the optimal one. It was shown much later by Rajgopal et al. (1994) that when this criterion is used, there exists an m* such that both (2.1) and (2.5) will be satisfied for all values of m _> m*, as long as c~ + fl < 1. Furthermore, it follows readily that the optimum m is given by m*. Virtually all of the subsequent work builds on the ideas in the two initial papers listed above. This work is now reviewed and the important results summarized. The review is based on the type of design problem solved rather than being in chronological order.

2.1. Series systems Of all the system configurations studied, these have received the most attention on account of their wide applicability as well as the relative simplicity of the expression for system reliability. The first extension was provided by Yan and Mazumdar (1986) who studied three different test procedures. These were based on (1) the total number of failures across all component types, (2) the number of failures for each component type and (3) the maximum likelihood estimator of system reliability. They showed that the first and third procedures led to identical results and based on a very limited comparison indicated that these results were in general superior to the second case from the point of view of costs. More interestingly, it was shown that for a series system the optimum policy called for testing all component types for the same length of time regardless of the test costs. The intuitive explanation for this comes from the fact that a series system is only as good as its weakest link. Thus, in the absence of any additional information there is no justification for testing a component any less than the others, nor anything to be gained by testing it any more than the others. In another paper Easterling et al. (1991) looked at a series system with a formulation along the same lines as the one above, but where the individual component failure data are of the pass-fail type following a binomial distribution.

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This paper had several important contributions. Perhaps the most important one lay in the fact that it was the first to contain a detailed discussion of the need for system-based component test plans by demonstrating the drawbacks associated with the ones that are used in practice. Typically, the practice has been to "allocate" the required reliability for a system among its individual components and then independently require each of the latter to show this level of reliability with some specified level of confidence. For example if a system reliability of 0.99 is required from a series system of two components, one common approach is to require each component to demonstrate a reliability of (0.99) °5. Using system OC curves, in their paper Easterling et al. demonstrate why this approach is erroneous and how it could lead to probabilities of Type I and Type II errors that could be vastly different from the advertised ones. Second, Easterling et al. also elaborated on the surn rule that Mazumdar had introduced earlier for accepting the system, along with a discussion of why with binomial failure data this rule made sense. In fact, it was not until a very recent paper by Mazumdar and Rajgopal (2000) that any formal justification was offered for the surn rule for the case with exponential failures. In this paper, the authors set up the mathematical program for computing the maximum likelihood estimator for system reliability and the Karush-Kuhn-Tucker conditions for the optimum result in expressions that involve the sum of the number of failures observed. The optimization problem formulation with binomial data was more complex than for the case with exponential lifetimes and involved techniques such as majorization and results from Schur-convexity. The results indicated that for a series system, the sample sizes for each component type should be the same (just as optimum test times should be the same with exponentially distributed failure times); the arguments justifying this are similar to the ones made in the previous paragraph. While the above results make sense from a statistical perspective, they could be quite unappealing to a practitioner or system developer who often has some a priori knowledge about the reliability of the various components. In such cases one would want to use this information in conjunction with the relative magnitude of the test costs in order to arrive at a more intuitively appealing test plan. Accordingly, the next major extension to be considered for series systems was the case where a priori information of some kind is available on the failure rates of the individual components (Altinel, 1992, 1994; Rajgopal and Mazumdar, 1995). Specifically, it was assumed that each 2j had an upper bound uj that was known (or estimated). The formulation of the problem is identical to Program P(m) above, with the exception that the sets $1 and So defined by (2.3) and (2.4) are now defined via

&={2+3~n]~-~2j<_-lnR1, J

uj>B_>OVj}

,

(2.9)

So= {2E~n[E# J

uj>_2j_>0Vj}

.

(2.10)

->-lnR°'

While it appears that this change is very minor, the resulting optimization problem is vastly more complicated. Suppose the vector of test times t is fixed.

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For the simpler case where Program P(m) used (2.3) and (2.4), the two inner stage optimization problems defined by the LHS of (2.7) and (2.8) have closed-form solutions. Thus it was possible to readily determine (at the outer stage) the optimum vector t* for which the total test costs are minimized. However, with a priori knowledge on failure rates (2.7) and (2.8) now make use of (2.9) and (2.10) to define the sets $1 and So and the resulting optimization problems yield nontrivial linear programming problems. Thus determining t* at the outer stage is a complex problem. One approach used to solve Program P(rn) was to restate it as a linear program with infinitely many constraints and then employ a cutting plane strategy to refine the feasible space in the region of the optimum. The interesting result here is that with a priori knowledge, the optimal test plan need not require all components to be tested equally, but rather for times that relate intuitively to the magnitude of the upper bound uj as well as the test cost cj. For instance, a component with a tight upper bound as well as a high test cost will be tested for less time than one with a loose upper bound and a low test cost. A detailed discussion of the algorithm and computational results may be found in Rajgopal and Mazumdar (1995). A few years later, Raghavachari (1998) examined the algorithm further and proposed a simpler procedure to solve the same problem. This method was based on results from linear programming duality; and the same numerical results were obtained but with far less computational effort. Rajgopal and Mazumdar (1995) also provided the formulation and a solution procedure for the case where the failure times follow a Gamma distribution. They assumed that the distributions shared a common, known shape parameter but the scale parameters were unknown. Once again, they considered both the case without and the one with known bounds on the failure rates. For the former case, they developed exact procedures which once again resulted in equal test times for all components. For the latter case, they used a Normal approximation to develop the optimization problem since the exact problem was intractable. As in the exponential case with a priori knowledge, the second stage problems are nontrivial linear programming problems. However, unlike the exponential case, the constraints for the first stage problem are now n o n l i n e a r in t. Fortunately, the feasible region is convex and once again, a cutting plane strategy was adopted with a sequence of nonlinear programs being solved at successive iterations. Each of these programs turns out to be a geometric program (a special kind of convex programming problem for which efficient algorithms exist). Finally, the series system was also studied by the same authors for the case where the component failure times follow a Weibull distribution (Rajgopal and Mazumdar, 1997a, b). Once again they used a Normal approximation to develop a solution procedure for the resulting formulation. The results for both the Gamma and Weibull distributions were consistent with those for the exponential distribution. The last paper was also important in that it was the first to introduce the notion of imperfect interfaces between the components of the system; thus far, all papers had assumed (for all system configurations) that interfaces were perfect and system failures occur only on account of component failures. In practice this is often not the case since systems often fail at an interface such as a weld, seam,

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solder, connecting wire, etc. Indeed this is one of the reasons why m a n y practitioners rend to be wary of component tests alone. Two recent papers (Rajgopal et al., 1999; M a z u m d a r and Rajgopal, 2000), consider this situation. The approach advocated in these papers is to consider both system testing and component testing. Rather surprisingly it turns out that in the absence of any prior information the optimum policy is to conduct only system tests. One explanation for this could be that with the "weakest-link" property of a series system there is no justification for testing a component any more or any less than some other component. Thus, since a system level test accounts for all the components equally, there is no reason to test components since such information would be redundant and simply add to test costs. Once again the authors consider the more c o m m o n situation when some prior information is available on the relative magnitudes of interface and component reliabilities. In this instance the authors considered the case where the combined reliability of the interfaces was bounded by some multiple of that of the components. Clearly, a weakness is that such a multiple is hard to estimate easily in advance. Therefore, the authors developed a procedure that attempts to provide a solution for a range of values of the multiple. Results are provided for several different cases and these indicate that with a priori information one could have only system tests in some cases or both component as well as system tests in other cases. All of the test plans for series systems that are discussed above replace components that fail during testing and use Type I censoring, i.e., testing is halted after a fixed interval of time and the number of observed failures is a random variable. In contrast, with Type II censored plans one would still replace failed components, but for each component t y p e j the number of failures to be observed (say rj) would be fixed and the time taken to observe this m a n y failures (Tj) would be a r a n d o m variable. In general, Type I censoring would appear to be preferable, especially with highly reliable systems, since it may take a very long time to observe eren a small number of failures. A Type II censored plan for a series system was studied by Rajgopal and M a z u m d a r (1996). The test statistic used was R = ~ j [(rj - 1)/Tj]; this quantity was shown to be an unbiased estimator of ~j2j. The decision rule was to accept the system as long as R < d, where d is a predetermined constant (similar to the integer m in Type I censored plans). Once again the objective was to minimize test costs but by now finding the optimum values of rj and d. I f the components are assumed as usual to have exponential failure times, the r a n d o m variables Tj each follow a G a m m a distribution. However, the distribution of R is unknown. Based on an approximation of this via the N o r m a l distribution the authors prove that similar to the case of Type I censoring, the optimum policy is independent of the number of components and requires the same number of components of each type to be tested. Motivated by the fact that the optimum d and r a r e independent of costs and the number of components, an exact procedure is then developed for a simplified problem with a single component and this leads to a closed form solution. While the approach and the results in this paper were interesting extensions to other distributions or to the case with prior information have not been considered thus rar.

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Finally, before moving on to other system configurations and specific extensions, we mention some interesting recent work by Sankar and Vellaisamy (2000) for a series system. In this paper the authors develop a two-stage test plan. The intuitive justification for such plans is similar to the logic behind double sampling plans that are commonly used in acceptance sampling. The purpose is to try to reduce the expected test times while retaining the same overall constraints on probabilities of Type I and Type II errors. The basic testing approach is similar to the earlier work cited, but now the plan has two numbers ml and m2 associated with it. The procedure is as follows: at the first stage each component j is tested for tlj units of time and the number of failures Xlj observed. If ~jXlj ~- mb the system is immediately accepted and if ~ j X l j > m2, the system is immediately rejected. If m l < ~ j X l j < m2 one resorts to second-stage testing where each c o m p o n e n t j is tested further for t2j units of time and X2i, the number of failures at this stage is observed and recorded. The system is finally accepted if [~jXlj + ~jX2y] < m2 and rejected otherwise. Note that the total test costs are no longer fixed for a plan with given test parameters since the total time on test is not deterministic any more. In attempting to arrive at optimum test times, the authors therefore minimize an expression for the maximum average cost subject to the usual constraints. The authors address both the case with and the one without prior information. For the latter case they adapt the simplified approach proposed by Raghavachari and for the former case where the problem is more complex, they use a meta-heuristic genetic algorithm for solving the optimization problem. The most interesting result of this paper is that for the same problem that was considered earlier by Rajgopal and Mazumdar (1995), the authors show that the maximum average cost is lower than the minimum cost for the single stage plan by a little over 10%. They speculate that the actual cost could be rauch lower than the optimized value of the maximum average cost.

2.2. Parallel systems Parallel systems are in general more difficult to handle than series systems on account of the somewhat more complex expression for the system reliability as a function of component reliabilities. All the papers that have addressed parallel systems have restricted their attention to the case of exponentially distributed failure times. Yan and Mazumdar (1987a) were the first to examine such systems where they compared several plans using Type I censoring. They examined a system of n components in parallel where the jth component had an exponentially distributed time to failure with parameter 2j and that the failures were independent. They also made the assumption that each 2j was "much smaller" than one. Given that each 2j is small they were able to approximate the system reliability expression as Rs = 1 - H [ 1 - exp(-2j)] ~ 1 - H 2 j J

.

(2.11)

J

This simplification allows the optimization problem to become tractable. In their paper, the authors examined three classes of plans: (a) a procedure based on the

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total number of failures across all component types with the usual sum-rule as the basis for system acceptance, (b) a procedure based on the number of failures of each component type ()(j) with the system accepted only if each Xj- _< rnj (given), and (c) a procedure based on the maximum likelihood estimator of system reliability (/)) with the system accepted only if Æ > d (given). In each case, the problem is to minimize test costs subject to satisfaction of constraints on Type I and Type II error probabilities. For all three classes of plans the authors make one crucial assumption given by the following inequality: )~Æ/(~j2j)_> a, for k = 1 , 2 , . . , n , where 0 < a <_ 1/n, i.e., that the failure rate of one particular component is not disproportionately larger than those of the others. The addition of this constraint was necessary to make the problem meaningful, because without a positive lower bound on each )~j there would be no possibility of occurrence of Type II error. Based on their somewhat limited computational analysis the authors found that the plans of type (c) fared much worse than the first two classes of plans, and that the plans of class (a) in general had a distinct advantage (in terms of costs) over those in class (b). More significantly, the authors found that the optimal test plans depend very strongly on the bounding constant a, which needs to be pre-specified. One suggested basis of comparison to neutralize the effect of a was to base the comparison on the same total number of component failures tolerated by each plan before rejection (e.g., the class (a) plan with rn = 2 and the class (b) plan with m l = 1 and m2 = 1 both tolerate a maximum of two failures). Comparison on this basis suggested that class (b) plans might be slightly better than class (a) plans. In a companion paper (Yan and Mazumdar, 1987b), the same authors also looked at a parallel system with Type II censoring. Once again it was assumed that component failure times are independent and exponentially distributed, and that unit test costs for each component type are different. However, in this approach testing of a particular component type j is halted when a pre-assigned number of failures (say kj) have been observed. The test statistic used was given by S = 1-IjTj where Tj is the total time on test for component type j, i.e., the time required to observe kj failures. The rule used was to accept the system as long as S > B, where B is a constant that is a function of the kj values. The objective is to determine the values of kj and B that minimize total test costs subject to the usual reqnirements that the error probabilities be suitably small. Based on the fact that each element of the product follows a Gamma distribution, the distribution of the product may be derived, but computation of the distribution function at specific points is rather complicated. The authors looked at a simple two-component system and presented numerical results for the same. In a subsequent paper by Rajgopal and Mazumdar (1988), a slightly different acceptance criterion was provided for the same design problem; this criterion was based on the sum of the logarithms of the test times. This vastly simplifies the process of computing the critical value for the test statistic; the authors provided several approximate procedures for this computation. A more comprehensive approach to test plans for parallel systems was provided recently by Rajgopal and Mazumdar (1998). As with all parallel systems,

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the authors assumed the system was "highly reliable" so that the reliability expression could be suitably approximated via (2.1 l). The authors discuss the quality of the approximation and conclude that in general it works well when (1) all 2j are oft roughly similar magnitude, (2) the number of components is relatively large, and (3) all 2j are small (say <0.1). The authors also derive the maximum likelihood estimator for system reliability and show that using this to accept the system is equivalent to requiring that 1-Ii (~/tj) _< d for system acceptance, where d is a constant to be determined. Equivalently this implies that S = ~ j in Xj. < In d + ~ j In tj. The Central Limit Theorem is then invoked to obtain a Normal approximation for the distribution of S and this is then used to formulare the usual optimization problem. The authors show that when this is done, the inner stage optimization problems are unbounded as a result of the approximation used in (2.11). This in turn renders the overall problem infeasible. To get around this problem, an assumption is made that each 2j is also bounded from above by some known upper bound. This, of course, is the same situation of a priori information examined with series systems, and in practice is a reasonable assumption. With this additional assumption the problem can now be solved. While a rule based on the M L E is desirable, it requires the approximation of the system reliability via (2.11) as well as the Normal approximation for the distribution of the test statistic in order to get useful results. Moreover, since the exact distribution of the test statistic is unknown it is impossible to actually compute the exact levels of protection from Type I and Type II errors. Given these drawbacks, the authors proceed to examine the familiar sum rule that was used with series systems. With this rule they show that in the absence of any a priori bounds on 2j it is impossible to protect against Type I error. However, if orte needs protection only from Type II error the authors provide an exact cutting plane algorithm for solving the optimization problem, as well as a simpler procedure based on approximating R s via (2.11). For the more general case, with both types of errors they consider the situation with a priori bounds on 2j. Once again, they provide a cutting plane algorithm where a succession of nonlinear (but convex) programs are solved at each iteration. Illustrative numerical examples are provided as well. The general conclusion here is that with the sum rule the test times rend to be somewhat more conservative (longer) than with the MLE-based approximate procedure, and that the critical value for use with the sum rule will increase as the upper bounds become looser or as the range R1-Ro becomes smaller. We conclude this subsection by noting that parallel systems in general appear to present more challenges than series systems in terms of analyzing different failure time distributions, as well as in selecting an appropriate test statistic and acceptance rule that will lead to a tractable optimization problem. 2.3. Other topics

In this last subsection we review some of the other work on system-based component testing where the focus is not specifically on a series or a parallel system.

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First, with respect to other system configurations it appears that there has been only one paper (Mazumdar, 1980) that explicitly looks at a different configuration; in this case a serial connection of parallel subsystems. Units in subsystem j are assumed to be identical with independent, exponentially distributed failure times with parameter 2j. The usual sum-rule is used for deciding on whether the system should be accepted or not. To solve the optimization problem a sequential approach is followed where for any given value of the critical value m the times are first determined with the Type II error constraint ignored; there is a closedform solution for this. Next the inner optimization problem for the Type II error constraint - the LHS of the constraint given by (2.8) - is solved for these times; a simple gradient-free algorithm (the method of Hooke and Jeeves) is used for this purpose. If the Type II error constraint is not satisfied, m is increased and the procedure repeated. The author provides numerical illustrations for two simple systems. Once again, the results indicate that the optimal test times are independent of the unit test costs. In most of the work reviewed thus far it was assumed that the component failure rates, while unknown, were constant and did not change over time. Altinel and Ozekici (1997) consider the situation where this is not necessarily true. For example, it might be that the failure rate is small during the early stages of the system's mission time but increases later on. Rather than look at failure rates that change continuously or arbitrarily, the authors restrict themselves to the simpler case where each component has a "piece-wise" constant failure rate function. That is, the lifetime is exponential with a failure rate that is constant over discrete intervals of time (which the authors refer to as "environments"); this constant takes on different values for different intervals in the mission time. The authors also assume that a priori information in the form of upper bounds on component failure rates is available. Assuming that the environment changes dynamically, each component j is tested in each environment x with replacement of failed components for tj{x) units of time and the familiar sum-rule is used with the total number of failures (across all components and all environments) being used as the test statistic. The authors formulare the optimization as a semi-infinite linear program and use essentially the same cutting-plane algorithm presented by Altinel (1994) to solve the problem. They provide an example of a serial connection of two subsystems working in three different fixed environments. In another paper that extends the ideas in the above reference (Altinel and Ozekici, 1998), the same authors consider a model where component failures are stochastically dependent, but that this dependence is due only to the random environment in which all components operate as a system. Essentially, the model is a fairly straightforward extension of their prior work in that the system is assumed to operate in a randomly changing environment identified as X = {Xt, t > 0} where Xt is the environment at time t. This variable X is assumed to follow an arbitrary stochastic process. The authors assume that some prior information on failure rates is available for each environment in the state space and use essentially the same solution procedure. They provide an example of a series system of three components operating in a two-stage randomly changing envi-

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ronment, i.e., Xt = 1 for 0 < t < min(1,U) and X~, = 2 for min(1,U) _< t < 1 where U is an exponentially distributed random variable. The a priori information is once again in the form of upper bounds on the failure rates 2j(x) for x = 1, 2. An interesting extension of system-based component testing is to the area of software reliability. The idea here is that a software system may be viewed as being composed of a collection of modules (e.g., functions or subroutines) tied together in some logical fashion. Clearly, the reliability of the system is dependent on the reliability of each module. The quantification of this relationship is based on the fact that software use may be characterized by some use distribution that captures the relative frequency of usage of the modules across various types of applications. Using this notion Cheung (1980) developed a Markovian model for the transfer of control between various modules in the system, which in turn allows the system reliability to be expressed as a polynomial function of the component reliabilities. In using this model, it was suggested by Poore et al. (1993) that the standard practice of allocating the system reliability goal reliability among its modules be followed; each module would then be tested to meet its own individual allocation at some specified confidence level. As mentioned earlier, this could of course lead to protection levels quite different from the advertised values. Rajgopal and Mazumdar (1998) suggest a system-based component test procedure. Only bounds on Type II error probability are considered and an optimization problem is formulated where the objective is to minimize the total number of test instances across all modules subject to this constraint. The decision rule used is the usual sum rule. In terms of the probability distribution, the authors note that the number of observed failures X:. from kj independent tests of module j (with reliability rj) is binomial with parameters kj and (1 - rj). Assuming that kj is small and rj is large, Xj is approximately Poisson and hence, so is ~ j X j with parameter ~-~~jki(1 - rj). Using this approximation the optimization problem is a two-stage problem once again, where the inner stage problem has a linear objective with a single non-linear constraint, while the outer stage problem is an integer linear programming problem. This leads to the usual cutting plane approach where the nonlinear program is solved to generate additional cuts for the linear program. The authors present solutions to several problems in the literature. Finally, in some recent work Jin and Coit (1999) address the use of component test plans to minimize the variance of the system reliability estimate for a series parallel system for binomial data. The question they have addressed is as follows: given the system configuration, how many times should each individual component be tested in order to minimize the estimate of the variance of the system reliability estimator? For this computation, they need a preliminary estimate of the individual component reliabilities, which they obtain from the test data.

3. Summary and future work

This paper has reviewed the literature on the design of component test plans for the purpose of making inference on system reliability. It should be clear from the

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preceding discussion that the design of these plans should take into account the system configuration and the system model that expresses the system reliability in terms of the component reliabilities. The formulation provided here demonstrates that the derivation of these plans requires a combined application of both statistical and optimization techniques. As our review shows, the derivation of the test plans is nontrivial. There exist many open areas for further investigation. First, from a practical point of view one needs to consider the set of criteria on which the test plans should be based. In this article we have considered a test of hypothesis framework and obtained test plans that guarantee protection against the Type I and Type II errors. This is in keeping with the guiding philosophy behind many reliability test plans and forms the basis for sample size determination in most statistical studies. However, other formulations are possible and should be considered. In the test of hypothesis framework that this paper has followed the main interest is in the system reliability; the component reliabilities are the "nuisance parameters". The method of optimization that we have adopted eliminates these nuisance parameters by considering the worst possible case, that is, by maximizing or minimizing over the entire parameter space. This leads to well-defined test plans, but they may be overly conservative. Therefore, an important question is how to remove the excessive conservatism from the test plans? This paper has given illustrations on how this can be achieved in some instances by restricting the size of the parameter space based on the prior information for the upper bounds on the component unreliability parameters. Another approach may be to consider a Bayesian framework. Here also an appropriate formulation will become necessary. There are also other possibilities for reducing the required test times such as considering multi-stage or sequential sampling. This will lead to a completely different research area. Another statistical issue that comes up in this problem relates to the question of how best to combine information from the component tests to make inference on system reliability. For example, in the discussion given in section 2.1 for the Type I censored component tests for a series system, the system parameter of interest is ~j2j. But the statistic that we have used is ~jXj., which is sufficient for the parameter ~j2jtj. If the tj's are different, what should be the best statistic? Finally, a challenging area for future research is the development of the mathematical programming techniques required to solve the optimization problem for various classes of plans. As seen in this review, the main problem in all cases has one constraint resulting from each of the Type I and Type Il error probabilities. However, one of the subproblems that arise from these is typically easy to handle (e.g., yields a convex programming problem) while the other is very difficult to solve. This in turn leads to the need for approximations or simplifying assumptions. The challenge is to develop techniques that can solve the exact design problem; perhaps through the application of some of the more recent meta-heuristic techniques such as evolutionary algorithms, tabu-search, etc.

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References Altinel, I. K. (1992). The design of optimum component test plans in the demonstration of a series system reliability. Comput. Stat. Data Anal. 14, 281~92. Altinel, I. K. (1994). The design of optimum component test plans in the demonstration of system reliability. European J. Oper. Res. 78, 97-115. Altinel, I. K. and S. Ozekici (1997). A dynamic model for component testing. Naval Res. Logistics 44, 187 197. Altinel, I. K. and S. Ozekici (1998). Optimum component test plans for systems with dependent components. European J. Oper Res 111, 175-186. Cheung, R. C. (1980). A user-oriented software reliability model. IEEE Trans. Software Eng. SE-6, 118-125. Easterling, R. G., M. Mazumdar, F. W. Spencer and K. V. Diegert (1991). System based component test plans and operating characteristics: binomial data. Technometrics 33, 287-298. Gai, S. (1974). Optimal test design for reliability demonstration. Oper. Res. 22, 1236-1242. Jin, T. and D. Coit (1999). Allocation of test units to minimize system reliability estimation variability. Rutgers University Industrial Engineering Department, Working Paper 99 122. Mann, N. R., R. E. Schafer and N. D. Singpurwalla (1974). Methodsfor Statistical Analysis and Life Data. Wiley, New York. Mazumdar, M. (1977). An optimum procedure for component testing in the demonstration of series system reliability. IEEE Tran. Reliab. R-26, 342-345. Mazumdar, M. (1980). An optimum component testing procedure for a series system with redundant subsystems. Technometrics 22, 23-27. Mazumdar, M. and J. Rajgopal (2000). Minimum cost test plans for a series system with imperfect interfaces. In Perspectives in Statistical Science, pp. 209-218 (Eds. A. K. Basu, J. K. Ghosh, P. K. Sen, and B. K. Sinha). Oxford University Press, New Delhi. MIL-HDBK-781 (1987). Washington: Department of the Navy, Space and Naval Warfare Systems Command, Washington DC 20363, July 14, 1987. Poore, J. H., H. D. Mills and D. Mutchler (1993). Planning and certifying software systems reliability. IEEE Software, 88-99. Raghavachari, M. (1998). A note on optimal component test plans for series system reliability with exponential failure times. Technometrics 40, 345-347. Rajgopal, J. and M. Mazumdar (1988). A type-II censored, log test-time based component testing procedure for a parallel system. IEEE Trans. Reliab. 37, 406-412. Rajgopal, J., M. Mazumdar and T. Savits (1994). Some properties of the Poisson distribution with an application to reliability testing. Prob. Eng. Inf. Sci. 8, 345-354. Rajgopal, J. and M. Mazumdar (1995). Designing component test plans for series system reliability via mathematical programming. Technometrics 37, 195-212. Rajgopal, J. and M. Maznmdar (1996). A system based component test plan for a series system, with Type-II censoring. IEEE Trans. Reliab. 45(3), 375-378. Rajgopal, J. and M. Mazumdar (1997a). System based component test plans for reliability inferences. In Frontiers in Reliability 1998, pp. 295-302 (Eds. Basu et al.). World Scientific Press, Singapore. Rajgopal, J. and M. Mazumdar (1997b). Minimum cost component test plans for demonstrating reliability of a parallel system. Naval Res. Logistics 44, 401-418, 1997. Rajgopal, J. and M. Mazumdar (2001). Modular Test Plans for Certification of Software Reliability. (to appear in IEEE Trans. Software Eng.). Rajgopal, J., M. Mazumdar and S. V. Majety (1999). Optimum combined test plans for systems and components. IIE Transactions 31(6), 481-490. Sankar, S. and P. Vellaisamy (2000). Two-Stage Component Test Plans for Testing the Reliability of a Series System. Department of Mathematics, Indian Institute of Technology, Powai, Mumbai 400076, India. Yan, J. H. and M. Mazumdar (1986). A comparison of severaI component testing plans for a series system. IEEE Trans. Reliab. R-3g, 437-443.

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Yan, J. H. and M. Mazumdar (1987a). A comparison of several component testing plans for a parallel system. IEEE Trans. Reliab. 11-36, 419~424. Yan, J. H. and M. Mazumdar (1987b). A component testing plan for a parallel system with Type II censoring. IEEE Trans. Reliab. R-36, 425-428.

N. Balakrishnan and C. R. Rao, eds., Handbook ofStatistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

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Life-Test Planning for Preliminary Screening of Materials: A Case Study

Jeff Stein and Necip Doganaksoy

Notation n ~s ~N

is(P) iN(E) A

sample size Weibull scale parameter for standard material Weibull scale parameter for new material Weibull shape parameter 100Pth Weibull percentile for the standard material 100Pth Weibull percentile for the new material true difference between the 10th percentiles of the failure time distributions for the standard and the new materials Maximum likelihood estimator

1. Introduction

An important consideration in the design of any experiment is sample size determination; the use of too many test specimens results in a waste of resources and inconclusive results may be reached if too few test specimens are chosen. Other relevant information should accompany the sample size determination question such as • • • •

the objective of the study (e.g., estimation vs. demonstration), the parameter of interest (e.g., mean or standard deviation), the desired level of precision, and an assumed statistical distribution of the measurements under study (e.g., the normal distribution is often assumed).

See Hahn and Meeker (1991) for a detailed review of sample size considerations in general settings. Sample size determination for life-test planning is complicated because • the response variable, "time to failure", does not typically follow a normal distribution, ~ 679

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• there are often censored (i.e., unfailed) units at the end of the test, and • percentiles of the statistical failure distribution are often of more interest than the mean or standard deviation. Nelson (1982), Lawless (1982), Meeker and Escobar (1998), and Tobias and Trindade (1995) provide a review of life data analysis methods that are widely used in industrial applications. The complications that arise in life-test planning restrict the use of standard formulas to calculate sample sizes for confidence intervals and hypothesis tests. For example, the formula n = (za/~) 2 is widely known and is used to determine the sample size, n, necessary to estimate the mean of a normally distributed process characteristic with a certain level of confidence (determined by the value of the standard normal variate z) within error, +e. The parameter, a, is the population standard deviation of the property of interest. If the distribution of the process characteristic is not normal (as in many life testing situations), one can invoke the Central Limit Theorem and still use this same sample size formula in such a case. However, the usefulness of this formula in lifetest planning is highly limited (assuming the characteristic of interest is the population mean time to failure) because this formula assumes all failure times are observed (i.e., no observations are censored) by the end of the experiment. In this chapter, we present a sample size determination approach to compare the time to stress corrosion cracking of two materials. Rather than calculate a single "optimal sample size" number, we provide useful insight, guidance, and rationale for evaluating tradeoffs between sample size and test duration. We begin this paper with background information on the case study in Section 2. Our lifetest planning approach is presented in Section 3. The life-test planning results for our case study are presented in Section 4. We end with a discussion of further considerations in Section 5.

2. Case study background An engineer sought our advice on sample size planning in a preliminary screening stage of an experiment to compare the time until stress corrosion crack initiation for a standard material (stainless steel) with the time until crack initiation of a new formulation of the standard material (a stainless steel alloy of a different zinc content and formulation). The engineer is most interested in estimating the "early" crack initiation failures. Therefore, a comparison of the two materials (standard and the new formulation) will be made at the 10th (P = 0.10) percentile of their respective life distributions. The experimental test specimens are stainless steel structural components of boiling water nuclear reactors. Specimens from both the standard and new materials will be tested simultaneously using a machine loaded stress test at a single high stress condition. The maximum number of units the engineer can test at one time is about 70-80 specimens. After the sample size is determined, an equal allocation scheme will be applied to test specimens of the two types of materials.

Life-test planning for preliminary screening of materials: A case study

681

The objective of the screening stage is to determine, within 12 months from the outset of the experiment, if the new material offers potential for further study. The engineer sought our guidance on the following issues before conducting the experiment: • How many units to test? • What is the duration of the test? Under normal operating conditions, the standard materials "survive" several decades. Therefore, the test conditions needed to be accelerated to ensure crack initiation failures without introducing a failure mode not usually experienced under normal operating conditions. Based on data collected from one historical study using the standard material, the engineer chose to use a constant stress load of 176 MPa consistent with an assumption that 20% (Case 1) to 40% (Case 2) of test specimens will fail in two months for the standard material. The failure proportions are principally controlled by the stress level applied to the test specimens. Although a higher stress level will decrease the test duration, the engineer thought other failure mechanisms might possibly be activated at higher stress levels. If the new material is qualified based on the outcome of this study, a future study will be conducted to evaluate this material under multiple stresses.

3. Proposed approach to life-test planning The two-parameter Weibull distribution was chosen to model the time to crack initiation. This distribution is characterized by a scale parameter, ~, and a shape parameter, fi. The scale parameter, also known as the "characteristic life", is always the 63.2nd failure percentile and has the same units as the response variable (i.e., months). The shape parameter is a unitless measure determining the shape of the distribution. The cumulative distribution function (cdf) and 100Pth Weibull percentile are given below in (3.1) and (3.2), respectively. F(t)=l-e

('/~)~, t > 0 ,

t(P) = c~(- ln(1 - p ) l/~) .

~,fi>0

,

(3.1) (3.2)

Data ffom one historical stress corrosion crack initiation study suggest the Weibull distribution as a model for this type of data. In this study, the new material will be further investigated if iN(0.10) -- iS(0.10) > 0 where iN(0.10) and ~s(0.10) denote the maximum likelihood (ML) estimates of the 10th percentiles of the new and standard material Weibull failure distributions, respectively. Although the corresponding Type I error of this "point estimate" approach is 50%, we are more concerned about failing to qualify a good material (Type II error) than incorrectly qualifying an

682

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improper material (Type I error). A large Type I error of 50% will be tolerated in order to reduce the Type II error (for a fixed sample size and test duration). The determination of the Type II error is critical to life-test design; a simulation-based approach will be utilized to estimate this error. Alternative approaches are discussed in Section 5. We prefer an approach based on point estimates described above because the engineer firmly believes the new material is an improvement, the screening of materials is only preliminary, and smaller sample sizes will be required. In other life-test planning situations it is orten necessary to find a sample size necessary to (statistically) demonstrate with a high degree of confidence that the difference between a specified percentile of two failure distributions is less than a specified value. Such demonstration testing will involve comparing confidence limits on the failure percentiles (to see if they overlap) rather than point estimates. The proposed approach can be easily adapted for design of a demonstration phase (see Section 5.2). The simulation approach taken in this paper is similar to the one outlined in Meeker and Escobar (1998, pp. 231-236) and also Nelson (1990, pp. 349 361). The output of our simulation are estimates of the probability of correctly qualifying the new material based on differences between the 10th percentiles of the life distribution for the new and standard materials. These estimates are computed using the empirically derived sampling distribution of differences between the 10th percentiles. 3.1. Simulation methodology

Our goal of the simulation is to generate sampling distributions of the differences between the estimates of the 10th percentiles of the life time distributions for the standard and new materials. The simulations are conducted under various scenarios of • assumed failure time distributions for the standard and new materials, • sample size (n), and • test duration. The sampling distributions from the simulation will be used to determine sample size and test duration combinations that are responsive to the needs of the engineer. We accomplish our goal by first randomly generating a data set of size n from the standard and new material failure time distributions (the next section describes a more detailed characterization of these distributions). The randomly generated failure times are censored at a pre-specified time (test duration); randomly generated failure times greater than the pre-specified time are set to the test duration valne and treated as censored in the estimation routine. The failure distribution parameters are estimated for both of these randomly generated data sets using the method of ML. The 10th percentiles are then estimated and the difference between these percentiles is computed. If the difference between the 10th percentile estimates is greater than zero (i.e., new material lasts longer) then

Life-test planning for prelirninary screening of materials'." A case study

683

it will be concluded that the new material warrants further investigation. This procedure is performed 2000 times (for fixed n and test duration), yielding a sampling distribution of the estimated difference in 10th percentiles between the failure distributions of the standard and new materials. Pseudo-code for this algorithm is listed in Appendix A using the Weibull distribution as an example; it is easily adapted for other failure time distributions. Figure 1 presents an example of the simulated sampling distribution of differences between the 10th percentiles. The design inputs for this example are: 20% of the standard material fails in two months and the new material represents a onefold improvement over the standard material (see Case 1, Table 1). A sample size of 40 specimens was used in this example and the test duration was specified as nine months. The proportion of observations above the vertical line drawn at zero is an estimate of the probability of correctly concluding the new material

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No change 1-fold improvement in the 10th percentile 2-fold improvement in the 10th percentile

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5.44 10.88 16.32

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684

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offers potential for further study (based on a comparison of point estimates). This probability will be used as our criteria for evaluating tradeoffs between sample size and test duration. The spread (uncertainty) in the sampling distributions of the difference in 10th percentiles will be controlled by, for a given distribution, the number of test specimens and test duration, and its mean will be determined by the specified "improvement" between the two materials. Assumptions regarding the form and parameter values of a material failure time distribution need to be input to the simulation to characterize the life properties of the two materials. We present these assumptions in the next section. 3.2. Simulation design

Because very few well-designed stress corrosion crack initiation life-tests have previously been conducted, sample size calculations are presented under various assumptions. The design factors considered in the simulation study are: • assumed failure time distributions for the standard and new materials, • sample size (n), and • test duration. Failure time distribution assumptions: The two-parameter Weibull distribution was chosen to model the time to crack initiation. An estimate of the shape parameter from an historical study is about 1.5; this shape parameter value was assumed in the simulation for all scenarios. We defined a "small" improvement between the two materials as a onefold increase of the 10th percentile of the time to crack initiation Weibull distribution for the new formulation over the same percentile of the standard formulation Weibull distribution. We defined a "large" improvement as a twofold increase in the 10th percentile of the respective time to crack initiation Weibull distributions. Based on inputs from the engineer, we considered two baseline conditions for the standard material: 20% (Case 1) and 40% (Case 2) failing in two months at the constant test stress level. The parameter assumptions for the Weibull failure distributions of the standard and new materials are displayed graphically in Figure 2(a) and (b) and numerically in Table 1. Formulas to determine the Weibull parameters for the cases considered are given in Appendix B. The results of the simulation rely on assumptions of population parameter values for the failure distributions. These assumptions should be representative of the failure mechanisms and failure proportions for both the standard and new materials. Inaccurate parameter specifications may result in over or under estimarion of appropriate sample size values. In our simulation, the parameters are estimated by the method of M L from the simulated data sets allowing the uncertainty in their estimation to be incorporated. Most commercially available software packages permit M L estimation from the type of data described here (e.g., SAS Proc Reliability, 1997; Minitab, 1997; SAS JMP, 1995; S-Plus, 1998). Sample size: In our simulation, we will consider sample sizes n = 10, 15, 20, 30, 40.

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686

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4. Life-test planning results for the proposed approach Figure 3 displays empirically derived estimates of power (probability of concluding a difference in the 10th percentiles between the failure time distributions of the standard and new materials) as a function of sample size and experimental test duration. The Type I error (i.e., the probability of incorrectly qualifying an improper material) has been set to 0.50 (i.e., one would reject the hypothesis of no difference between the 10th percentiles of the material failure time distributions if the estimated difference between these percentiles, tN(0.10)- ts(0.10), is larger than zero). The values plotted in Figure 3 are analogous to the proportion of observations greater than zero in Figure 1. As one would expect, the lines for larger sample sizes are above the lines for smaller sample sizes. Also, notice the estimates of power are roughly constant for a fixed sample size after a test duration of about nine months. This phenomenon is a result of a large proportion of standard and new material test specimens "failing" in nine months in the simulation under the various scenarios. Most specimens in the sample have failed by this time so we are not observing any further failure times. Also note the estimated power cannot be determined for a sample size of n = 10 for Case 1. This is because, for some simulation scenarios, all but one simulated failure time was censored. Both Weibull parameters cannot be estimated by the method of M L for these scenarios. Based upon these graphs the engineer arrived at answers to his questions: 1. How many units to test? The engineer chose 30 specimens from each of the standard material and the new material formulations to test. This sample size, chosen from a practical and statistical standpoint, guarantees at least 94% power. 2. What is the duration of the test? The information in the graphs guided the engineer to feel reasonably confident that by the end of a 12-month experiment, he would reach the correct conclusion. Through an iterative exchange of engineering knowledge and statistical guidance, the simulation approach adopted in this paper addressed a non-standard sample size determination question and provided valuable information necessary for an engineer to design a preliminary screening life-test plan with a relatively small number of costly experimental test specimens. Although we address a specific sample size question in this case study, the methods presented here can be generalized to other life-test planning situations which may require different statistical failure time distributions, different population parameter values, or a comparison of more than two populations.

5. Further considerations The simulation-based approach adopted in this paper met the goals of the engineer and was designed to be easily modified as the engineer's needs and

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assumptions changed. In the following sections, we present other life-test planning approaches and point out consequences and effects of different parameter and failure time distribution assumptions. In Section 5.1, we present an approach to sample size determination using the asymptotic normality property often assumed for MLEs. Section 5.2 lists other life-test plans and approaches. Parameter estimation and distribution assumptions are explored in Sections 5.3 and 5.4, respectively. Bayesian approaches are touched in Section 5.5.

5.1. Asymptotic normaEty of MLEs Our initial effort to evaluate sample size requirements as a function of test duration was based on tables in Meeker and Nelson (1977) for the Weibull distribution. This approach uses an ML method to derive asymptotic (largesample) variances and covariances of the Weibull parameter estimates that can subsequently be used to derive approximate confidence limits for distribution percentiles. Such confidence limits are based on the large-sample normality of ML estimators. Both failed (uncensored) and unfailed (censored) test units are allowed at the end of the life-test in this approach. By following this approach initially, we generated a graph (see Figure 4) to summarize sample size estimates as a function of test duration by comparing differences in the 10th percentile between a standard and improved material. In this example, we assume 10% of the standard materials fails in two months and the new material is a onefold improvement over the standard material. L o w e r 2.5th P e r c e n t i l e of E m p i r i c a l S a m p l i n g Distdbution for the D i f f e r e n e e in 10th P e r c e n t i l e s

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Life-test planning for preliminary screening of materials: A case study

689

The dashed lines in Figure 4 are lower 2.5th empirical percentiles of the sampling distribution of the difference in the 10th percentiles between the failure time distributions of the standard and new materials as a fnnction of sample size and experimental test duration. The interpretation of these empirical percentiles is: Assuming a true difference of A months between the 10th percentiles of the failure time distributions of the standard and new materials, 97.5% of the random samples of size n will yield differences, Ä, in the 10th percentiles larger than the lower 2.5th empirical percentile. The solid horizontal line (at t = 2 months for a ù small" improvement) is plotted to represent the "true" difference in the 10th percentiles between the failure time distributions of standard and new materials. Test durations and sample sizes corresponding to lower 2.5th percentiles greater than zero could be considered appropriate planning values to use in the experiment in order to determine if the new material warrants further consideration. In such cases, there is at least 97.5% probability that the point estimate of the 10th percentile for the new material will exceed the point estimate for the standard material. Upon observing the evolving nature of the problem definition, we decided to adopt the simulation-based approach which is more flexible, avoids the need to refer to tabled values, and can be easily adapted for specific needs of the program in the future (e.g., design of a multiple stress accelerated life test) where formulas are not readily available. We also believe the output from the simulation (i.e., the probability of concluding a difference in the 10th percentiles between the failure time distributions) is easier to interpret than the empirical lower 2.5th percentiles that were generated from the large sample M L approach. An estimate of the power to detect a difference can also be generated using the large sample M L approach, but one must assume the sampling distribution of the differences in the percentiles follows a normal distribution. For finite samples, it has been shown that the normality assumption is questionable (see, for example, Doganaksoy and Schmee, 1993; Ostrouchov and Meeker, 1988). 5.2. Other approaches to life-test planning

A conventional (i.e., confidence interval) approach to determine the sample size of a life-test for one population is to select a sample size such that a percentile of a statistical failure distribution is estimated with a certain level of precision. Formulas for calculating confidence intervals for a single population based on M L estimators can be adapted for the two sample comparison situation to determine appropriate sample sizes (see e.g., Nelson, 1982 or Lawless, 1982). The performance of this approach can be assessed via the simulation approach described here. Doganaksoy (1995) presented a method to determine duration of a life-test to compare two samples while the test is underway and there are unfailed (i.e., censored) units. However, that approach assumes the sample sizes have already been determined. A common type of life-test is a "zero-failure" demonstration plan. In this case, a reliability demonstration is deemed successful if all of the n units survive a

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pre-specified time without any failures. Construction of such test plans is discussed in Meeker and Escobar (1998). 5.3. Parameter assumptions

The case study of this paper dealt with a life-test with time (Type I) censoring. In Type I eensoring, the life-test is stopped after a predetermined test time. With this type of censoring the sampling distributions depend on assumed parameter values of the underlying failure time distributions. One might, therefore, wish to consider different parameter values in simulations to assess sensitivity of results. From a practical perspective, Type I censoring is offen appealing since it makes planning of test resources (i.e., equipment, labor, material, and time) easier. Type II (failure) censoring is the case where a life-test is stopped after a prespecified number of failures is observed. Even though Type II censoring is seldom used in applications, it offers certain advantages. For example, if the main interest of the investigation is to estimate a low percentile (say, the 10th percentile) of the failure time distribution, there may be no further advantage to continuing the test after 15% of the units in the sample have failed (Meeker and Escobar, 1998, Chapter 10). Type II censoring also offers a salient statistical advantage for estimating failure time distributions that belong to the location-scale family (such as the smallest extreme value and the normal distributions that are associated with the Weibull and lognormal distributions respectively). This advantage arises from the existence of pivotal quantities whose sampling distributions do not depend on any unknown parameters (Bain and Engelhardt, 1991). Orte can then simulate the distribution of the pivotal quantity with any assumed location and scale parameter values. The test plans will not depend on the assumptions made about the parameter values. 5.4. Distributional assumptions

We did not know for certain whether the time to stress corrosion crack initiation is better characterized by a Weibull or lognormal distribution or another statistical distribution. The simulation approach can be adapted for other distributions as well. After the failure time data are obtained, the analyst can determine the appropriate distribution form for the subsequent analyses. See Meeker (1984) for consequences of incorrectly assuming a lognormal distribution when the underlying distribution is Weibull and vice versa. 5.5. Bayesian test planning

Limited historical data led us to develop a flexible approach that produced sample sizes under various assumptions. In this case study, we assumed a known shape parameter value for the Weibull distribution. A sample size determination method that is robust to parameter specifications as well as distribution assumptions employing Bayes estimation techniques is an area of future research.

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For Bayesian methods used in reliability data analysis see Martz and Waller (1982).

Acknowledgments

The authors thank Tom Angeliu for providing the background of this case study.

Appendix A: Pseudo-code for simulation algorithm assuming a Weibull failure time distribution I

Below, we denote the Weibull scale parameter for the failure time distributions of the standard and new materials by C~s and eN respectively. The Weibull shape parameter, /~, is assumed to equal 1.5 for all simulation conditions. Define Parameter Values for Standard Material (es,/3) Define Parameter Values for New Material (c~N,/3) Do the following for each sample size value (n - 10, 15, 20, 30, 40) Do the following for each of k = 2000 simulation runs Generate n Weibull random numbers from the Weibull distribution (Standard) Generate n Weibull random numbers from the Weibull distribution (New) Do the fonowing for the test duration in months, t = 9, 12, 15, 18, 21 Censor the n Weibull random numbers at month t from the Weibull distribution (Standard) Censor the n Weibnll random numbers at month t from the Weibull distribution (New) Estimate Weibull parameters(c2s,/~) from the n Weibull random numbers (Standard) Estimate Weibull parameters(~N,/~) from the n Weibull random numbers (New) Compute the 10th percentile of the fitted Weibull distribution (Standard) Compute the 10th percentile of the fitted Weibull distribution (New) Calculate the difference between these two 10th percentile estimates and store End End Do the following for each month t = 9, 12, 15, 18, 21 Compute the proportion of observations larger than 0 and store End End

Appendix B: Derivation of the scale parameter («) for a fixed shape parameter (~) of a Weibull distribution

The cdf, F(t), for the Weibull distribution is the fraction of the population failing by time, t, and is given in (3.1). For a fixed shape parameter value,/~ = 1.5, the scale parameter, c~, associated with 20% of the standard material failing by time t = 2 months is obtained by solving the following equation for c« 1 T h e s i m u l a t i o n s w e r e r u n u s i n g S-Plus, V e r s i o n 4.5 (1998).

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0.20 = 1 - e -(2/c~)1'5

or

c~ =

2

= 5.44 .

(-- in(0.80)) 1/1"5 F r o m (3.2), a o n e f o l d i m p r o v e m e n t in the 10th p e r c e n t i l e r e q u i r e s d o u b l i n g the scale p a r a m e t e r w h i c h yields 10.88. Scale p a r a m e t e r s a s s o c i a t e d w i t h 4 0 % o f the s t a n d a r d m a t e r i a l failing in t w o m o n t h s a n d w i t h failure d i s t r i b u t i o n s o f the n e w m a t e r i a l a r e o b t a i n e d similarly.

References Bain, L. J. and M. Engelhardt (1991). Statistical Analysis of Reliability and Life-testing Models. 2nd edn. Marcel Dekker, New York. Doganaksoy, N. (1995). Determining the duration of a demonstration life-test before all units fail. IEEE Trans Reliab. 44, 26-30. Doganaksoy, N. and J. Schmee (1993). Comparisons of approximate confidence intervals for distributions used in life-data analysis. Technometrics 35, 175-184. Hahn, G. and W. Q. Meeker (1991). Statistical Intervals: A Guidefor Practitioners. Wiley, New York. Lawless, J. F. (1982). Statistical Models and Methods for Life Time Data. Wiley, New York. Martz, H. F. and R. A. Waller (1982). Bayesian Reliability Analysis. Wiley, New York. Meeker, W. Q. (1984). A comparison of accelerated life test plans for Weibull and lognormal distributions and Type I censored data. Teehnometrics 26, 157-171. Meeker, W. Q. and L. A. Escobar (1998). Statistical Methods for Reliability Data. Wiley, New York. Meeker, W. Q. and W. Nelson (1977). Weibull variances and confidence limits by maximum likelihood for singly censored data. Technometries 19, 473~t76. Minitab (1997). Minitab User's Guide 2." Data Analysis and Quality Tools, Release 12. Minitab, State College, PA. Nelson, W. (1982). Applied Life Data Analysis. Wiley, New York. Nelson, W. (1990). Accelerated Testing: Statistieal Models, Test Plans, and Data Analyses. Wiley, New York. Ostrouchov, G. and W. Q. Meeker (1998). Accuracy of approximate confidence bounds computed from interval censored Weibull or log-normal data. J. Stat. Comput. Simul. 29, 43-76. S-Plus Statistical Sciences (1998). S-Plus User's Manual, Version 4.5. Statistical Sciences, Seattle. SAS Institute (1995). JMP User's Guide, Version 3.1. SAS Institute, Cary, NC. SAS Institute (1997). SAS/QC Software." Changes and Enhaneements for Release 6.12. SAS Institute, Cary, NC. Tobias, P. A. and D. C. Trindade (1995). Applied Reliability, 2nd edn. Chapman & Hall/CRC, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook ofStatistics, Vol. 20 © 2001 ElsevierScienceB.V. All rights reserved.

'~P7

Analysis of Reliability Data from In-House Audit Laboratory Testing

Rekha Agrawal and Necip Doganaksoy

In manufacturing situations, it is often critical to be able to detect early warnings of potential reliability and field failure issues after product release. These field issues may be ones that occur later in the life of the product and thus are often quite difficult to predict during product development, and even in the early stages of manufacturing. In this chapter, we will use an estimate of the mean cumulative function (MCF) to analyze audit laboratory test data on toasters. We will show how the M C F analysis can be used as an analysis technique to monitor in-house testing of manufactured units, providing the required early warning. We will conclude the chapter with some general observations on issues encountered in applying such a technique to situations in industry.

1. Introduction

Companies all over the world are finding that in today's business environment, they must focus on delighting their customers to stay competitive. One of the biggest contributors to customer satisfaction is product reliability. Thus, a key for many companies is assuring reliability of their offerings. Improving the reliability of products is a multi-faceted process applied at all the different periods in the product life cycle. Modern reliability programs require quantitative methods for predicting and assessing product reliability, and for providing information and early signals on root cases of failure. This will typically involve the collection and analysis of reliability data from many diverse sources at all phases of product design, development, manufacturing as well as use in the field. At the design stage of the product, the reliability goals are determined at the product (system) level, and then "flowed down" to the subsystem, component and sub-component level. The current design is assessed, "benchmarked", and compared to the new design where applicable. Gaps between the reliability goals for the new design and the current design are identified. In this phase, a major thrust is to identify the weaknesses of the new design and eliminate them. This stage makes use of a number of reliability engineering tools such as failure mode effects 693

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analysis (FMEA), highly accelerated life testing (HALT), highly accelerated stress and strain testing (HASS) and failure reporting and corrective action system (FRACAS). O'Connor and Patrick (1991) provide useful background on these broader considerations for reliability improvement. Also see Meeker and Hamada (1995) for types of reliability testing and appropriate data sources throughout product development and field use. Once the product has passed the design stage, prototype units are built and tested to assess the reliability level. After the internal qualification of the product, prototype units are then taken out to the field for testing. Often, issues can be found in the field due to circumstances that the product designers were unable to anticipate, and therefore were unable to create in a laboratory environment. These field tests will most likely detect issues that are apparent in the early life of the product, since they often cannot be run for long, due to resource constraints. We know of one situation in which cockroaches in Alabama were accessing the circuit board of a product, and causing short circuits. This was not a situation that had been anticipated in the internal tests for the product. If no major issues appear in the field test then the product is ready for launch and regular manufacturing. In the situations that we have encountered, the design is intended to be "fixed" by this time. However, usually continuous on-going minor changes are being made to the product over time, either in an attempt to improve the product or because of cost constraints. We will return to this point later. Once the product is released for field use, there are usually two primary sources of data for monitoring reliability. • In-house audit test data: Manufactured product is sampled regularly and

subjected to some kind of internal accelerated life test, in an audit lab. • Field data: In an industrial context, we have found that field data can be generally classified into two periods of interest: the in-warranty period and the longterm life of the product. The reason for making this distinction is that usually the amount of information available in the two periods is substantially different. Meeker and Escobar (1998), Lawless (1982), and Nelson (1982) are useful references on statistical analysis of various types of reliability and lifetime data. In this chapter, we will discuss the use of the mean cumulative function (MCF) as a tool to analyze audit lab data. The outline of the chapter is as follows: We first describe the role of audit testing in reliability. We then present a particular case study that we were involved in and explain why we felt a more thorough analysis of the data was beneficial; we describe the M C F technique and we then show how it applies to the case study. Finally, we give some general observations and recommendations.

2. The role of audit testing in reliability

Reliability evaluations present a ¢hallenge beyond that normally encountered in quality evaluations because there is usually an elapsed time between when the

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product is built and when the reliability information from the field is forthcoming. If the elapsed time is long compared to the manufacturing cycle of the product, a large number of units may be produced before it is possible to react to a reliability issue. Therefore, an earlier warning of potential reliability issues is desirable. An audit lab is used in a manufacturing situation to continually assess the quality and reliability of product being produced. Often, a small number of units are sampled from production and subjected to extensive testing. The units are then observed to determine how well they withstand testing. This cycle of sampling and testing is done with regular frequency, and significant changes in results may be an early indication of changes in the quality (and the reliability) of the product being manufactured. If failures and/of unexpected behavior of the product occur in the audit lab, then this can be carefully studied by design and manufacturing engineers to understand root causes. This is an important function of such a lab. To häve an in-house audit lab it is usually necessary to be äble to accelerate the life of the product in the laboratory, at an acceleration rate that is approximately known. This will make it possible to obtain reliability information in the lab at a fastet rate than in the field. One way of doing this is to cycle the product laster then we would expect it to be cycled in the field. Another way would be to subject the product to elevated levels of stress, for example high temperature or high humidity. Of course, it is always dangerous to assume that we can understand and reproduce the in-use conditions that a product observes in the laboratory. There taust, however, be a reasonable level of comfort that every attempt has been made to reproduce those factors that have a significant effect on the product life. Nelson (1990) and Meeker and Escobar (1998) provide in-depth discussion of accelerated testing and associated considerations. Why would we be interested in monitoring reliability at the audit lab? A valid question to ask hefe is, "Isn't it too late to be assessing reliability at the audit lab stage?" Clearly, if this was the only place in the product life cycle where we were assessing reliability, then the answer to that question would be yes. As part of a comprehensive reliability effort, however, it is important to recognize that the audit lab is our last defense before we find field problems as they are occurring in the field. Such problems may be a result of manufacturing issues, or because of ' minor" design changes that may have been caused by a change of supplier, raw materials, or other modifications. Judicious use of audit lab data could potentially mitigate millions of dollars in product service and replacement costs, not to mention the higher toll that might be paid in consumer dissatisfaction and loss of brand name, due to a major field issue. Further, there are also potential positive scenarios that can be capitalized on. While a monitoring tool can trigger a flag if the situation has changed for the worst, it can also trigger a flag if the situation has significantly improved, which might cause us to drill down further to capture and maintain the improvement. Such an improvement might otherwise have been missed. We have seen many situations in which audit labs can be used for the purpose described above.

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(1) As in the case study of this chapter, an appliance manufacturer can use an audit lab on its toasters to monitor reliability. In this particular instance, accelerating the life of the product is relatively easy. Since a regular consumer might only use his toaster a couple of times a day, exercising the toaster almost continuously could quickly simulate a long period of life. Caution would be required in this case to not induce artificial failure modes - say, for example, in the overheating of coils due to constant use. (2) Manufacturers of power generation equipment are concerned with the reliability of dielectric insulation used on generator armature bars. This is a fairly mature technology and the nominal life expectancy of such insulation is on the order of several decades. However, unexpected changes in manufacturing or raw material conditions can result in premature failure of the insulation. In order to provide early notification of such problems, a small number of bars are sampled from production periodically - typically one or two bars a week due to relatively low volume of production. These bars are pur on a high stress voltage endurance life test to accelerate their failure times. Failure time data (often censored data since there are unfailed units) from different production periods (say, quarterly) are compared to detect changes in failure time distributions. (3) Thermoplastic resin manufacturers are concerned about changes in the color of automotive exterior body panels, molded from their product. Sample parts, molded from resin sampled during different production periods, are placed under highly intense UV light to detect possible changes or degradation of color.

3. Case study background We look now at a manufacturing process producing "toasters". We were primarity interested in the first year life of the toaster, since that was the warranty period of the product. We were also interested in the ten-year life of the product, since that was roughly how long consumers expect their toasters to last. An audit lab had existed at the toaster manufacturer's for some time, and there was a data collection system in place. The sampling scheine was such that five toasters were sampled randomly each week (typically in a systematic manner) from the manufacturing line, and put on "one year test". Also, approximately one toaster a week was taken and put on "ten year test". The reason for the discrepancy between units on one-year test and on ten-year test was simply a resource issue. Since these toasters were being manufactured in large quantity (about 80,000 toasters a month), this sampling represented a negligible amount of the overall population. The major costs were associated with testing and measurement rather than the cost of the units. The audit lab tests that were done were simply automatic rapid cycling tests of the toasters, allowing for sufficient cooling. Finally, we had some data that told us that on average, consumers use their toasters 400 times a year. One-year life of the toaster could be simulated in the lab in about two to three weeks.

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I f a failure occurred on the toaster during its testing period, the component that caused the failure was repaired or replaced and the testing was continued. In this case, eight different failure modes (i.e., failing components) were observed on the toaster, the eighth category being "other". For the discussion that follows, it is assumed that a failure in a certain category always implied that the same component was replaced in the toaster. Although this was not always the case, it is a reasonable assumption for the categories of failure that were not "other". A segment of data gathered during a particular audit period is shown in Table 1. Even though there was a total of 30 toasters in this group, the table displays repair data from five units. Toaster T140 failed at 200 cycles (due to failure mode 3), at which point the failing component was repaired (or replaced). The unit was retired from testing after 425 cycles. Toaster Tl41 completed 410 cycles without any failures at which time it was taken oft test. Prior to our involvement in the audit lab, the data that resulted from the rapid cycling tests were used mainly for engineering analysis of the failure modes observed during testing. When a component on a toaster failed, the engineer responsible for that component was notified. Time permitting, the engineer would inspect the part and in some cases, do a root cause analysis. The test results were also plotted: the percent defect level vs the m o n t h of manufacture. This was also done by particular failure mode. It was felt that the analysis of this data could be strengthened. The largest issue was that the effect of age was not being accounted for in the current analysis. This is due to the fact that failing components were replaced with new ones. The failure rate estimates described above were based on number of toasters on test and did not take the age of the components into consideration. It was impossible to tell, therefore, whether there appeared to be an increasing failure rate, or a decreasing failure rate. Clearly, the former case would cause serious concern, while the latter case may be dealt with differently. Also, no quantification was being given to the variability in the data. Without such a quantification, results could not highlight

Table 1 Sample data. A failure mode of 0 here indicates that the unit was taken oft test Toaster number

Cycles

Fail mode

T140 Tl40 T141 T142 T143 T144 T144 T144 T144 T144

200 425 410 400 390 350 580 793 30t2 4011

3 0 0 0 0 1 6 7 1 0

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significant changes compared either to other manufacturing periods or to an absolute standard. Further, the tradeoffs in precision made between the one-year and the ten-year tests were not being conveyed. We saw an opportunity to make better use of this data, and the resources that are being put into the audit lab to generate it. In the next section, we discuss a non-parametric estimate of M C F as a monitoring tool for this data.

4. Using the mean cumulative function as a monitoring tool

4.1. Types of reIiability data The two common types of reliability (failure time) data are: (1) The time of failure for non-repairable units or components. Since a nonrepairable component can fail only once, time to failure data from a sample of non-repairable components consist of the times to first failure for each component. In most instances involving non-repairable components, the assumption of independent and identically distributed failure times is a reasonable one and suitable lifetime distributions (such as the Weibull or lognormal) are used to describe the distribution of failure times. Meeker and Escobar (1998), Lawless (1982), Nelson (1982) are useful references on statistical analysis of various types of reliability and lifetime data. (2) A sequence of reported system repair times for a collection of repairable systems. Data typically consist of multiple repair times on the same system since a repairable system can be placed back in service after repair. In some cases one can analyze time between repair times for a repairable system using analysis methods devised for non-repairable components. However, this approach makes the important assumption that after a failure of a system component, the act of repair (e.g., replacing the failing component) restores the system, with respect to that failure mode, to as good as new. When a single component or subsystem in a larger system is repaired or replaced after a failure, the distribution of the time to the next system repair will depend on both the overall state of the systems at the time just before the repair and the nature of the repair. Thus, repairable system data, in many situations, should be described with models that allow for changes in the state of the system over time or for dependencies between repairs over time. Since the toasters were being treated as repairable systems in the audit lab, the standard tool that is often used for reliability data, (e.g., fitting of a Weibull or lognormal distribution to times between failures), would not be directly applicable. 4.2. Mean cumulative function Repairable system data are viewed as a sequence of repair times. At a particular age t, each population unit has accumulated a number of repairs. One can

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envision a population distribution of the cumulative number of repairs at age t, since different systems accumulate different number of repairs by age t. The distribution at age t has a mean M(t), the population M C F for the number of repairs. For a large population, M(t) increases smoothly with age t. It is useful to regard M(t) as a continuous function assumed to have a derivative ra(t) = dM(t)/dt. Here, ra(t) is the mean rate at which the number of repairs accumulated at age t. For the number of repairs, m(t) is called the "instantaneous repair rate function". It is also called the "recurrence rate" or "intensity function" if some other type of recurring event is observed. It is expressed in repairs per unit time per system, for example repairs per month per toaster.

4.3. Estirnation of mean cumulative function The model for such recurrence data is sometimes called a point process. Parametric methods used to analyze repairable system data assume a parametric form for M C F (of instantaneous repair rate function). See Meeker and Escobar (1998), Tobias and Trindade (1995), Engelhardt (1995), and Ascher and Feingold (1984) for discussion of parametric models for repairable systems. We will use a non-parametric approach to estimate MCF. Given a collection of repairable systems, a simple estimator of the M C F at time t would be the sample mean of the available cumulative number of system repairs for the systems still operating at time t. This estimator is appropriate if all the systems are still operating at time t (i.e., no censoring in the data). For example, for the data in Table 1, the estimate of M C F at 200 cycles is 1/5 (assuming there are only these five units in the group). Likewise, at 350 cycles, the M C F is 2/5. However, the estimation of M C F at 580 cycles is not as clear cut since the number ofunits is no longer five at this time. The data are multiply censored as a result of units retiring at different ages. An unbiased estimate of M C F allowing for multiple censoring is described in Nelson (1995), Nelson and Doganaksoy (1989), Meeker and Escobar (1998), and Tobias and Trindade (1995). These references also provide a discussion of assumptions underlying the non-parametric estimate of M C F and associated confidence bounds. Doganaksoy and Nelson (1998) extended this method to compare MCFs of two samples from two different populations of systems.

5. Application to the case study The analyses described in this section were done using SAS (1997) RELIABILITY P R O C E D U R E which is part of the QC software. The MCF, and associated confidence bounds, for three different periods of manufacturing, in chronological order are shown in Figure 1. The x-axis on these plots represents the number of cycles that the toaster has seen. The y-axis is the average number of repairs (or replacements). All three plots have been put on the same scale, for the purposes of comparison.

R. Agrawal and N. Doganaksoy

700

Period 2

Period 1 5 4.5 4 3.5 3

õ

] Upper95% CB ~

f ~ - -

~~

2.5

õ

5 4.5 4 3.5 3 2.5

2

2 1,5

1 0,5 500 1000 1500 2000 2500 3000 3500 4000 4500 CYCLES

1,5 1 0.5 0

«

500 1000 1500 2000 2500 3000 3500 4000 4500 CYCLES

Equivalent10 yearlife

Period 3

I

5

4.~ 3.5

2.~ 1.5

°5

~~

---__~__2 0 500 1000 1500 2000 2500 3000 3500 4600 4500 CYCLES

Fig.

1. M C F

estimates for three different manufacturing periods.

A preliminary look at this data appears to indicate that period 1 has a higher number of replacements than the other two periods. We also notice the tightness of the confidence intervals for the first 400 cycles, relative to those bounds for the remaining life of the toaster. This is a consequence of the sampling scheme discussed earlier. Let us assume that we are considering the manufacturing period 1 as the baseline for all subsequent manufacturing periods. Then, as we were testing period 2 manufacturing, we could be comparing it to period 1 by looking at the difference in the two periods. This plot is shown in Figure 2. Since the 95% confidence bounds for about the first 400 cycles of life are negative and do not include zero, we can conclude that manufacturing period 1 performs better in the warranty period than period 2. If we were doing this comparison as we were testing period 2 manufacturing, this might cause us to drill down further to particular modes, to understand where we are doing worst, and why. Note also that Figure 2 indicates that for the period of life when the toaster has seen more than 2000 cycles (a five-year equivalent), period 2 is better than period 1. Thus, we have improved the long-term reliability of the toaster. In between 400 and 2000 cycles, there is no apparent difference between the manufacturing periods. Similar comparisons can be done between period 1 and period 3, and manufacturing periods 2 and 3. These are shown in Figures 3 and 4, respectively. In Figure 3, we see that for about the first 1600 cycles (four-year equivalent), there is no apparent difference between the two manufacturing periods. After that, however, it seems that period 1 is worst. In Figure 4, we see that period 2 is

Analysis of reliability data from in-house audit laboratory testing

701

I Upper95%CB I

2500

0

3000

I Lower95% CB CYCLES

Fig. 2. MCF differences between period 1 and period 2.

/

2000

2500

3000

3500

4000

4500

5000

CYCLES

Fig. 3. MCF differences between period 1 and period 3.

slightly worst than period 3 for about the first 900 cycles, but then there appears to be no difference. One of the attractive features of M C F is that it can be applied in the same way to particular failure modes of the system, not just to the total number of replacements. When doing this, we ignore replacements due to failure modes other than the one of interest. The implicit assumption here is that replacing another component in no way affects the life of the failure mode of interest to us. We have

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Æ

CYCLES

Fig. 4. MCF differencesbetween period 2 and period 3. plotted the M C F s for one particular failure mode of interest for each of the three different manufacturing periods in Figure 5. These are plotted on the same scale as in Figure 1, to allow an understanding of how much of the total replacement rate is due to this failure mode. Differences between the manufacturing periods are shown in Figure 6. There appears to be very little difference with respect to this mode in periods 2 and 3 - both appear better than period 1 in the mid-tolong-term life of the toaster. We can see, then, how these analyses allow us to monitor reliability in-house, and give us an early warning of potential field issues. This is especially true for those issues that might appear later in the life of the product. The audit lab information can signal a change in long-term reliability in this case within weeks of the actual manufactured time. This allows a much faster response than if we wait to discover the issue when our customers are experiencing them in the field. By using the audit lab data as an early warning indicator, we can potentially avoid or mitigate field issues. The potential impact of this system are reduction in the dollars spent on field issues, as well as an overall increase in customer satisfaction and loyalty to brand name.

6. Observations, recommendations and conclusions We have encountered a few issues when trying to implement the reliability monitoring system described here. We discuss some of these issues briefly.

Analysis of reliability data from in-house audit laboratory testing Period 1

703

Period 2

5 4.5 4 3.5

5 4.5 4

3 2.5 2

3 2.5. 2 1.5

3.5.

1.5 1 0.5 0 500

1000 1500 2000 2500 3000 3500 4000 4500 5000 CYCLES

0.5 ~ ..... ~ ..... 0 ,~ ---Y ........ T- - - ~ - - " " ~ ~ ' . 0 500 1000 1500 2000 2 5 0 0 3 0 0 0 3 5 0 0 4000 4500 5000 CYCLES

Period 3 5 4.5 4

8 = ®

3.5 3

2.5 2

0.5 0 500

1000 1500 2000 2500 3000 3500 4000 4500 5000 CYCLES

Fig. 5. M C F estimate for a particular failure mode for three manufacturing periods.

Period 1 and 3

Period 1 and 2 5 4

8

a

u.

~oo

o ;"%~'--t~'Sg00"~õ06

2500 30i10 3500 4000 4500 5000

-1

CYCLES

CYCLES

Period 2 and 3

8 ®

-õ u.

f ~- 5 " õ ~ ~ 4 0 0 0

4500 5000

CYCLES

Fig. 6. M C F differeuces for a particular failure mode.

704

R. Agrawal and N. Doganaksoy

The first obstacle we encountered when trying to implement this system was in justifying the M C F as the appropriate methodology. The people who were interested in this data were engineers, for the most part, who had some familiarity with reliability. They were acquainted, for example, with Weibull plots and standard mean time to failure calculations for an exponential model. There was some reluctance to embrace a new tool that was not generally familiar. The second issue we encountered was an organizational one, related to the fact that there were numerous people who had a stake in the audit lab results. It was owned by the people in manufacturing, and the focus there was on the in-warranty short-term issues. The people in engineering, however, were focusing on improving reliability and were in large part the customers of this early warning reliability system. Thus, it was difficult for us to achieve buy-in on changing the current system, since the people who would have most benefited from such a change were not the same people who could directly influence it. Another issue we encountered is that it was perceived that working on the audit lab was not going to make a large impact on the reliability of our products. It was felt that energy was better spent in the upfront design of the product. Again, we emphasize the importance of a complete reliability program that encompasses all stages of the product life cycle. Finally, this system is only as good as the data that go in to it, and we found that while the current data collection system was serving the current needs well, some changes could be made to it to improve the applicability of the results. We have discussed how the M C F can be used as a reliability monitoring tool for in-house testing. Such an early warning system allows us to understand when significant changes have occurred in a product's life. Especially for issues that occur later in life, these early warnings allow us to dramatically reduce our response time, thereby averting potential field issues that may have a negative effect on customer satisfaction.

Acknowledgements We would like to thank Gerry Hahn, Ed McInerney, Mike Ali, Jim Meyer, Paul Raymont, Todd Heydt and Carl Peterson for valuable assistance in doing the work that was described as the case study for this chapter.

References Ascher, H. and H. Feingold (1984). Repairable Systems Reliability: Modeling, Inference, Misconceptions, and Their Causes. Marcel Dekker, New York. Doganaksoy, N. and W. Nelson (1998). A method to compare two samples of recurrence data. Life Data Anal. 4, 51-63. Engelhardt, M. (1995). Models and analyses for the reliability of a single repairable system. In Recent Advances in Life-Testing and Reliability, pp. 79-106 (Ed., N. Balakrishnan). CRC Press, Boca Raton, FL.

Analysis of reliability data ßom in-house audit laboratory testing

705

Lawless, J. F. (1982). Statistical Models and Methodsfor Lifetime Data. Wiley, New York. Meeker, W. Q. and M. Hamada (1995). Staöstical tools for the rapid development and evaluation of high-reliability products. IEEE Trans. Reliab. 44(2), 187-198. Meeker, W. Q. and L. A. Escobar (1998). Statistical Methodsfor Reliability Data. Wiley, New York. Nelson, W. (1982). Applied Life Data Analysis. Wiley, New York. Nelson, W. (1990). Aceelerated Testing: Statistical Models, Test Plans, and Data Analysis. Wiley, New York. Nelson, W. (1995)~ Confidence limits for recurrence data - applied to cost or number of product repairs. Teehnometries 37, 147-157. Nelson, W. and N. Doganaksoy (1989). A Computer Program for an Estimate and Confidence Limits for the Mean Cumulative Function for Cost or Number of Repairs of RepairabIe Systems. GE CRD Technical Information Series, 89CRD239, Schenectady, New York. O'Connor and D. T. Patrick (1991). Practical Reliability Engineering, 3rd edn. Wiley, New York. SAS (1997). SAS/QC Software: Changes and Enhancements for Release 6.12. SAS Inst., SAS Campus Drive, Cary, North Carolina. Tobias, P. A. and D. C. Trindade (1995). Applied Reliability, 2nd edn. Chapman & Hall/CRC, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 200t Elsevier Science B.V. All rights reserved.

~')

z.ù K.y

Software Reliability Modeling, Estimation and Analysis M. Xie and G. Y. Hong

1. Introduction

Our society has become increasingly dependent on computer systems. Many system failures occur due to latent software defects encountered as the software executes various input combinations during operation. Software failure process is a random process and a systematic approach is needed to predict, measure and manage software failures so that the reliability of software can be quantified and improved. Software reliability is defined as the probability of failure-free operation of a computer program for a specified time in a specified environment. Software reliability parameters, measures and metrics, described by software reliability models (SRM), offer the possibility of evaluating and monitoring software failures quantitatively during the verification phases of a product. The probabilistic models for software failures and their frequency of occurrence can be described and measured by software reliability functions and mean time between software failures. In this chapter, a number of important software reliability models are reviewed. The focus is on the presentation of some typical models of different type with the emphasis on failure process modeling and reliability analysis. The review is not exhaustive as there are a number of published works reviewing the history of software reliability modeling, for example, Shooman (1984), Musa et al. (1987), Xie (1991), and Lyu (1996). Xie (1993) provides an annotated bibliography of 100 significant papers in software reliability and interested readers are also referred to the individual papers for detailed description of each model.

2. Classification of software reliability models

Many different models have been proposed, modified and adapted in the past few decades. The understanding of their differences and interrelationships can help the user to select appropriate model for different applications. The model clas707

708

M. Xie and G. Y. Hong

sification scheme is very important since it can give a clear picture of the existing software reliability models, signifies the relationships among different models, and aids the efficiency of model comparisons. Some existing model classification schemes are discussed in this section. Ramamoorthy and Bastani (1982) classified software reliability models into four types according to the phase of the software development: debugging phase, validation phase, operational phase and maintenance phase models. The authors also proposed another classification scheme according to the debugging strategy where software reliability models can therefore be divided into software reliability growth models, sampling models and seeding models. These schemes are appropriate for software engineers and developers so that suitable data are collected for model application. According to the nature of the failure process, Goel (1985) classified the existing approaches into four models: time between failure models, failure count models, fault seeding models and input domain-based models. This classification scheme is useful for practitioners as it clearly indicates the type of data needed for the analysis. Xie (1991) classified the software reliability models according to their probabilistic assumptions. Under this scheine, the existing models were grouped as: Markov model, nonhomogeneous Poisson process (NHPP) model, Bayesian models, statistical data analysis methods, input domain-based models, seeding and tagging models and software metrics models. This classification scheine will be adopted here. Nonhomogeneous Poisson process models a r e a group of models that has received most attention by researchers and practitioners. It is this type of models that will be emphasized here. Markov models contain a number of earlier models in the development of software reliability growth models and they will also be summarized. A number of Bayesian approaches to modeling and data analysis will also be mentioned separately. In fact, an important task in software reliability analysis is reliability prediction and data analysis, and they will be covered in a separate section. According to different criteria, software reliability models are classified into different categories. However, for a good software reliability model, it should have several important characteristics as follows: (1) it gives good prediction of future failure behavior, (2) computes useful quantities, (3) is simple to use, (4) is widely applicable, (5) is based on sound assumptions. It is well known that no single model will work in all situations and alternative models are usually needed. 3. Some important NHPP software reliability models

3.1. Concept of NHPP Denote by N(t) the number of failures occurred by time t. The process {N(t);t >_O} can be modeled by a NHPP. Because software reliability is in-

Software reliability modeling, estimation and analysis

709

creasing during the testing phase, this type of models is also commonly called software reliability growth model (SRGM). The general assumption for S R G M described by an N H P P is as follows: 1. 2. 3. 4.

A software system is subject to failures at r a n d o m caused by software faults. There are no failures experienced at the time t = 0. N o new faults are introduced during the testing. The probability that a failure will occur in an interval At is 2 At + o(At), where 2 is the failure intensity, which may depend on t. 5. The probability that more than one failure will occur in an interval At is o(At).

The function 2 (t) is called the failure intensity function and it is usually easier to work with the so-called mean value function given by

/0 t 2(s)ds

#(t) =

.

(1)

Note that #(t) represents the expected number of failures that are experienced at time t, i.e., #(t) = EIN(t)]. Divide the time 0 to t into a number of intervals, to = 0, q , . . . , ti-1, t l , . . . , tk = t. The number of faults in the /th interval, Ni, is a Poisson random variables with the mean E(Ni) = #(ti) - #(ti 1). N H P P models can be classified into two gronps: finite failure models and infinite failure models. The commonly known G o e l - O k u m o t o and S-shaped models belong to the finite failure model family. M u s a - O k u m o t o , Duane and logpower models can be classified into infinite failure family. As these models are representative and also widely used, we will give more emphasis to these models in the following. Software reliability models contain a number of parameters and they are usually unknown. To estimate the parameters of these models, the m a x i m u m likelihood method can be applied. Given the nnmber of failures in each interval, nj, we have that

P(Ni = ni) = {#(tl) - #(ti-1) } ni exp{-[#(ti) - #(ti-1)]} . ni! For model parameter estimation, m a x i m u m likelihood method can be applied. The likelihood function is given by

Models [NHPPSoftwareReliability E

1

I

[ FinitëFN,ureModels 1

I InfiniteFailureModels

T - -

.___[__

I

I

[ Goel-Okumoto [---S--Shaped ] [-Musa-Okumotõ] r L Model Modet , ~odo, J ,

I

Duane Model

Fig. 1. Some common NHPP models.

II

Log-power Model

M. Xie and G. Y. Hong

710

k

L ( n l , . . . , nk) = I ~

{#(ti)

-

# ( t l _ l ) } ni

exp{-[#(t/) - #(ti-l)l}

(2)

ni!

i=1

By taking logarithm of both sides of equation, we have

--#--(ti-1)]niexp[-(#(ti) - #(ti 1))]

lnL = ~-~ln i=1

v

nj!

k =

~{n

i In[#

(tl) -- # (tl-l)]

- - E#

(ti)

-

#

(ti-1)]

-

In nil } .

i=1

The above formula can be simplified to k

ln L

= ~~~_~{niln[# (ti) - # (t,-l)] - In ni!} - #(&) . i=1

In general, to find out the maximum likelihood estimates, we can take the derivative of this equation and equate it to zero. It is also useful to obtain the confidence interval for the estimated parameters. Assuming that there are two parameters, a and b, we can use the Fisher information matrix to obtain their confidence intervals. The asymptotic variances Var(ä) and Var(/~), the covariance Cov(ä,/~) of the MLEs of the parameters are also needed for the calculation which can be derived following the standard a p p r o a c h . The Fisher information matrix is given by

I-E[ô21nL/~ 2 a I F = [_E[ô21nL/ôaôb ]

-E[ô21nL/aaôb] l -E[ô21nL/ô 2 bi

The asymptotic covariance matrix V of the M L estimators for parameters a and b is the inverse of the Fisher information matrix: V = F -1 =

[Var(ä) LCov(ä,/~)

Cov(ä,b)] Var(b)

(3)

The two-sided approximate 100a o'/o confidence limits for the parameters a and bare The above conclusions should be:

au = ä + Z~/2

V~ßär((~))

»u = b + z ~ / 2 ~

aL =

ä - Z~/2 ~

,

eL = b - z ~ / 2 ~ ,

where Z~/2 is the (1 - c~/2) quartile of the standard normal distribution. Given the mean value function #(t), the software reliability R(x[t) is defined as the probability of a failure-free operation of a computer software for a specified time interval (t, t + x] in a specified environment. Given the mean value function #(t), we have that

R(x[t) = e x p [ - { # (t + x ) - # (t)}] .

(4)

Software reliability modeling, eslimation and analysis

711

It should be noted that the above is testing reliability concept, and when mission reliability after the release is concerned. The proper concept is the operational reliability and this was recently pointed out in Yang and Xie (2000). The testing reliability concept is adopted here as illustration. With (4), the asymptotic variance Var(R) can be determined as

?R~ 2

Var(/~) = , e R , 2 Var(ä) +

+

Var(/~)

2(~R'~( ~R) a=a,»=äC o v ( ä , ~ ) k,8aJ ~

,

The two-sided approximate 100Œ% confidence limits for R is then given by

Rv=Æ+Z~/2~)

and

R L=Æ-Z«/2v/Var(R)

Similarly, the asymptotic variance for the failure intensity estimate is given by Var(2) =

ä(8)L'2~ Var(ä) ÷ ()2 \~aßa= ~62 b=$ Var(/~)

The two-sided approximate 100e% confidence limits for the true value 2 is

2u = 2 + Z~/2[Var(2 )] 1/2 and

)oL = 2-Z~/2[Var(~)] 1/2 •

3.2. Descriptions of specific models 3.2.1. The Goel-Okumoto model Goel and Okumoto (1979) developed a software reliability model, also called exponential growth model. It is characterized by the following mean value function #(t)=a(1-e

-bt)

a>0,

b>0

,

where a = # (oc) is the number of inherent faults and b indicates the failure occurrence rate. The failure intensity function 2 (t) is il(t) = abe -»t . A typical plot of the p(t) and 2(t) for the Goel-Okumoto model is shown in Figure 2. The ML estimates are the solution of the following equations:

712

M. Xie and G. Y. Hong 50

220

45 40

170

35

£ 30 O) -_=

120 1

.__=20

70 >

k1_

25 t-

ii O

15

20

10 5 0

-30 0

2

4

6

8 10 12 14 16 Time t (hour)

18 20

Fig. 2. A typical plot of the # (t) and 2 (t) for the Goel Okumoto model. {

k ^ ä = i~=lni/(1 - e-bt») , k

-

k

^

S-'( "' -21=~"------2');t.e-~t~ - t i _ l e-bti-1) = 0 \ e z;',-1-e -l;', 1-e-";~'k )t, The negative second partial derivatives of the lol likelihood function can be derived using the M L estimators ä and b of the G o e l - O k u m o t o model as follows: k

(_ô2 lnL/ßS a)== a = Z

nilä2 '

j=l

k (-521nL/52b)b=[~ = Z n i

i=1

( t i - ti 1)2e D(t~+ti 1) ^ (e-bti i - e &/)2

~_.2_-Dtk ätôe-Dt0 atkc +

and ^

^

(__~2 lnL/Sa~b)~=a,»=D = tke -btk - toe bto The software reliability R(xlt ) at the time interval (t, t + x] can be derived as R (x]t) -- exp[ae bt(e-óX - 1)l . The asymptotic variance Var(R) is obtained as Var(/)) = e 2ae ~"(e D~_ 1) × {e-2l;ti (e-& _ 1)•Var(ä) + ä2 [(tl + z)e -;(ti+~) - tie -[~t*]2Var(;) + 2äe -;t~ (e -& - 1)[tie [~ti_ (tl + z)e-b(ti+~))Cov(ä,/~) } •

Software reliability modeling, estimation and analysis

713

The two-sided approximate 100«% confidence limits for R can also be derived. Similarly, Var(,~) is obtained by using the maximum likelihood and local estimates of the variances and covariance: Var(,~) = b2e-2&Var(ä) + ä2e-2&(1 - bt)2Var(/;) + 2äße 2/;t(1 -- t ~ t ) C o v ( ä , / ~ ) . The two-sided approximate 100«% confidence limits for the true value of failure intensity )~ can be derived. 3.2.2. The S-shaped models Yamada and Osaki (1984) discovered that some faults may be covered by other faults and cannot be detected at the beginning of testing. A model that can deal with this case is the delayed S-shaped N H P P model which has the mean value function #(t)=a(1-(l+ót)e-ót),

b>0

,

(6)

where a is the total number of faults eventually to be detected and b is a faultdetection rate parameter. The failure intensity function is 2(t) - d#Ds(t) -- ab2te-ót dt A typical plot of the mean value function/~(t) and failure intensity function 2(t) for the S-shaped N H P P model is shown in Figure 3. The MLEs of the parameter a and b can be derived as the solutions of the following two equations: 30

220

25

170 t-0

20

"6

12o

g

IJ _

70 "

10

20

'

~

'

'

'

2

4

6

8

10

~

'

12 14

'

'

-30

16 18 20

Time t (hour)

Fig. 3. A typical plot of the # (t) and 2 (t) for the S-shaped N H P P mode1.

M. Xie and G. Y. Hong

714

ä=

~~=1 n~ (1 - (1 + Dtk)e -[~t~'

ù_~(

~~('~e-~"-t~-~e-~~~~) ,, ) = ä t Z e il + ~,t,-_~)e-»',-, - (1 + bt~)e

b~~

~',/

The software reliability R(x]t) at the time interval (t, t + x] can be derived as

R(x[t) = exp{ae-bt[(1 + bt + bx)e -bx - (1 + bt)]} . The asymptotic variance Var(R) is derived as

?R , 2

Var(Æ) =

+ \aa/

Var(ä) +

~

?R~ 2

Var(/;)

a=a,»=~

The asymptotic variance for the failure rate function is Var(2) =/;4fle 2btVar(ä) + ä2/~2t2e-Sbt(2 -- bt)2Var(/;) + 2ä/;3t2e 2b(2 -/~t)Cov(ä,/~) .

3.2.3. The Musa-Okumoto model Considering the possibility of infinite number of faults in the software, Musa and Okumoto (1984) proposed another NHPP model called the logarithmic Poisson model. It has a mean value function that contains two parameters /~(t) = aln(1 + bt) .

(7)

This lnodel reflects that faults with larger size are found earlier and it provides good results in modeling many software failure data. The failure intensity function is derived as

ab

;~(0 =

1 + bt

A typical plot of the mean value function and failure intensity function for the Musa-Okumoto model is shown in Figure 4. The maximum likelihood estimates of the parameters a and bare the solutions of the following equations: ä=

k ln(l+/~tk) '

k

li~l ll[~ti b =

ktk

(l+btk) ln(l+b&)

--0.

The software reliability R(x]t) at the time interval (t, t + x] is

Software reliability modeling, estimation and analysis 50

350

45 >., c c

715

300

40

cO

35

250 õ E

30

200 ti-

25 150

20

e100 0~

15 10

/

5

FailureIntensity X (t)

0 2

i

i

i

i

i

i

i

4

6

8

10

12

14

16

Time t

50 i

18 20

(hour)

Fig. 4. A typical plot of the # (t) and 2 (t) for the Musa Okumoto model.

~1~ R(xlt) = [1 +l+bt b(t + x Then the asymptotic variance Var(R) is derived as

1 +b(t + x ) J

Var(ä)

1 +~+x)

ä(l+/~t)2~-2 Var(/~)_2{ B [1 +[~(t+x)] 2~+2

l+/~t

}a

1 +~(t+x)

~ln{1 -FffQ~-x) 1+~~}äl~+~~lat«ovlä,~ [1 +[~(t+x)] a~+l

The asymptotic variance of the failure intensity function can be obtained as follows: /~2

ä2

äß

Var(,~) - ¢ t .+~ (b) 1 Var(ä) -~ (1 + t~t)4 Var(b) + . t . ~+ ( b) 1 Cov(ä, b) .

3.2.4. The Duane model The Duane model was one of the earliest models proposed for hardware reliability (Duane 1964). Duane noticed that if the cumulative failure rate versus the cumulative testing time was plotted on log-log paper, it tends to follow a straight line. Later on, Crow and Singpurwalla (1974) observed that it can also be referred to as a Weibull process. It is an NHPP in which the failure intensity function has the same form as the hazard rate for a Weibull distribution. This model is sometimes called the power model since its mean value function for the cumulative number of failures by time t is taken as the power of t, which is

M. Xie and G. Y. Hong

716

# ( t ) = a t b,

a>0,

b>0

.

(8)

For this model, the cumulative number of failures observed at time t will tend to be on a straight line on a log-log scale plot. The slope of the fitted line gives us an estimate of in a. This graphical interpretation provides us a simple way to estimate the parameters and it is very useful in reliability analysis (Donovan and Murphy, 1999). This model has the failure intensity function

),(t) = abt b-1 A typical plot of the mean value function #(t) and failure intensity function 2(t) for the Duane model is shown in Figure 5. The MLEs of the parameters a and b a r e of close form and they are given by (Crow and Singpurwalla, 1974) k,

ä

k

4

D

~~-j1 ln(&/ti)

The software reliability R(xlt ) at the time interval (t, t+ x] can be derived as

R(xlt) = exp[at b - a(t + x) bI . The asymptotic variance Var(R) is Var(/~)= { S R ) 2 V a r ( ä ) + (/aR'~ 2 Var(b)

\ ~ / «:ä

\ e b/»=»

~~'~(~~)

~ov~ä,~~

==a,b=~

+ kCa) ~

50 380

45 40

¢O

>~ 35

330 =

30

Lt.. 13)

_c 25

280

20 ij_

c-

15

IlJ

230

10

'

'

'

2

4

6

'

'

.

.

.

8 10 12 14 Time t (hour)

. 16

180 18 20

Fig. 5. A typical plot of the/x (t) and )~(t) for the Duane mode1

Software reliability modeling, estimation and analysis

717

Then the asymptotic variance for the failure rate function is Var(~)

b2t2/;-2Var(ä) + [ät~-1 + äb(D - 1)tb-2]2Var(b)

=

+ 2ä/~t2~-2[1 + / ~ ( b - 1)t-l]Cov(ä, b) .

3.2.5. The log-power model Modifying the Duane mode1, Xie and Zhao (1993) proposed an N H P P model called log-power model. It has the mean value function #(t)=alnb(l+t),

a>0,

b>0.

(9)

An important property is that the log-power model has its graphical interpretation. If we take the logarithmic on both sides of the mean value function, we have ln#(t) = lna + b l n l n ( 1 + t) . If the cumulative number of failures is plotted versus the running time, the plot should rend to be on a straight line on a log-log scale. A first-model-validationthen-parameter-estimation approach is also discussed in Xie and Zhao (1993). This model has the following failure intensity function ,~(t) :

ab in b-~ (1 + t) l+t

A typical plot of the mean value function/~(t) and failure intensity function B(t) for the log-power model is shown in Figure 6.

30

280

Mean Value function

25

~t (0

260 240 = O

B 20

22o o~

~-15

2oo g

lt.

180 ~160 ~ 140 0

i

i

i

3

5

7

i

9

i

I

i

i

120

11 13 15 17 19 21

Time t (hour) Fig. 6, A typical plot of the #(t) and )~(t) for the log-power model.

M. Xie and G. Y. Hong

718

The MLEs of the parameters a and b a r e ä-

k lnb(1 + t k ) '

b=

k

klnln(1 +tk)

- 2~=~ lnln(1 +tl)

The software reliability R(x)) at the time interval (t, t + x] can be derived as

e(xl«)

= exp{alnb(1 + t) - alnb(1 + t + x ) } .

Then the asymptotic variance Var(R) can be calculated using the following equation: Var(Æ) = , 2

Var(ä) +

\8a /la=ä

÷

Var(/;)

\ 8 b J D=D

2//~"' ~( 8") Cov(~,~;) ~Ôaß ~ b a=ä,b=[,

.

Then the asymptotic variance for the failure rate function is Var(i) =

;2 ln2~-2(1 + t) Var(ä) (1 + t) 2

+

ä2 W»-2(~ + t) [~ + blnln(1 + t)]2Var(/~) (l + t) 2

÷

2ä/;ln2;-2(1 + t)[1 + »lnln(1 + t)]Cov(ä,b) . (1 + t) a

3.3. A case study N H P P models are widely used by practitioners. Usually point estimates are used and there are many examples available. An example is given in this section to illustrate the applications of the interval estimation of parameters and reliability prediction. A piece of software was developed and then tested for 28 weeks. The complete failure data were recorded and given in Table 1. Table l Number of failures per week from a large communication system Week

Failures

Week

Failures

Week

Failures

Week

Failures

1 2 3 4 5 6 7

3 3 38 19 12 13 26

8 9 10 11 12 13 14

32 8 8 11 14 7 7

15 16 17 18 19 20 21

7 0 2 3 2 5 2

22 23 24 25 26 27 28

3 4 1 2 1 0 1

Software reliability modeling, estimation and analysis

The Goel-Okumoto

719

m o d e l is u s e d h e r e . T h e M L e s t i m a t i o n s b e c o m e s t a b l e

only when an adequate amount of data are accumulated. The MLE results of the l a s t f e w w e e k s a r e l i s t e d i n T a b l e 2. The 95% confidence intervals for parameters a and bare calculated using the m e t h o d i n t r o d u c e d i n t h e p r e v i o u s s e c t i o n s a n d t h e r e s u l t s a r e l i s t e d i n T a b l e 3. The 95% confidence intervals for the software failure intensity and reliability p r e d i c t e d i n t h e f o l l o w i n g w e e k s a r e l i s t e d i n T a b l e 4.

Table 2 The estimation of parameters a and b using ML methods Week

Failures

CMF

a

b

20 21 22 23 24 25 26 27 28

5 2 3 4 1 2 1 0 1

220 222 225 229 230 232 233 233 234

258.9 256.3 256.3 268.5 255.8 255.3 256.3 250.2 249.2

0.09472 0.09578 0.095597 0.083328 0.095575 0.095751 0.092254 0.099146 0.099855

Table 3 A 95% confidence intervals for parameters a and b Week

a

ac~

aL

b

bv

bL

20 21 22 23 24 25 26 27 28

258.9 256.3 256.3 268.5 255.8 255.3 256.3 250.2 249.2

224.7 222.6 222.8 233.7 222.7 222.5 223.4 218.1 217.3

293.2 290.0 289.8 303.3 288.9 288.2 289.2 282.3 281.1

0.09472 0.09578 0.09560 0.08333 0.09558 0.09575 0.09225 0.09915 0.09986

0.07397 0.07519 0.07544 0.06548 0.07626 0.07682 0.07407 0.08058 0.08159

0.1155 0.1164 0.1157 0.1012 0.1149 0,1147 0,1104 0,1177 0,1181

Table 4 A 95% confidence intervals for the failure intensity and software reliability predicted for the following weeks Week

)~

)~u

)~L

R

Ru

RL

28 29 30 31

1.52 1.38 1.24 1.13

2.11 1,92 1,75 1,59

0.93 0,83 0.74 0.66

0.74 0.76 0.78 0.80

0.83 0.84 0.86 0.87

0.65 0.68 0.70 0.73

M. Xie and G. Y. Hong

720

3.4. Some extensions The models described in the previous sections are simple and commonly used ones. However, they are based on a number of assumptions. Depending on actual situation, some assumptions should be relaxed and there a r e a number of N H P P models developed for specific applications. Considering software faults of different type, Yamada et al. (1985) developed two-type of failure. Pham (1996) extended it by considering three different error types: type 1 error (critical) - very ditficult to detect; type 2 error (major) difficult to detect; type 3 error (minor) - easy to detect. The model also allows for the introduction of any of these errors during the removal of an error, which is commonly known as imperfect debugging. The mean value function of the software reliability model is 3

#(t) = Z

api [1 - exp(-(1 - fii)bit)]

i=1 1 -/~i

where pi is the proportion of type i fault. Yamada et al. (1986) incorporated the concept of test effort in the GoelOkumoto model (Goel and Okumoto, 1979) for a better description of the failure phenomenon. Later, Xia et al. (1992) incorporated the concept of a learning factor in the same model. They have considered the effects of test effort and learning process to describe the failure process independently, but in reality, test effort and learning process are dependent on each other. Chatterjee et al. (1997) presented a software reliability growth model which incorporates the joint effect of test effort and learning factor. The overall mean value function has the following form:

#(t)=a

1-exp

f 2kb

-bLl" ~ 2kb2a ) In

~ c~

~ J=+pt kb2 "

where a, b a r e the same notations as in the Goel-Okumoto model, k is proportionality constant, and c~, fi are constants. The parameters a, b, cq/3 and k can be estimated by using the traditional maximum likelihood method. Although the parameter estimation based on N H P P models can be carried out in a standard way, Knafl and Morgan (1996) presented the results of general twoparameter N H P P model. Zhao and Xie (1996) discussed the problem of parameter estimation in more details. Model comparison and validation is another interesting statistical problem. In most of the papers presenting a new model, some results comparing with other models are usually presented. Gaudoin (1998) studied the test for the Duane model and it can also be used for the log-power model.

Software reliability modeling, estimation and analysis

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4. Markov software reliability models Software failure process can be considered as a counting process and software reliability can be studied under the framework of Markov assumption. Markov models are very useful in studying software fault-removal processes. Especially, many earlier studies tend to focus on this type of models.

4.1. The Jelinski-Moranda (JM) model One of the earliest software reliability models is the Jelinski and Moranda (1972) model. The underlying assumptions of the JM-model are: (1) There a r e a fixed number of unknown initial faults within the software; (2) When a fault is detected, it is removed immediately and no new fault is introduced; (3) Times between failures are independent, exponentially distributed random quantities; (4) All remaining software faults contribute the same weighing factor to the software failure intensity. Denote by No the number of software faults in the software before the testing. The initial failure intensity is then equal to N0~b where q5 denotes the failure intensity contributed by each fault. Hence after removing the kth fault, there are (No - k ) faults left, and the failure intensity becomes ~b (No - k ) . Let ti be the observed time between the ( i - 1 ) s t and the /th failure. Then T/, i = 1 , 2 , . . , N 0 , are then exponentially distributed random variable with parameter

)~(i)=~(No-i+l),

i=l,2,..,N

0 .

The distribution of Tj is given by P(T/
i=l,2,..,N0

.

This model has an order statistic interpretation. Successive failure times constitute order statistics of No independent random variables from an exponential distribution with parameter ~b. The parameters ~b and No in the JMmodel can be estimated by maximum likelihood technique. The JM-model has been generalized by various authors. For some recent references, see E1 Aroui and Lavergne (1996). In the following, we will introduce some more direct generalizations. The number of faults detected is denoted here by n which will be called the sample size. Suppose that the failure data set ~= {tl, t 2 , . . ,th; n > 0} is given, the parameters q~ and No in the JM-model can easily be estimated by maximizing the likelihood function. The likelihood function of the parameters No and ~b is given by

M. Xie and G. Y. Hong

722

L(tl, t 2 , . . , th; No, ¢) = 11 ¢(N0 - i + 1). exp[-¢(N0 - i + 1)ti] i=l

=q5 n

-i+l

.exp -

-i+l)ti

By taking the partial derivatives of this log-likelihood function with respect to No and ¢, respectively, and equating them to zero, we get the following likelihood equations: 8 lnL_~ 1 ~ No i=1 No --i + 1 lnL -n ~ ê~b

~i=1 (N0

~--~~q~ti = 0, i=1

1 -i+1

) ti=O .

Usually numerical procedures have to be used to solve these two equations. However, the equation system can be simplified as follows. By solving q5 from the second equation above we ger -1

B=n

-i+1

ti

and by inserting this into the first equation, we obtain an equation independent of ¢. The estimate of No can then be obtained. 4.2. A general DFI formulation

An assumption with the JM-model that is widely discussed is that all software faults are assumed to be of the same size. In fact, some faults are more easily detected than the others. A modification of the JM-model is presented in Xie (1990). Basically, the JM-model can be modified by using other intensity function 2(i). A failure intensity function 2(i) is said to be decreasing failure intensity (DFI) if 2(i) is a decreasing function of i. Software failure process under Markov assumption can be described as a Markov process model with DFI function. The theory for continuous time Markov chains can then be applied such as that the collection of probabilities {pi(t)=P[N(t)=iJ;

i=0,1,2,..,N0,

t_>0}

satisfies the so-called Kolmogorov's differential equations. The power-type DFI Markov model assumes that the failure intensity 2(i) is a power-type function of the number of remaining faults, that is 2 ( i ) = ¢ [ N o - ( i - 1 ) ] ~,

i= 1,2,..,N0

.

Software reliability modeling, estimation and analysis

723

As we expect that 2(0 decreases fastet at the beginning, it is reasonable to assume that 2 ( 0 is a convex function of i. Hence, ct is likely to be greater than one. Another type of function that satisfies this requirement is the exponential-type Markov DFI model. It assumes that the failure intensity is an exponential function of the number of remaining faults. It is characterized by the failure intensity function 2(0:-qS[exp{-fi(N0-i+l)}-l],

i= 1,2,..,N0 .

The parameters in 2(0 can be estimated by maximizing the likelihood function given by L(t, 2(.)) = ~

2(0 e x p { - 2 ( i ) t i }

.

i=l

4.3.

The Shanthikumar

general Markov

model

The JM-model has been generalized by using a general time-dependent transition probability function by Shanthikumar (1981). This model assumes that the failure intensity function as then number of faults removed is given by 2(n, t) = ~o(t)(No - n) , where qS(t) is a proportionality factor. Under the Markov assumption, the forward Kolmogorov's differential equations can easily be obtained and solved together with some standard boundary conditions. This model reduces to the JM-model when ~b(t) = 1. Furthermore, a special case of this model is known as the Schick-Wolverton model (Schick and Wolverton, 1978). It has a linearly increasing function, i.e. qS(t~) = cti. That is, for the Schick-Wolverton model, the failure intensity function after detecting the /th fault is 2(ti) = O(No - i + 1)tl . Note that the failure intensity function of the Schick-Wolverton model depends both on i, the number of removed faults and ti, the time since the removal of last fault. The probability distribution function of N ( t ) is denoted by Pù(t). Under the Markov assumption, we have that the forward Kolmogorov's differential equations are given as follows,

aPo(t) ~t

--

-No~(t)Po(t),

8Pn(t) _ (No - n -F 1)~)(t)Pn_l(t) - (No - n)~)(t)Pn(t); at

1 < n < No •

M. Xie and G. Y. Hong

724

Using the boundary conditions P,(0)= 0, n > 0 and P0(0) = 1, this system of differential equations can easily be solved and the solution is given by

P~(t)=(N°)[a(t)]N°-'[1-a(t)]';

l
,

where a(t) is defined as

For this general model the number of remaining faults at time t is Binomial distributed with parameter a(t), that is

P[No - N(t) = k] = (N°)[a(t)]k[1 _ a(t)]N0-k;

1 < n < No •

It should be noted that the assumption that all fäults contribute the same amount to the failure intensity is still used here. Hence this model suffers from the same critique as the JM-model.

4.4. Some further remarks In this section, some general software reliability growth models based on Markov process assumption are described. There are many similar models and extensions. Detailed statistical inference has also been studied in a number of papers. Because of the complexity of such models, it has not been widely adopted compared with the non-homogeneous Poisson process models. However, it should be pointed out that Markov models are very general, and it can be used in many reliability related context. For example, for combined software-hardware systems, reliability measures can be directly evaluated by available tools for numerical processing of the Markov chains (Kanoun et al., 1993). It can also be used for availability studies of complex systems, see e.g., Goseva-Popstojanova and Trivedi (2000), Whittaker et al. (2000) and Laprie and Kanoun (1992).

5. Some Bayesian approaches Parameter estimation is one of the difficulties encountered in using the existing Markov and N H P P models. Many Bayesian models have been proposed for the analysis of software failure data with knowledge of previous releases.

5.1. The Littlewood-Verrall (LV) model One of the most well-known Bayesian model was proposed by Littlewood and Verfall (1973). The LV-model assumes that times between failures are exponen-

Software reliability modeling, estimation and analysis

725

tially distributed with a parameter that is treated as a random variable, which is assumed to have a Gamma prior distribution. The exponentiality is natural and justified by the random testing condition. The choice of Gamma distributed prior is mainly due to its flexibility and mathematical tractability. Specifically, the successive times between failures, tl, i = 1 , 2 , . . ,n, are assumed to be independent, exponentially distributed random variables with density function f(til2i) ---- )Liexp{--)~iti)

i= 1,-2,..,n

,

where 2~ is an unknown parameter whose uncertainty is due to the randomness of the testing and the random location of the software faults. The uncertainty of 2~ is described by a Gamma distribution with parameter c~ and O(i), that is ù c~ c~-i

f(2ilc~,O(i) ) _ IN(0] 2i

exp{_O(i)2i)

i= 1,2,..,n

,

where c~is the shape parameter and O(i) is the scale parameter depending on the number of detected faults. By taking different forms for O(i), different test environments may be described. The conditional likelihood function of observation t~, can be shown to be

f(tilc~, O(i)) -

c~[~t(i)]~ [ti q-

for ti > 0 .

@(i)ff+l

The posterior distribution of t~, is then given by

F(ti[~,q/(i))=

l-

[ ~t(i) ]~ Lti + ~t(i)J "

The posterior failure rate function, give~ « and O(i), is simply expressed by ¢(

)o(tilc~, t)(i) ) - ti ÷ &(i)

1

It can be noted that for the LV-model, times between failures have a Pareto distribution. In fact the times between failures have a strictly decreasing failure rate. Further analysis requires specific functional form for O(i). Usually, O(i) describes the quality of the test and it is a monotonically increasing function of i. This condition implies that Æi is stochastically decreasing in i, i.e. P ( 2 ~ < 2 ) >P(2i_l_<2)

for a l l i > 1 .

As an example we may specify O(i) to be a linear function of i, that is O(i) = flO + fli i ,

which is also originally proposed by Littlewood and Verrall (1973). In the above, B0 and fia are model parameters to be determined.

M. Xie and G. Y. Hong

726

5.2. The L a n g b e r g - S i n g p u r w a l l a model and modifications

Another important Bayesian model is presented in Langberg and Singpurwalla (1985). In the original paper, a shock model interpretation of software failure is given as a justification of the JM-model. In the Langberg-Singpurwalla Bayesian model the parameters in the JM-model are treated as random variables. Denote by (b the failure intensity per fault and let N be the number of faults initially in the software. Given ~b and N, the times between failures are exponentially distributed with parameters • and N, that is P(T~. > tiIN,+) = e x p [ - + ( N - i + 1)til

i > 1

and we can assign a prior distribution to the pair (N, ~). The resultant posterior distribution may be obtained by the Bayes theorem. In the original paper by Langberg and Singpurwalla (1985), three cases by assigning different prior distributions to the parameters ~b and N are considered. The most general model, which includes the other two as special or limiting cases, is the following. Due to the discrete nature of the parameter N, the prior distribution for N is allowed to be any specified discrete distribution given by k= 0,1,2,..,

7zk=P(N=k)

and • is assumed to have a Gamma prior distribution with scale parameter a and shape parameter b, i.e. ab + b- l e-~~ P(+)

-

+ >- o

r(b)

.

In this case, the joint posterior distribution of N and ~b given the data set t is the following +b+n- ~e-~(a+Tù,k) P ( N = k, cb = +It-) =

C

rck

k »_ n

where C is a normalizing constant given by

c:r(b+n)~~/_-n)!.jt

(a + T,"«,.~-ó-" ~j

J=n

and Tù,k is the total time on test defined by Tn,~=~~~(k-i+l)ti

k>n

.

i=1

The posterior probability of ~bgiven N = k is also Gamma distributed, Gamma (d, b~), with revised shape and scale parameters a' and b' which are given by aI = a + Tn~ß,

bl = b + n ,

Software reliability modeling, estimation and analysis

727

respectively. The posterior marginal probability of N, is shown to be k! ( a + Th,k) b-n ~k (k-n)! P ( N = klt) = x-'~ J~ (a -b-n z_ùj=n (j-n)! ~ + Th,k) ~j

k >_ n .

Jewell (1985) presented a further Bayesian formulation of the JM-model following that of Langberg and Singpurwalla. As for the Langberg-Singpurwalla Bayesian model, the parameter • is assumed to be G a m m a a priori and the number of initial software faults N is assumed to be a r a n d o m variable having a Poisson(A) distribution. The difference between the Jewell's formulation and that of Langberg and Singpurwalla is that A is further assumed to have a G a m m a prior distribution. Some further discussion can be found in Bergman and Xie (1991).

6. Some forecasting methods 6.1. Time series models Software failures are successively observed and the failure data can be treated as a time series. As time series models are very useful in forecasting which in this case is the forecasting of failure behaviour, they can be used in software reliability analysis. Some typical models are the r a n d o m coefficient autoregressive process model and a Fourier series model by Crow and Singpurwalla (1984). An autoregressive process is a sequential process by which we mean that the value of interest at instance n, tù, depends on the previous values of the series together with a noise factor. Denote by f the functional relationship between the value tn and the previous observations, th-i, i < k, we have that tn = f ( t n - l , t n - 2 , . . , t n - k )

+ en ,

where en is a r a n d o m variable denoting the random fluctuation of the observed value. A simple model can be based on weighted sum of past k observations: k tn = ~

aitn-i + en

for all n .

i--1

A more general class of time series models is so-called autoregressive integrated moving average ( A R I M A ) models. An A R I M A (p, d, q) model can be defined as %(B)v~xt = o«(B)e, , where Op(B) = (1 - q~lB - q~2 B 2 . . . . . OpBp) is an autoregressive polynomial of order p. Oq(B) = (1 - O1B - 02 B2 . . . . . OqB q) is a moving polynomial of order q, and V is the backward difference operator, B is the backshift operator, and et is a sequence of normally and independently distributed random "shock" with mean zero and constant variance a z.

M. Xie and G. Y. Hong

728

A model based on a first-order autoregressive process with a random coefficient 0~ can be obtained as follows. Denote by T/the time to failure of the software after i changes, i= 0 , 1 , . . , n. Assume that T~ is related to T~_I which is reasonable because the software is subjected to only minor changes when a detected fault is corrected. In order to reflect this fact, the dependence may be written as T/ = 6iT/~l,

i = 1,2,..,n

,

where Õi is a coefficient and 6i is an uncertainty factor. The relation is recognized as the power law in reliability and biometry. The model will describe reliability growth or decay depending on whether the value of 0i is greater or less than unity. Different choices of models with respect to 0, i > 1, can be made by using different K a l m a n filter models. It has been observed that software failures usually occur in clusters. For this type of problem, a Fourier series model is proposed in Crow and Singpurwalla (1984) which has shown that the Fourier models can be used successfully for analyzing clustered failure data, especially those with cyclic behavior. Denote by et the disturbance term of the process with mean zero. Ler t~, i ~- 1 , 2 , . . . , n, be the times between failures, the model assumes that there is a cyclic function g(i) such that

ti=g(i)+si,

i= 1,2,..,n

.

This means that the times between failures are deterministic functions of the number of failures caused by r a n d o m disturbances. More specifically, for odd n, the Fourier series model assumes that g(i) is a linear combination of sine and cosine terms, that is, for all i,

,~1,~~{ (~:)

g(i) = ~o + ~

o~(kj)cos

kji

+fl(kj) sin(2~ßkji)) • The estimates of the parameters c~0, c~(kj) and method of least squares. They are given by

fl(kj)

n

~(kj)

B 2tl t/2rc..~ = 2..a--COSl--Kj// i=l

H

\H

~/

j = 1,2,..,q

may be obtained using the

729

Software reliability modeling, estimation and analysis

6.2. Regression models Another important type of statistical technique that has a great potential in analyzing software failure data is the regression analysis. Usually software reliability characteristics depends on a number of factors such as software size and development metrics. It is assumed that there is a relationship between the software reliability and other factors during software development, and the information should be made use of in software failure prediction. Let y denote the software reliability attribute of interest and assume that x is the explanatory variable which may be a vector consisting of a number of metrics. If the relationship between y and x may be described by a systematic functionf(x) subject to a random fluctuation e, that is y = f(x)

+ ~ ,

where e is a noise term with zero expectation, then this relationship can be used for future projects development under similar environment. A special type of regression model is the Cox proportional hazard model. It assumes that the failure intensity is an exponential function of the explanatory variables. That is, the following holds

r(t, x)

=

ro(t) exp{fllXl +

f12x2 @ . . . -- flmXm} ,

where xi's are the explanatory variables and/3i's are the corresponding regression coefficients. In the above, ro(t) is a baseline failure intensity function that expresses the failure intensity when all explanatory variables are set to zero. Historically, the Cox proportional hazard model has been successfully used in analyzing biomedical data. Since software reliability is strongly affected by many factors during the software development, proportional hazard models are helpful tools also in analyzing software failure data (Slud, 1997). A direct generalization of the simple proportional hazard model is to allow the explanatory variables xi to be a function of time which is normally the case. The general formulation is that the relation between the failure intensity and the time varying explanatory variable is given by

r( t, x(t))

= ro (t) exp{fllXl (t) +

f12x2(t)

-r-

" " "

--

flmXm([) }

.

In order to estimate the baseline failure intensity function r0 (t), we may use either a parametric approach or a distribution-free approach that can be found in many statistical texts.

7. Final remarks

Among the many models proposed, N H P P models are the most widely used one by software engineers and developers. However, it is recognized that most of the models do not give satisfactory results. This is partly because of the lack of understanding of the stochastic nature of failure process and partly because of the

730

M. Xie and G. Y. Hong

lack o f d a t a a n d i n f o r m a t i o n a v a i l a b l e for the analysis. I n fact, it is i m p o r t a n t to i n c o r p o r a t e a d d i t i o n a l i n f o r m a t i o n a v a i l a b l e to the d e v e l o p e r such as p a s t p r o jects a n d metrics for c u r r e n t project. It s h o u l d be p o i n t e d o u t t h a t o t h e r t h a n failure p r e d i c t i o n a n d analysis, issues such as release t i m e m o d e l i n g ( K a p u r et al., 1994; C h a t t e r j e e et al., 1997; a n d Xie a n d H o n g , 1998) a n d t e s t i n g r e s o u r c e a l l o c a t i o n ( Y a m a d a et al., 1995) are i m p o r t a n t to p r a c t i t i o n e r s as well. S o f t w a r e reliability m o d e l s p l a y a n imp o r t a n t role in this type o f studies. A s s o f t w a r e d e v e l o p m e n t is a h i g h - r i s k v e n t u r e , it is i m p o r t a n t to also m a k e m o r e use o f i n t e r v a l e s t i m a t i o n in v a r i o u s analysis.

References Bergman, B. and M. Xie (1991). On Bayesian software reliability modelling. J. Stat. Plan. Infer. 29, 33-42. Chatterjee, S., R. B. Misra and S. S. Alam (1997). Joint effect of test effort and learning factor on software reliability and optimal release policy. Int. J. Syst. Sei. 28, 391 396. Crow, L. H. and N. D. Singpurwalla (1984). An empirically developed Fourier series model for describing software failures. IEEE Trans. Reliab. R-33, 176 183. Donovan, J. and E. Murphy (1999). Reliability growth a new graphieal model. Q. Reliab. Eng. Int. 15, 167-174. Duane, J. T. (1964). Learning curve approach to reliability monitoring. IEEE Trans. Aerospace 2, 563-566. E1 Aroui, M. A. and C. Lavergne (1996). Generalized linear models in software reliability: parametric and semi-parametric approaches. IEEE Trans. Reliab. 45, 463-470. Gaudoin, O. (1998). CPIT goodness-of-fit tests for the power-law process. Commun. Stat. Theory Meth. 27, 165 180. Goel, A. L. (1985). Software reliability models: assumptions, limitations, and applicability. IEEE Trans. Software Eng. 11, 1411-1423. Goel, A. L. and K. Okumoto (1979). Time-dependent error-detection rate model for software reliability and other performance measures. IEEE Trans. Reliab. 28, 206-2i 1. Goseva-Popstojanova, K. and K. S. Trivedi (2000). Failure correlation in software reliability models. IEEE Trans. Reliab. 49, 37-48. Jelinski, Z. and P. B. Moranda (1972). Software reliability research. In Statistical Computer Performanee Evaluation, pp. 465-484 (Ed. W. Freiberger). Academic Press, New York. Jewell, W. S. (1985). Bayesian extens]ons to a basic model of software reliability. IEEE Trans. Software Eng. 11, 1465-1471. Kanoun, K, M. Kaaniche, C. Beounes, J. C. Laprie and J. Arlat (1993). Reliability-growth of faulttolerant software. IEEE Trans. Reliab., 42, 205-219. Kapur, P. K., S. Agarwala and R. B. Garg (1994). Bicriterion release policy for exponential softwarereliability growth-model. RAIRO-RECH OPER, 28, 165 180. Knafl, G. J. and K. Morgan (1996). Solving ML equations for 2-parameter Poisson-process models for ungrouped software-failure data. IEEE Trans. Reliab., 45, 42-53. Langberg, N. and N. D. Singpurwalla (1985). A unification of some software reliability models. S I A M J Sci. Stat. Comput. 6, 781-790. Laprie, J. C. and K. Kanoun (1992). X-ware reliability and availability modelling. IEEE Trans. Software Eng. 18, 130 147. Littlewood, B. and J. L. Verrall (1973). A Bayesian reliability growth model for computer software. Appl. Star. 22, 332-346. Lyu, M. R. (1996). Handbook of Software Reliability Engineering. McGraw-Hill, New York.

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Musa, J. D., A. Iannino and K. Okumoto (1987). Software Reliability." Measurement, Prediction, Application, McGraw-Hitl, New York. Musa, J. D. and K. Okumoto (1984). A logarithmic Poisson execution time model for software reliability measurement. In Proceedings of the Seventh International Conference on Software Engineering. Orlando, pp. 230-238. Pham, H. (1996). A software cost model with imperfect debugging, random life cycle and penalty cost. Int. J. Syst. Sci. 27, 455-463 Ramamoorthy, C. V. and F. B. Bastani (1982). Software reliability status and perspectives. IEEE Trans. Software Eng., 8, 35~371. Schick, G. J. and R. W. Wolverton (1978). An analysis of competing software reliability models. IEEE Trans. Software Eng. 4, 104~120. Shanthikumar, J. G. (1981). A general software reliability model for performance prediction. Microelectron. Reliab., 21, 671-682. Shooman, M. L. (1984). Software reliability: a historical perspective. IEEE Trans. Reliab. 33, 48-55. Slud, E. V. (1997). Some applications of counting process models with partially observed covariates. Telecommun. Syst., 7, 95 104. Whittaker, J., K. Rekab and M. G. Thomason (2000). A Markov chain model for predicting the reliability of multi-build software. Inf. Software Technol. 42, 889-894. Xia, G., P. Zeephongsekul and S. Kumar (1992). Optimal software release policies for models incorporating learning in testing. Asia Pacißc J. Oper. Res. 9, 221-234. Xie, M. (1990). A Markov process model for software reliability analysis. Appl. Stochas. Models Data Anal. 6, 207~14. Xie, M. (1991). Software Reliability Modelling. World Scientific, Singapore. Xie, M. (1993). Software reliability models - an annotated biography. J. Software Verif. Test. Reliab. 3, 3~8. Xie, M. and G. Y. Hong (t998). A study of the sensitivity of software release time. J. Syst. Software, 44, 163 168. Xie, M. and M. Zhao (1993). On some reliability growth-models with simple graphical interpretations. Mieroelectron. Reliab. 33, 149-167. Yamada, S., S. Osaki and H. Narihisa (1985). A software reliability growth model with two types of errors. R.A.LR.O. 19, 87-104. Yamada, S., H. Ohteria and H. Narihisa (1986). Software reliability growth models with testing-effort. IEEE Trans. Reliab. 35, 19-23. Yamada, S. and S. Osaki (1984). Nonhomogeneous error detection rate models for software reliability growth. In Stochastic Models in Reliability Theory, pp. 120-143. Springer-Verlag, Berlin. Yamada, S., T. Ichimori and M. Nishiwaki (1995). Optimal allocation policies for testing-resource based on a software reliability growth model. Math. Comput. Model. 22, 295-301. Yang. B. and M. Xie (2000). A study of operational and testing reliability in software reliability analysis. Reliab. Eng. Syst. Saf. 70, 323 329. Zhao, M. and M. Xie (1996). On maximum likelihood estimation for a general non-homogeneous Poisson process. Stand. J. Star. 23, 597-607.

N. Balakrishnan and C. R. Rao, eds., Handbook ofStatistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

")Q

Bayesian Analysis for Software Reliability Data

Jorge Alberto Achcar

In this paper, we consider Bayesian Analysis for software reliability data using Markov Chain Monte Carlo methods and some usual modeling approach for reliability data: modeling the interfailure times between successive failures or the number of failures up to a given time. The methods are illustrated with two numerical examples.

1. Introduction

Software reliability is the probability of a computer program to be free of error in operation during a specified period of time. The software failures are related to errors in syntax or logic (see for example, Singpurwalla and Wilson, 1994). Once such an error is found, it can be corrected and does not give rise to any more failures. Software randomly that justify the use of stochastic models for software reliability is related to a program receiving many different inputs, each with its own path through the software and so capable of bringing different errors in light. These different inputs arrive to the software randomly, which implies in detection of errors in a random way. The literature in statistics and especially in software engineering presents many different models for software reliability (see for example, Singpurwalla and Wilson, 1994; or Mazzuchi and Soyer, 1988). Among these stochastic models, we have two different strategies:

Type I strategy: Modeling times between successive failures of the software. Type H strategy: Modeling the number of failures of the software up to a given time. The type I strategy models can be derived from considerations of the failure rates of the software (type I-1 strategy) or to define a stochastic relationship between successive failure times (type I-2 strategy). A popular model of type I-1 strategy is introduced by Jelinski and Moranda (1972). Suppose that the total number of bugs in the program is N, and suppose that each time the software fails, orte bug is corrected. 733

734

J. A. Achcar

The failure rate of t h e / t h time between failures T/, is assumed to be a constant proportional to N - i + 1, which is the number of bugs remaining in the program. Thus, the failure rate for T/is given by, rr,(tIN, A ) = A ( N - i +

1) ,

(1)

where i = 1,2, 3 , . . , t _> 0, for some constant A. This means that if N and A are known, then T/has an exponential density, f ( t i l 2 i ) = 2ie -xit~ ,

(2)

where 2i = A ( N - i + 1). Some modifications of JM model (1) are presented in the literature. Moranda (1975), supposed that the fixing of bugs that cause early failures in the system reduces the failure rate more than the fixing of bugs that occur later, because these early bugs are more likely to be the bigger ones. Thus, Moranda (1975) assume that the failure rate should remain constant for each T/, but that it should be made to decrease geometrically in i after each failure, that is, for constants D and K, rr, (tID, K ) = D K i-1 ,

(3)

wheret>0,D>0and0
(4)

When p = 1, we get the JM model (1). Schick and Wolverton (1978) assume that the failure rate is proportional to the number of bugs remaining in the system and the time elapsed since the last failure. Thus, rr~(t[N, A ) = A ( N - i + 1)t .

(5)

Other models also are proposed in the literature following type I - 1 strategy. An alternative to model time between failures is to define a stochastic relationship between successive failure times (type I-2 strategy). As an example, let Tx,..., T/,... be random variables denoting the length of time between successive failures of the software, with the relationship, T/+I = DTi @ Si ,

(6)

where p _> 0 is a constant, and ei is an error term. Jelinski and Moranda (1972) and Joe and Reid (1985) resort to maximum likelihood methods to estimate the number of bugs in the software. A more direct approach, as compared to modeling the interfailure times, is to model the number of failures M ( t ) discovered in the interval (0, t] (type II strategy).

Bayesian analysis for software reliability data

735

The counting process most commonly used to model software errors is the Non-homogeneous Poisson process (NHPP), though there is no reason other counting processes may not be used. M(t) is modeled by an N H P P with the mean function given by m(t) = E[M(t)], a non-decreasing function of t. The intensity function is given by 2 ( t ) = (dm(t)/dt). The probability function is given by PIM(t) = n] = ([m(t)]~/n!)e -ra(t), where n = 0, 1 , . . For a constant 2(t),M(t) reduces to a Homogeneous Poisson process. Suppose there are an unknown number of faults N, at the beginning of the debugging stage. When the software failures are modeled as the first n-order statistics taken from N i.i.d, observations from a common density f and cdf F, with support in R +, the statistical model is referred to as a General Order Statistics (GOS) Model. In addition if N is distributed as a Poisson variable with mean 0, then M(t) is N H P P - I and ra(t) = OF(t). Goel and Okumoto (1979) with F(t) = 1 - e -~t and Goel (1983) with F(t) = 1 - e -~t~ are common examples of these processes. Achcar et al. (1998), consider a supermodel given by the generalized gammaorder statistics model which incorporates somes of the standard models discussed in the literature and the log normal-order statistics model. Observe that m(t) -+ 0 as t ---+ec. When new faults are introduced during debugging, the Record Value Statistics (RVS) model replaces the GOS setup. The software failures are modeled as record breaking statistics of unobserved i.i.d, outcomes from a density f . Here ra(t) ~ oc as t ~ oc. A special case of the mean function is ra(t) = - 1nil - F(t)] which yields the Musa and Okumoto process (1984) with 2(t) = a/(t + fi), the Duane process (1964) with 2(t) = c~fiff-~, and the Cox and Lewis process (1966) with 2(t) = exp(~ + fit). Kuo and Yang (1996) developed a unified approach for the software reliability growth models incorporating both the GOS and the RVS cases. They proposed the use of Markov chain Monte Carlo methods to model the software failures. Kuo et al. (1996) also developed Bayesian inference for the Ohba et al. (1982) process by considering a further extension given by the N H P P - G a m m a - K model with F(t) = 1 - e pt ~k-ò j ~ ((~t)J/j!) An alternative to monotonic intensity functions, is to consider the superposition of several independent N H P P with simple intensity functions. Kuo and Yang (1996) develop Bayesian inferences and model selection methodologies for the superposition model. The points of failure of a superposition process are defined to be the union of the points of failure from several component point processes. Let Mj(t) denote the N H P P for the failures f r o m j t h component in (0, t), with intensity function 2j(t[Æj), where the function form of #(tl/?j) is assumed to be known with unknown ~ärameter fij that may be a vector. We also assume that the number of failures from the jth dömponent Mj(t),j = 1 , . . ,J is independent. A process M(t) = ~J=l Mj(t) which counts the number of failures in the interval (0, t) for the superposition model, is also a N H P P with intensity function 2(t]_fi) = 2jJ_l 2](t[fij), where fi_ = (fi_2,'", fiJ)"

736

J. A. Achcar

With superposition of NHPP, we can have many different forms for the intensity function: bathtub-shaped functions, polynomials with peaks and valleys, and simple monotonic functions. Some special cases are given by, (i) 2(t)

~iflllal 1 _~_~2P2ot~)-l_ (two Weibull intensity functions),

=

(il) 2(t) =

c~~ + ~2t + ~3 (Gaver and Acar, 1979), t+/~l (iii) 2(t) = fio + fll t + " " + fimtm

(7)

Usually, we can have great difficulties to get standard classical inference for software reliability models. This motivated the use of Bayesian methods which have the additional advantage of allowing software engineers to utilize information from similar testing. The use of Gibbs sampling with Metropolis-Hastings algorithms (see for example, Gelfand and Smith, 1990) has been used by many authors to ger Bayesian inference for software reliability models (see for example, Kuo and Yang, 1996; Yang, 1996; Kuo et al., 1996; Achcar et al., 1998). 2. A B a y e s i a n analysis for a special type-I strategy model: The Goel and O k u m o t o model

Assuming the GO model (4), the likelihood function for 2~o,p and N (with a failure truncated model) is given by n A (N,p) e x p { - 2 G o B ( N , p ) } L(2Go,N,p) = 2GO

(8)

where A ( N , p ) = I-[i~_lIN - p(i - 1)] and B(N,p) = ~i~=l IN - p(i - 1)lti. Since ti = x; - Xi*l, where x; denotes the ordered epochs of failure time, we have B(N,p) = p ~~-1 x; + (N - np)x*~. Assuming prior independence, consider the following prior densities for 2GO,P and N. (i) 2~o ~ F[al, bl], (ii) N ~ P(01),

(9)

(iii) p ~ B[a2, »2] , where al, bi, a2, b2 and 01 are known constants, P(O) denotes a Poisson distribution with parameter 0, F(a, b) denotes a gamma distribution with mean a/b and variance a/b 2 and B(a, b) denotes a Beta distribution with mean a/(a + b) and variance ab/[(a + b)2(a + b + 1)]. The joint posterior density for 2~o,p and N is given by n+a1-1

~z(2~o,N,pID ) oc

xexp

200

{[ ~ -

bl + p

i=1

A(N,p)O~I _p)b2-1 N! Pa2-1 (1

J}

x*i + (N - np)x~* 2 a o

,

(10)

Bayesian analysisfor software reliability data

737

where 2Go > 0; N = n, n + 1 , . . , and 0 _

I

(i) 2 ~ o I N , p , D ~ F

n+al,bl

+p

x;+(N-np)x*~

i=1

e-010~

(ii) 7c(NI2GO,p,D) oc Nr.

~

]

,

7Sl(N'p'2G°)'

where

(11)

01 (N, p, 2GO) = exp{ln A ( N , p ) - (N - np)x*~Äco } and

(iii) ~(pIN, 2Go, D) o( pa2 where

1(1 _ p)b2-1O2(N,p ' )~GO),

/

~b2(N,p, 2GO) = exp l n A ( N , p ) - p 2 6 o Z

n

x; - (N - np)x;2GO

/

.

i=1

Observe that, the variables N and p should be generated using the MetropolisHastings algorithm (see for example, Chib and Greenberg, 1995).

3. Bayesian inferenee for NHPP-I software reliability models Let us assume that the mean value function ra(t) is indexed by the unknown parameters 0 and 0, where 0 is possibly vector-valued. There can be two possible scenarios here. Time truncated testing refers to the case when the testing is continued till a given time. On the other hand, when we monitor the software till a specified number of failures occur we are looking at the failure truncated situation. Given the time truncated model testing until time t, the ordered epochs of * The likelihood function the observed n failure times are denoted by x *1, x *2 , . . ,x n. is given by n

LN•pp(0, WID) = ]-I[),(x2)]e ra(c)

(12)

i=1

where D = { n ; x l , x 2 , . . ,Xn, t} is the data set (Cox and Lewis, 1966 or Lawless, 1982). For the failure truncated model a similar expression can be applied with t replaced by x*. For Bayesian inference of such software reliability models, we consider the use of the Gibbs Sampler (Gelfand and Smith, 1990; Casella and George, 1992), and the Metropolis-Hastings algorithm (Chib and Greenberg, 1995). The presence of the expression ra(t) = OF(t) in the likelihood function for NHPP-I in Eq. (12) usually prevents us from specifying a convenient form for the conditional density

738

J. A. Achcar

of 0 and 0 given D, needed in the Gibbs sampling. Therefore, we introduce a latent variable N~= N - n which has a Poisson distribution with parameter 0[1 - F ( t l 0 ) ] where 0 indexes all the parameters involved in the cdf F(tl0 ) (Kuo and Yang, 1996). Now the posterior distribution p(O, OID) can be obtained from the joint density p(O, O,N~ID) by marginalization. The joint posterior density p(O, 0, N'IDt) is approximated from the Gibbs samplers drawn from the following conditional densities: p(N']O, 0, D),p(O]N', 0, D), and p(OIN', O,D). As a special case, assume that F(t) is the distribution function of an inverse Gaussian distribution (see for example, Chhikara and Folks, 1989), given by

F(t) = •

- 1

+ e2)#"e -

1 +;

,

(13)

where #(x) denotes the distribution function of a standard normal N(0, 1) distribution. From (13), the intensity function 2(t) which is the derivative of ra(t) is given by

x ( t / = ~0vÄ t

-3/2 2#2/ C-~)2J~ , exp { ')~(-t

.

(14/

The inverse Gaussian distribution represents a wide class of distributions, ranging from a highly skewed distribution to a symmetrical one. This gives a great flexibility for the shape of the intensity function (14), which can be very useful to model software reliability data. The likelihood function for 0, # and 2 is given by,

L(O, lA,~) --

on2n/2 {T'I" , 3/21

(27c)n/2

li=1 lxi

f

)~ ~ ( X ; -- #)2}

- -2#2 - i=1 )? expl-OF(t) ~,

x* (15)

where F(t) is given in (13). For a Bayesian analysis of this model, we consider the use of Metropoliswithin-Gibbs algorithms. Considering the introduction of a latent variable N ~ = N - n, where N is the number of bugs in the beginning of test, we assume the following prior densities for N ~, 0, # and 2: (i) N' ~ P[0[1 - F(t)]],

F(al, bi); al, bi known, (iii) 2 ~ F(a2, b2); a2, ba known, (iv) 0 ~ F(a3, b3); a3, b3 known .

(ii) # ~

(16)

We further assume independence among the parameters 0,# and 2. We choose gamma prior distributions to represent our beliefs on #,2 and 0 since there are different shapes for this distribution and the parameters #, 2 and 0 are positive.

739

Bayesian analysis for software reliabißty data

The joint posterior density is given by 7"c(N/, #, Ä, O/D) ~c ONt+n+a3-1.~n/2+a2 l#al 1~1-~ }Nr

x exp -

• xi

-

-

where D = {n,x*l,... ,x*~,t} is the data set and F(t) is given in (13). For the failure truncated model, similar expressions to (15) and (17) can be applied with t replaced by x~, We assume the failure truncated model. The conditional posterior densities for the Gibbs algorithm are given by, (i) N'/#, 2, O,D ~ P[0[1 - F(x*)]], (ii) 0/#, )~,N',D ~ F[N' + n + a3, b3 -}- 1],

(18)

(iii) re(#~2, O,N',D) o( #as 'e b~#Tl(#,2) , where TI(#,2)={1-F(x*)}

exp-~g2i=1

xi

j,

and (iv) ~z(,~/0, #,N',D)o(Äa2-1e-b2"~~2(#,2), where

t/j2(#,; ) = ~n/2{1 ù F(xn) , }N' exp --~ß2

X*

The variables # and 2 should be generated using Metropolis-Hastings algorithm. The choice of independent gamma prior distributions usually incorporate expert knowledge and engineering information on the software. Other choices for the prior densities could be considered to develop Bayesian analysis for the N H P P models. Campodonico and Singpurwalla (1995) consider a choice of prior distribution for N H P P models based on the information of an expert as the number of modules, the operational profile and other factors. From this information we can consider a joint prior distribution for the mean value function, ra(t) for different values of t.

4. Bayesian inference for the superposition model Let ~ denote the data set observed until time t. It consists of n observed ordered epochs (X*l,...,x~£), where 0 <x~ < . . - < x ; < t. Assuming that the

J. A. Achcar

740

point process of the ordered epochs follow the superposition model, and the prior distributions for fi, j = 1 , 2 , . . , J are independent, the posterior density for fi is given by ~J =(fi[~) c<

2j(x;)

e x p - ; ~ - 1 mj(t)

j=lH~ZJ(fiJ) "

(19)

To simplify the conditional distributions needed for the Gibbs sampling algorithm, we consider the introduction of latent variables (see Tanner and Wong, 1987), Ii = (Iil,..., Ii«), where Iij = 1 if the ith failure is caused by thejth type and Iij = 0 õtherwise, i = 1 , . . , n. Observe that ~J=~ 1,7 = 1 for i = 1 , . . , n. Assume that the conditional distribution for Ii given fi and ~ is a multinomial distribution MN with parameters 1 and cell probabilities (Pil,... ,PiJ), where

;v(x;)

Pij --

E;=I "~S(X* )

Therefore, we have, J

~(~1~,~)

r-r

o~ H P u

« r #(x;)

(20)

j=l

In this way, the posterior density of_fi given I = ( / 1 , . . , In) T, (an n × J matrix), and ~ is,

That is, J

J

~z(fi_lI,~ ) o( H H )v(x;) H e x p ( - m j ( t ) ) j=l i:/,7=1

j=l

J

H~J(fij ) •

(21)

j=l

As a special case, consider the superposition of a Musa and Okumoto (1984) process with intensity function 2~ (t) = (c~/fi1 + t) and an exponential process with intensity function 22(t) = fi2t. Therefore, 2(t) = ~/(B1 + t) + fl2t and mean value function m(t) = c~ log(1 + (t/fil)) + (fi212/2). Assume the following prior distributions for c~,fll and/~2.

O~ ~ _F(al,bl) , B1 ~ F(a2, b2), B2 ~ F(a3,b3),

al,bl known, a2, b2 known,

ag,bs known .

We further assume independence among the parameters. From (21), we have the joint posterior distribution for ~1, fll and fi2,

(22)

741

Bayesian analysis for software reliability data n

n

.al-l+~--~. , Iil oa2-1 o n+a3-1- Ei=l Iil ~z(~,fil,fl2l/,~ ) c( ~ . . . . Pl P2

Filz,,< (/~~+xT)

*

//

X*2~

"~

~e,p( ~~~l (~,+'og('+~l))~ V~3+2J~2J (23) The conditional distributions for the Gibbs algorithm are given by (i) Construct I given fi and ~ by generating independent variables Iil from the Bernoulli distribution with parameter 0{ Pil

+ P2(P~ +x*)x;

(ii)

(

°

(

*

(iii)

7c(fl210~,fll,/,~ ) ~ r

(

a3 4- n - ~-~ fil;b3 q-

,=1

x'*2~ 5-J'

(24)

7c(flllc~,fl2,/,~ ) oc fi~2-1e b2/~*0(a, fil,fi2 ) , where 0(c~, fil, f12) = e x p ( - c t log(1 + (x*/fl 1)) - E i L 1 [il log(fll -}- x*)). Observe that, we need to use the Metropolis-Hastings algorithm to generate the variable fll.

5. A superposition model in the presence of a covariate

Assume now that we have a covariate w which could be related to type of the input or to different programmers. In the presence of a covariate w, we assume the superposition model with intensity function J

A(t[~, •,w)

= ~ Oj#(t[[3•) , j=l

(25)

where fl = (fll,..-, flJ),Z = ( ~ 1 , - . , 7x) are parameters related to the incidence probabilities Oj with ~jJ-10j = 1. Logistic regression links could be used for the incidence probabilities Oj,j= l,...,J.

742

J. A. Achcar

For the special case

J = 2,

we have, (26)

B~(¢i) = OlißI (tl) ÷ 02iß2(li) , where e7--'cwi

OIi -- 1 + e ?+zwi 02 i z 1 - Oli . Observe that in this case, 7 = (7, r) It is interesting to o b s e ß e that Y a m a d a and Osaki (1984, 1985) considered the superposition of two G o e ~ O k u m o t o (1979) models, each with intensity function 2j(t) = ~ Ojfije-~Jt,j = 1,2, where { > 0, fij > 0, 0 < Oj < 1 and 01 + 02 = 1, but not including covariates. Different choices for ~.j(t),j = 1,2 in (26) could be considered. As a special case, consider a Musa and O k u m o t o (1984) with intensity function ) q ( t ) = o~/(fll ÷ t) and an exponential process with intensity function

B2(t) = fi2t. Assuming a failure truncated model, the likelihood function for ~, fll, fi2, 7 and is (from (12)) given by,

( ~-[((~

O~

0

fl2x;02i))

×exp(--Oln~log(l+~l)-O2n~2x~2-),

(27)

where 01n

eV+~wn 1 + e~+~w~ '

02n

1 - 01~ .

Assume the following prior distributions for c~,fil, fi2, 7, and,

~~r(a~,bl), B1 ~'~ß(a2,62), (28)

f12 ~'~ß(a3,b3),

7NN(dl,cl), ~NN(d2,c2)

,

where al, aa, a3, bi, b2, b3, cl, c2 are known and N(#, er2) denotes a normal distribution with mean # and variance «2. Also assume prior independence among the parameters. Also assuming the introduction of latent variables Ii = (Iil, Ii2), where Iij = 1 if the ith failure is caused by t h e j t h type and//j = 0 otherwise, j = 1,2, i = 1 , . . . , n,

Bayesian analysis for software reliability data

743

we obtain the joint posterior distribution for O~,fl,f2,7 given by

and r (see (21)),

(Ui~~t=I Oll)(Hi~Ii2=lX*02i) " f 1 q7 X*) Hi:Iil=l(

X ca

~,=1

Pl

P2

c

×exp(-(bl-kOInl°g(l+~l))C~

-- (b3 + ×exp(

(?-&)2 2«1

X;2\

(z-d2)2) 2-«7 -J '

(29)

where

O~i-

e?+Zwi 1 + e~ +~w, '

02i = 1 -- Oli .

The conditional distributions for the Gibbs algorithm are given by (i) Construct I given c~,fl, f2, 7, z and 9 by generating independent variables//1 from the Bernoulli distribution with parameter O~Oli Pil ~ ~Oli -~- f202ix*(fl + x*)

where/i2 = 1 -//1 (ii)

=(~lfl, f» 7, ~,z, 9)

F al +

~~1;bi+ 01nlog 1 +

.

i=1

(iii) ~(f2[ ~, fit, 7, ~,I, 9)

B

a3 -t- n --

(30)

/11; b3 qi=1

(iv) 7z(flIŒ, f2,7, z,I, 9 ) 0< fl~2-1e b2flll]/l(0~,fl,f2,7,

T) ,

744

J. A. Achcar

where

(

(~1) _ Z"i i ~

01 (c~,fil, 82, 7, r) = exp --O~Olnlog 1 + x•

log(81 -t- xf)

i=1

)

(v) (

(ù~ -- dl) 2

~(WIc~,S1,82, z,I,=@) oc exp

("c - d2)2"~ , , 2-c2 ") ~'/j2('~,81' 82' ')2»27)'

2Cl

where B2( ~, 81,82, 7, T) = exp

(~

Iil

log Oli

J[i2log 02i

-~-

\i=1

-01nc~ log 1 +

i=1 *

02

Xn _

Observe that we need to use the Metropolis-Hastings algorithm to generate the variables 81,7 and z. Similar results could be obtained considering more than one covariate w.

6. Bayesian inference and model determination

We can use the Gibbs samples to ger inferences on the parameters of the software reliability model or on functions of these parameters. In this case, we could approximate posterior moments of interest. As a special case, consider the mean value function ra(t). A Bayes estimator of ra(t) with respect to the squared error loss function is given by the posterior mean E(m(t)]~). Similar inferences could be obtained for 2(t). For model selection, we could consider (see Raftery, 1996) the marginal likelihood of the whole data set N for a model M, given by

v.<= J L (nMl-)-.<(8.<)%,

(31)

where 8M denotes the set of all unknown parameters in the model M, and ~M(fi_M) is the prior distribution. The Bayes factor criterion prefers model M1 to model M2 if VM2/VM1 < 1. A Monte Carlo estimate for the marginal likelihood VM is given by 1 T

(32) I t=l

\--

/

Bayesian analysis for software reliability data

745

where /~(~,t = 1 , 2 , . . , T could be generated using importance sampling. The simplest such estimator results from taking the prior as importance sampling function (see Raftery, 1996).

7. Some examples

7.1. Example 1 In Table 1, we have a software reliability data set introduced by Jelinski and Moranda (1972). The data consists of the number of days between the 26 failures that occurred during the production phase of a software (NTDS data - Naval Tactical Data System). From the data of Table 1, we have n = 26 and x~*= x ~ 6 = 250. Assuming the GO model (4), we consider (from (9)) the prior densities N~P(30),2GO ~ Y[0.2,20] and p ~ B[2.5,2.5]. From (11), the conditional distributions for the Gibbs-within-Metropolis algorithm are given v-~26 , by 2GOIN,p, Dn ~ F[26.2, 20 + p 2-,i=1 xi + 250(N - 2@)], ~(N[2GO,P, Dn) o( (e-3° 30N/NI)Ol (N,p, 2GO) where 01 (N,p, 2GO) = exp{ln A(N,p) - 250(N- 26p)2GO}, and ~(pl2co,N,Dù) o~p2S-l(1 - - p ) 2 " 5 - 1 ~ 2 ( N , p , 2GO), where ~2(N, p, 2GO) = exp{ln A(N,p) --P2GO ~ ~ 6 x] -- 250(N- 26p)2GO}. Generating 5 separate Gibbs-within-Metropolis chains each of which with 1000 iterations, we selected for each parameter, the 205th, 210th, 215th,...,995th, 1000th iterations, which for 5 chains yields a sample of size 1000. In Table 2, we have the obtained posterior summaries for the parameters )~GO,Pand N.

7.2. Example 2 The data of Table 2 were simulated from an NHPP, with intensity function

Table 1 NTDS data (ti = x,.* - xi*l)

i

ti

x*

i

ti

x*

i

ti

x*

1 2 3 4 5 6 7 8 9

9 12 11 4 7 2 5 8 5

9 21 32 36 43 45 50 58 63

10 11 12 13 14 15 16 17 18

7 1 6 1 9 4 1 3 3

70 71 77 78 87 91 92 95 98

19 20 21 22 23 24 25 26

6 1 11 33 7 91 2 1

104 105 116 149 156 247 249 250

J. A. Achcar

746

Table 2 Posterior summaries for the GO modeI

2~o N p

Mean

Median

S.D.

95%Credible interval

0.00634 28.272 0.62189

0.00582 28 0.64403

0.00257 5.091 0.17523

0.002980; 0.0126824 19; 39 0.241834; 0.905575

Table 3 Generated data from the superposition of a Musa Okumoto process and an exponential process with ¢~= 20, fll = 100, f12 = 0.0005, 7 = 1.0,~ = 2.0 Xi

Wi

Xi

Wi

Xi

Wi

Xi

Wi

1.6590 3.4440 4.3809 22.1900 25.2149 37.6644 39.9029 43.4150 43.5898 59.3859 61.9592 04.1077 70.5231 73.6860 82.1966

3 2 2 4 4 3 3 3 2 3 2 3 4 2 3

88.3958 89.2106 92.9258 85.6049 97.4439 105.7311 106.8248 122.2828 131.7938 135.8979 140.5997 145.1028 145.4650 151.4792 160.3269

2 2 3 4 3 3 3 2 2 3 2 2 4 3 2

165.0632 172.1504 175.2387 182.6241 184.8264 186.5892 190.4671 193.0431 195.1684 198.6102 224.8489 227.2276 230.9596 235.2451 247.9304

4 3 3 3 4 4 3 2 2 3 4 4 4 2 4

252.9478 253.5369 256.0304 272.1693 283.5702 288.3712 291.4028 293.3793 297.1773 298.3395 306.5081 314.0550 329.2328 345.7896 355.1561

4 2 2 4 4 4 3 2 4 4 3 2 4 4 2

considering Œ= 20, fi 1 = 100, fl 2 = 0.0005,7 = - 1 . 0 and r = 2;60 observations were generated considering the covariate w with values 2, 3 and 4, respectively (20 observations for each level o f the covariate w). For a Bayesian analysis o f the software reliability data o f Table 3 first we assume the superposition of a M u s a - O k u m o t o process and an exponential process without including the covariate wi with the prior distributions for c~, fiI and fi2 given in (22), where al = 400, bi = 20, a2 = 2500, b2 = 25, a3 - 0.25, and b3 = 500.

F r o m the conditional distributions for I, cz, fil and 02 given in (24), we generate 10 separate Gibbs chains each o f which ran for 1200 iterations. In order to decrease the effect of the starting distribution, we discarded the first 200 elements of each chain. We monitored the convergence o f the Gibbs samples using the G e l m a n and Rubin (1992) m e t h o d that uses the analysis o f variance technique to determine if further iterations are needed. For each parameter, we considered every 10th draw, that is, a sample of size 1000. In Table 4 we have the obtained posterior summaries o f the parameters ~, fil and fi2. W e also have in Table 4, the estimated potential scale reductions R (see

Bayesian analysis jor software reliability data

747

Table 4 Posterior summaries (superposition of Musa-Okumoto and an exponential process) Parameter

Mean

S.D.

95% Credible interval

Öl /32

20.285905 99.948741 0.000395

1.0103 1.9856 0.0001

(18.3325;22.1793) 1.0021 (96.0156;103.7820) 1.0002 (0.000198;0.000625) 0.9983

Table 5 Posterior summaries (superposition of Musa-Okumoto and an exponential process in the presence of a covariate w) Parameter B1

B2 7

Mean

S.D.

95% Credible interval

21.4154 98.7085 0.00031753 -0.9838 2.0780

1.0056 2.0312 0.000468 0.2984 0.2899

(19.5014;23.4389) (95.8152;103.6534) (0.00000023;0.0017) (-1.5523; -0.3891) (1.5553;2.6867)

1.0019 0.9995 1.0042 1.0002 1.0021

G e l m a n and Rubin, 1992) for all parameters. In this case, the considered n u m b e r o f iterations were sufficient for approximate convergence ( x / ~ < 1.1 for all parameters). N o w , let us consider the superposition o f the M u s a O k u m o t o process and an exponential process in the presence of a covariate w, assuming values 2,3 and 4, with intensity function (26). F o r a Bayesian analysis consider the prior distributions (28) with al = 400, bi = 20, a2 - 2500, b2 = 25, a3 = 0.25, b3 = 500, cl = 0.3, c2 = 0.5,dl = - 1 and d2 = 2 . F r o m the conditional distributions for I,c~,/31,/~2, 7 and r given in (30), we generated 10 separate Gibbs chains each o f which ran for 1200 iterations. We eliminated the first 200 elements o f each chain, and we selected every 10th element in each chain. Thus we obtained a sample o f size 1000. In Table 5, we have the obtained posterior summaries o f the parameters c~,/31,/~2, 7 and r. We also have in Table 5, the estimated potential scale reductions R. W e observe convergence for all parameters. Considering M1 for the model with a covariate w and M2 for the model not including the covariate, we find M o n t e Carlo estimates for VM (see (31)) considering the generated Gibbs samples. Since VM2/VM1 < 1, we conclude that the best fit for the software reliability data of Table 3 is given by the superposition of a M u s a - O k u m o t o process and an exponential process including a covariate w.

References Achcar, J. A., D. K. Dey and M. Niverthi (1998). A Bayesian approach using nonhomogeneous Poisson process for software reliability models. In Frontiers in Reliability, pp. 1-18 (Eds. A. P. Basu, S. K. Basu and S. Mukhopadhyay). World Scientific, Singapore.

748

J. A. Achcar

Campodonico, S. and N. D. Singpurwalla (1995). Inference and predictions from Poisson point processes incorporating expert Knowledge. J. Amer. Statist. Assoc. 90, 429, 220-226. Casella, G. and E. I. George (1992). Explaining the Gibbs sampler. Amer. Statistician 46, 167 174. Chhikara, R. S. and J. L. Folks (i989). The Inverse Gaussian Distribution. Marcel Dekker, New York. Chib, S. and E. Greenberg (1995). Understanding the Metropolis-Hastings algorithm, Amer. Statistician 49, 4, 327-335. Cox, D. R. and P. A. Lewis (1966). Statistical Analysis of Series ofEvents. Methuen, London Gaver, D. P. and M. Acar (1979). Analytical hazard representations for use in reliability, mortality and simulation studies. Comm. Stat., Simulation Comput. B8, 9i Ill. Gelfand, A. E. and A. F. M. Smith (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoe. 85, 398-409. Gelman, A. and D. B. Rubin (1992). Inference from iterative simulation using multiple sequences (with discussion). Statist. Sci. 7, 45~511. Goel, A. L. (1983). A guidebook for software reliability assessment. Technical Report RADCTR-83-176. Goel, A. L. and K. Okumoto (1978). An analysis of reccurrent software failures on a real-time control system. In Proceedings o f A C M Conference, pp. 496 500. Washington, DC, USA. Goel, A. L. and K. Okumoto (1979). Time-dependent error detection rate model for software reliability and other performance measures. IEEE Trans. Reliab. R-28, 206-211. Jelinski, Z. and P. B. Moranda (1972). Software reliability research. In Statistical Computer Performance Evaluation, pp. 465~497. (Ed. W. Freiberger). Academic Press, New York. Joe, H. and N. Reid (1985). Estimating the number of faults in a system. J. Amer. Statist. Assoc. 80, 222-226. Kuo, L. and T. Y. Yang (1996). Bayesian computation for nonhomogeneous Poisson processes in software reliability. J. Amer. Statist. Assoc. 91, 763-773. Kuo, L., J. Lee, K. Choi and T. Y. Yang (1996). Bayesian inference for S-shaped software reliability growth Models. Technical Report 96-05, Department of statistics, University of Connecticut, Storrs, USA. Lawless, J. F. (1982). Statistieal Models and Methodsfor Lifetime Data. Wiley, New York. Mazzuchi, T. A. and R. Soyer (1988). A Bayes empirical-Bayes model for software reliability. IEEE Trans. Reliab. R-37(2), 248358. Moranda, P. B. (1975). Prediction of software reliability and its applications. In Proceedings of the Annual Reliability and Maintainability Symposium, pp. 327-332, Washington, DC. Musa, J. D. and K. Okumoto (1984). A Logarithmic Poisson execution time model for software reliability measurement. In Proceedings of Seventh International Conference on Software Engineering, pp. 230-238. Orlando, FL. Ohba, M., S. Yamada, K. Takeda and S. Osaki (i982). S-shaped software reliability growth curve: How good is it? COMPSAC82, pp. 38-44. Raftery, A. E. (1996). Hypothesis testing and mode1 selection. In Markov Chain Monte Carlo in Practice, pp. 163-i87 (Eds. W. Gilks, S. Richardson and D. J. Spiegelhalter). Chapman Hall, London. Schick, G. J. and R. W. Wolverton (1978). Assessment of Software Reliability, Proc. Oper. Res., pp. 395~422. Physica-Verlag, Wirzberg-Wien. Singpurwalla, N. D. and S. P. WiIson (1994). Software reliability modeling. Internat. Statist. Rev. 62, 3, 289-317. Tanner, M. and W. Wong (1987). The calcnlation of posterior distributions by data augmentation. J. Amer. Statist. Assoe. 82, 528-550. Yamada, S. and S. Osaki (1984). Nonhomogeneous error deteetion rate models for software reliability growth. In Reliability Theory, pp. 120-143 (Eds. S. Osaki and Y. Hatoyama), Springer, Berlin. Yamada, S. and S. Osaki (1985). Software reliability modeling: models and applications. IEEE Trans. Software Eng. 12, 1431 1437. Yang, T. Y. (1996). Computational approaches to Bayesian inference for software reliability. Ph.D. Thesis, Department of Statistics, University of Connecticut, Storrs, USA.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 ElsevierScienceB.V. All rights reserved.

~ J

k./

Direct Graphical Estimation for the Parameters in a Three-Parameter Weibull Distribution

Peter R. Nelson and K. B. Kulasekera

There are very few graphical procedures that can be used to estimate the parameters of a three-parameter Weibull distribution. The Weibull probability paper designed by L. S. Nelson (1967) can be used for this purpose, but it requires iterating on the estimate of the threshold parameter until the resulting graph looks linear. W. Nelson (1972) discusses similar graphical estimation in terms of the hazard function. The only other graphical procedure is that of Cohen et al. (1984) (call this (CWD)); which uses modified moment estimators (MMEs). The CWD procedure only works with complete samples. The graphical procedure discussed in this paper uses three order statistics to estimate the three parameters, it works for both complete and right-censored samples, and does not require any iteration. We started by picking X(1) and the largest available order statistic (call it X(r3)), and chose the third order statistic (call it X(r2)) based on the asymptotic variance of the resulting ~ and its effect on the other estimates. It turns out that r2 = [_r3/2~ works fairly well. Sets of curves are provided based on using these three order statistics for estimating the three parameters. Comparison of this procedure with the Weibull probability plot and CWD indicates that it is competitive. It tends to overestimate 3, while the probability plot tends to underestimate 3, and CWD also tends to overestimate 3. Our procedure is worse in this regard for large ~ and large n, and CWD is worse for 6 = 0.5 and/or small n. Bias and variance for the estimates of the two other parameters differ rauch less, except for ~ = 0.5 and r3 _< 50 where the Weibull plot estimator of fi does a poor job in terms of both bias and variance. Our procedure seems better for ~5= 0.5 and/or small n, and the other two are better for large Õ and large n. Notation

f(x; ö, fi, 7)

the density function of a three-parameter Weibull distribution the shape parameter 749

750

fi 5

? X ~ Weib(eS, fi, 7)

x(r) Y

»(y) &Cv) Y(~) u(y) r(x) C(n, r, a) F1 »F2, F3

T

g(a) ~r ~3

r~

P. R. Nelson and K. B. Kulasekera

the scale parameter the threshold parameter an estimate of c~ an estimate of fi an estimate of 7 X has a Weibull distribution with parameters Ô, fi, and 7 the rth order statistic from a random sample of n X~ ~,, Weib(a, fi, 7) a Weib(& 1,0) random variable the density function of Y the survival function of Y the rth order statistic from a random sample of n B ~ Weib(5, 1,0) the density function of Y(r) the expectation of the random variable Y the gamma function elfe)I/r(1 + 1/5) Three integers such that ?'i < r2 < F3 (X(F2) -- X(rl))/(~~-(r3) -- X(y2) ) EG(/'/» F2, a) -- C(n, ?'1, a)]/[C(/'/, •3, a) -- C(n, r2, ô)]

the asymptotic variance of T the asymptotic variance of ä the value of r2 that minimizes ~ ä for fixed r3 and rl = 1

1. Introduction A great deal of research has been done on estimating the parameters for a Weibull distribution, and a very good summary of this work can be found in Johnson et al. (1994). We were interested in graphical estimation procedures, and there is relatively little material on that topic. All that we could find consisted of some early papers on Weibull probability plotting, the paper by Nelson (1972) on Weibull hazard plotting, and the more recent paper where Cohen et al. (1984) proposed modified moment estimators (MMEs) for the three parameters of a Weibull distribution and investigated their properties (mostly numerically). Their MMEs, however, require a complete sample. In the situation where a sample is right censored we have considered alternative estimators for the three Weibull parameters based on three order statistics. Our goal was to provide a graphical estimation procedure that would work for both complete and right-censored data, but that did not require the trial and error iteration needed when using Weibull probability paper. The procedure we developed seems to be competitive in most situations.

Direct graphical estimation for the parameters in a three-parameter Weibull distribution

751

2. Parameter estimation

The Weibull density function is B (X f ( x ; a , fi,•) = fi-ä

-- 7) 6-1 exp { - [ ( x - 7)/fi] 6} ,

(2.1)

where a > 0, fl > 0, and 7 < x < aa; and zero otherwise. The parameters a, fi, and 7 are the shape, scale, and threshold parameters, respectively. We will denote a r a n d o m variable X having the density (2.1) as X ~ Weib (a, fi, 7). Ler X(r) represent the rth order statistic from a sample of size n and consider E[X(r)I. If Y(~) is the rth order statistic from a Weib(a, 1,0) distribution, then

X(r) = fiY(~) + • .

(2.2)

Thus, we can da most of our work by starting with Y N Weib(a,l,0). Let fr(Y) represent the density of Y, St(y) = exp [ - x a] represent the corresponding survival function, and n~

K(n,r) = ( r - 1 ) ! ( n - r)! "

(2.3)

fr(, I (y) = K(n, r)[1

(2.4)

Then - & ( Y ) ] ~ 1 [ & ( y ) ] n-~ f ~ ( y )

and

E[Y(r)]=K(n,r)

~00°°

= K(n, r)

y [ 1 - Sy(y)] r l [ S y ( y ) ] n - r f y ( y ) d y

Õya[1 - exp (_ya)]~-I exp [-ya(n - r + 1)ldy .

(2.») Alternatively, one can re-write Eq. (2.4) as

fYl,.)(Y) = K ( n , r ) ~

~1(; / r

1 (_l)r

1 k[Sy(y)] n k-,fy(y)

k=0

Noting that [Sy(y)]

n Æ-1

fr(y) = =

Ôya-1

exp [-yä] exp [-ya(n - k - 1)]

Ôya-1 exp [-ya(n - k)]

(1)~~~~

.

752 where

P. R. Nelson and K. B. Kulasekera

ck = (n -- k) -1/~, it follows that fr(,)(y)=K(n,r) Z

r-1

k=0

f(y;O,«k, 1) .

(_l)~_l k

k

Using the fact that (see, e.g., Johnson et al., 1994, p. 632)

E(Y)

=r(1 +~)

one obtains E[Y(r)] = K ( n , r ) Z

=F(1

k=O 1"~

E(Y)

r 7 1 (_1) ~ 1-k

r , (r_l~(_l)r_l

+ ~)K(n,r)~

~/

k(

= r ( l+?1) C(n'r'Ô) ' where

~1()

C(n,r,Ô)=K(n,r)E

k=0

r-1

Ck "]

\ ù - ic

(2.6)

(1)r_l_k ( ck ~

k

-

F r o m Eqs. (2.5) and (2.6), it is clear that

(2.7)

\n- - ~ k )

C(n, r, 3) can also be expressed as

C(n, r, 5) - r K(n,r) ( 1 + 1) fO °° @a[1 - exp (_ya)]r-] exp [-ya(n - r + l)]dy and making the change of variable x = ya

K(n,r) C(n, r, 6) = ~(1 T ~ ) J 0f~xlla [1 - exp (-x)] r-1 exp[-x(n - r + 1)]dx . (2.8) Eq. (2.7) is better for computing when n is small and Eq. (2.8) is bettet for larger values of n. Also note that when r = 1, both Eqs. (2.7) and (2.8) reduce to

C(n, 1, (~) = n -1/a Eqs. (2.6) and (2.2) imply that

EIX(r)] - 7 = fiF(l + ~)C(n,r.,(~) •

Direct graphical estimation for the parameters in a three-parameter Weibull distribution

753

Using three different values of r (rl < r2 < r3) and setting the corresponding sample quantiles equAal to their expected values, one can solve simultaneously for estimators ~/,/~, and 6. The first step is to numerically solve the equation T -- X(r2) - X(r~) _ C(n, r2, 6) - C(n, rl, 6) = ~ ( 6 ) X(r3) -- X(r2) C(rt, r3, 6) - C(r/, r2, ~) to obtain ~ ~- ~

(2.9)

~(T). The estimator ~ is then obtained from

- X(r~) B = / ' ( 1 + 1/~)[;}2;,r2,~)--«(n,

(2.10)

rl,~)]

and finally, the estimator ~" is obtained f r o m 1

A

(2.11)

EXAMPLE 2.1 C o h e n et al. (1984). p r o p o s e d M M E s for the three-parameter Weibull distribution and used the following data on m a x i m u m flood levels (millions of cubic feet per second) for the Susquehanna River at Harrisburg, PA over 20 four-year periods from 1890 to 1969 (see Table 1). With the order statistics 1, 10, and 20 T -

0.402 - 0.265 - 0.4053 . 0.740 - 0.402

Since n is small, Eq. (2.7) is used to c o m p u t e C(n, r, 6). Using a search routine to find Y - ~ (0.4053), one obtains 6"= 1.499. F r o m Eqs. (2.10) and (2.11) one obtains ß =

0.402 - 0.265 = 0.216 (0.9028)(0.8373048 - 0.1355402)

and 7 = 0.265-- (0.216)F [ 1 +

1 ](0.1355402)

= 0.265 - (0.216)(0.9028)(0.1355402) = 0.239 .

Table 1 Maximum flood levels of the Susquehanna River at Harrisburg, PA 0.654 0.402 0.268 0.416

0.613 0.379 0.740 0.338

0.315 0.423 0.418 0.392

0.449 0.379 0.412 0.484

0.297 0.3235 0.494 0.256

754

P. R. Nelson and K. B. Kulasekera

Table 2 Comparison of estimators for Example 2.1 Estimator

Nelson and Kulasekera

Cohen et al. (1984)

3" B

1.500 0.216 0.239

1.479 0.202 0.241

For comparison the estimates of Cohen et al. (1984) corresponding to out values are given in Table 2.

3. Choice of rl, r2, and r3

One of the advantages of this procedure over, for example, the M M E s of Cohen et al. (1984) is that it works with any three order statistics, so censored data are not a problem. However, the question arises as to which order statistics provide the best results. It seems reasonable to choose rl = 1 and r3 equal to the largest order statistics available. It is less clear how to choose r2~.We decided to choose 1"2 such that it minimizes ~ ~ , the asymptotic variance of b. Let ~ ~ be the asymptotic variance of T. Then

= ¢~r

Fd

1~

L

J x=~7(5)

[~ ~-(x ~1 ~ x=Ô

1

= e T [5,(~)]2 - -

(3.1) •

3.1. The asymptotic variance of T The asymptotic variance U v can be obtained as follows. For notational convenience let X(ri) = ~ . In addition, let

B = T/+ 1 -- Ti,

02 = ~3 - ~2,

A

--

~2 - ~1 -

-

~3 - ~2

Direct graphical estimation for the parameters in a three-parameter Weibull distribution

755

Then Y1 q- v/nO1

T-A--

01

Y2 q- ~02 O2 02171 -- 011/2

(r2 + v~02)02 and 02 Y 1 -- O1Y2

~(r

- A) = O22+ ( 0 2 r ~ / ~ )

Since 02Y2 P÷0

it follows that x/~(T - A) and O2Y1 - O1Y2 _ 02(T2 - Tl) - 01(T3 - T2)

o~

0~

converge in distribution to the same r a n d o m variable. Also, (Tl,T2, T3) Z (N1,N2,N3) ~ N ( O , X ) ,

where

s = {«~A, p~(1 - p j )

for i < j ,

(3.2)

a~J - fr(«i)fr(~.j)

Pi = FY(«i) = 1 - exp(-{~) . Therefore, 02(T2 - Tl) - 01(T3 - r2) ~ 02(N2 - N 1 ) - 01(N3 - N 2 ) = N *

o2

0~

and 1 ~UT = Var(N*) = 0~'~~ 2i~~~ aiajaij ,

where a l = --02 a2 = 03 a3 = --01

and ~rij is given by Eq. (3.2).

(3.3)

P. R. Nelson and K. B. Kulasekera

756

3.2. The asymptotic variance of 6 Eq. (3.3) provides a means for computing ~//~r, and ~ ' ( 3 ) can be obtained in terms of ~ C ( n , r, 6) = C'(,,, r, 3)

.

The actual computation is made somewhat easier by noting that any factor of C(n, r, 3) that does not contain r does not effect Y(b). Therefore, we can ignore the factor F(1 + 1/6) in Eq. (2.8) and obtain ~ ' ( 3 ) in terms of

C,l(n,r, 3) - K(n,r)~-~~(r3 2 ~ ~ 1) (_l)r_l_ k ln(n - k)ck

(3.4)

k=0

(for small values of n) or

c;(n, r, 3)

-K(n, r) 3

ln(x)xl/a[1 - exp (-x)] "-~ exp[-x(n - r + 1)]dx (3.5)

(for larger values of n). Thus, ~U~ can be computed using Eqs. (3.1)-(3.3) and either Eq. (3.4) or (3.5).

3.3. Behavior of ~ ä We computed the asymptotic variance /f3 for n = 10,20,30,50,100; r3 = 0.5n, 0.67n, and 0.75n; 3 E [0.5, 4]; and values ofr2 between rl + 1 and r3 - 1. The value of r2 corresponding to the minimum variance (call it r~) is an increasing function of 3 and can be predicted quite accurately using the equation A~

r 2 = L-0.727 + 3.6313 + 0.072n + 0.086r33 - 0.81732~ ,

where LxJ is the greatest integer in x. This might be useful as part of a two-stage numerical procedure that first estimated b and then chose r2 based on that estimate, but for a graphical procedure such changes in r2 do not produce smooth estimation curves. However, just considering two ranges of 3's could result in a workable graphical procedure. For 3 > 1 the minimum variance occurred at r~ - r3/2. The value r~ was slightly less than r3/2 for values of 3 closer to 1 and increased monotonically to about r3/2 as 3 increased. The asymptotic variance ~ S changed very little for r2 values near r~. For ~ _< 1 it appeared that better results (in terms of smaller variance) would be obtained with smaller values of r2. However, for larger values of r3, values of r2 close to rl = 1 result in ~ ( 3 ) - 0, which is of no use in estimating 3.

Direct graphical estimation for the parameters in a three-parameter Weibull distribution

757

As a compromise, we tried using r2 = Lr3/4J when 6 _< 1. Unfortunately, while this procedure resulted in reasonable estimates for 6, using r2 = ~r3/4J sometimes produced terrible estimates for/?. This led us to consider using only r2 = Lr3/2J. As a check on how well the asymptotic variance predicted behavior for small n, on how the bias in the estimation of 6 behaved, and on how robust the choice of r2 = ~3/2J was to different 6's; we simulated the expected value and variance for 6. We generated Weib(g, 1,0) samples of size n = 20,30, and 50 for values of 6 representing both increasing and decreasing hazard rates. R a n d o m numbers were generated using the S U N performance library routines, simulations were done in F O R T R A N , and the samples were simulated 500 times for each combination of 6 and n. The value of rl was fixed at 1; different amounts of censoring corresponding to r3 =0.5n, O.67n, O.75n,n were used; and both r2 = lr3/2J a n d r 2 = ~r3/4J were studied. For each sample the statistic T was calculated and 6 was obtained using Eq. (2.9) for all the r3,r2 combinations. The sample means and the sample variances of these estimates of 6 are given in Tables A1-A7. These tables suggested that using just r2 = ~r3/2j would be reasonable.

4. A graphical solution While it would be possible to create a computer program to perform the estimation procedure described above, out interest was in providing a procedure that required only a few graphs. The first sets of graphs (Figures 1-4) are graphs of B ( 6 ) and are used to obtain 6. The four figures correspond to values of r3 = 0.5n, 0.67n, 0.75n, n. Estimates of/~ are obtained using Figures 5 8. Again, the figures correspond to values of r3 = 0.5n, 0.67n, 0.75n, n. Given n, r3, and 6 (r2 is a function of r3); the figures provide the denominator in Eq. (2.10). Finally, Figure 9 provides the coefficient of ~ i n Eq. (2.11), from which Nis obtained. EXAMPLE 4.1. Reanalyzing the data in Example 2.1 graphically, one would proceed as follows. In this example rl = 1 and r3 = n = 20, and therefore, one would use r2 = L20/2J = 10. As in E xample 2.1, ones computes T = 0.4053 and from Figure 4 the estimate of 6 is 6 = 1.5. F r o m Figure 8 (with 6"= 1.5) one obtains the denominator for Eq. (2.10) as 0.63 and ».

B-

O.4O2 O.265 - 0.217 . 0.63

Finally, from Figure 9 one obtains the coefficient for ~" in Eq. (2.11) as 0.12 and ~ ' = 0.265 - 0.217(0.12) = 0.239 .

758

P. R. Nelson and K. B. Kulasekera

2.5

2.0

1.5 LI-

I

~~~I~!t I111t 1

ù,,

g:,.--.

/Q;

1.0 -

I'. ,'."" 1.O

0.5

i i

,/ 0.0

i

I

2

1

0

3

4

5

6

Fig. 1. Graph of ~ ( 6 ) for P3 = 0.5n and c~E [0.5, 7].

~I~I~!!/~

2.5

2.0

/ ù ,, :.";.

1.5

.

.

.

.

.

.

, ' la.

)~I" ....... ù ."

1.0

B.,"l"

!~ ~ I

I

"-" "---'-

0:5

/.7' 0.0

i

i

I 0

1

2

3

Fig. 2. Graph of ~ ( 6 ) for r 3

4

5

0.67n and 6 c I0.5, 7].

6

7

Direct graphical estimation for the parameters in a three-parameter Weibull distribution

2.0

- -

nlO n20 n30 n50 n lO0 I

1.5

/ "::;ù--'12.... ......

L tl.

1.0

! ù: 0.5

0.0

0

I 1

t 2

I 3

I 4

5

6

Fig. 3. G r a p h of ~-(&) for r3 = 0.75n and & E [0.5, 7].

1.2

0.8

0.4

0.0

0

1

2

3

4

5

6

Fig. 4. G r a p h of ~(&) for r3 = n and & C [0.5, 7].

6

7

759

760

P. R. Nelson and K. B. Kulasekera

nlO n20

/,

n30 n50

0.4

/ P

/ .,/'" ..........

nlO0!

Ot

v

o

¥

s' , , , . . . . . . . . . . . . . '" . . . . .

ù,i/:/,' J'*s"

0.3

,f «

0.2-

:/

0.1 -

'

i.ùæ..

'',.

f

f

0.0 -

2

3

4

5

6

7

G

Fig. 5. G r a p h of the denominator in Eq. (2.10) for estimating beta when r3 = 0.5n and ~ E [0.5, 7].

0.5 ------

nlO n20 n30 n50 nlO0 ~°.**,

¢- 0.4 0

,

,,

J

I

',,,, 0,3

--

ù",,,,,,, ",,,,,,

¥

f

v

0.2 -

/

0.1 -

2

:3

4

5

6

G

Fig. 6. Graph of the denominator in Eq. (2.10) for estimating beta when r3 = 0.67n and ~ c [0.5, 7].

Direct graphical estimation for the parameters in a three-parameter Weibull distribution

o6!

I

761

I nlO n20

- - ~ ........

J

n30 n50 , nlO0

.,

/.'i

L*"

"*%

0.5

.~.

,~.~..

,rt-

*%% .%

O I

ù,

~'~*.% ~.%. '%'**°

~ ~~.~.

'",

~ 0.4-

+ ~" o.3 ", I_: 0.2

2

3

4

5

6

7

5

Fig. 7. G r a p h of the denominator in Eq. (2.10) for estimating beta when r3 = 0.75n and 6 c I0.5, 7].

I

I

/

"%%

0.7 -

'% '%~. ",%. ù , ' '%~ *%*% ,% '%%% %*%. ~'~. ,~

~~~,'........ .,% ~-

0.6-

(J

,-~ c

ù,

\

0.5

o

~v

'%~ *'**% ,% =%**

,,

%'~

\

0.4 nlO n20 n30 n50 nlO0

•~ -

0.3 --~

0.2

I 0

1

2

3

4

5

6

7

B

Fig. 8. G r a p h of the denominator in Eq. (2.10) for estimating beta when r 3 = n and 6 E [0.5, 7].

P. R. Nelson and K. B. Kulasekera

762

J 0.6 -

- -

~

I ......... I .....

I.....

I

nlO

"2°1 n301

/

nS°l

ù,

/

'-- 0.4 /

v

¥ 0.2

""'/ i 0.0

2

3

4

5

6

7

Fig. 9. G r a p h of the coefficient for # in Eq. (2.11) for estimating ~/when 6 c [0.5, 7].

5. Comparison with other graphical procedures

5.1. Censored data The only other graphical procedure that works with right-censored data is a Weibull probability plot, so we compared our procedure (referred to as N & K in Tables 3-6) with the Weibull probability paper developed by Nelson (1967) (referred to as LSN in Tables 3 6). For the comparison we generated values from a Weib(6, 1, 0) distribution, and used 1000 trials for each 6, n, r3 combination. The results are summarized in Tables 3-5. Obviously, it was not feasible to plot all the generated values by hand to obtain estimates using Weibull probability paper, so we wrote a program to do that estimation for us. Recall that when plotting data on Weibull probability paper, one taust first shift the data so that y - 0 by subtracting an estimate of y from each data value. This is a trial and error procedure, and one would generally start by plotting the unshifted data to see if there was any indication (i.e., curvature) that y was not zero.

Out first attempt at this computerized estimation transformed the (x,y) coordinates of each point so that the transformed points would rend to fall on a straight line if the data were Weib(6, /3, 0) (this mimics what the scales for the x and y axes on the probability paper do), fitted a line to the transformed points using

Direct graphical estimation for the parameters in a three-parameter Weibull distribution

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764

P. R. Nelson and K. B. Kulasekera

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Direct graphical estimation for the parameters in a three-parameter Weibull distribution

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P. R. Nelson and K. B. Kulasekera

766

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Direct graphical estimation for the parameters in a three-parameter Weibull distribution

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P. R, Nelson and K. B. Kulasekera

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Direct graphical estimation,for the parameters in a three-parameter Weibull distribution

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P. R. Nelson and K. B. Kulasekera

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Direct graphical estimation for the parameters in a three-parameter Weibull distribution

771

least squares, and computed the R 2 value associated with the line. The appropriate amount ~" to subtract was obtained iteratively by repeating the procedure, each time subtracting a slightly larger value for 7 and comparing the resulting R 2 values. If at the first step R 2 decreased, we assumed ~"= 0 was appropriate. Otherwise, the amount subtracted was increased until R 2 reached a m a x i m u m value. Using this computerized estimation led to some very bad estimates of 6, and we discovered that in m a n y instances (particularly when only a few points were being plotted) the values X(1) and/or X(2) seemed to unduly influence the slope of the fitted line. If one were actually fitting a line by hand, they would tend to compensate for this by allowing these first two points to fall further from the line. As a counterpart to that we had the program check to see if deleting X(1) and refitting the line resulted in an estimate of X(1) that was too far away from the actual value (more than 25~). I f it was, the point X(1) was deleted. IfX(1) by itself did not have undo influence, the program checked the pair )((1) and X(2) by deleting them both, fitting a least squares line to the remaining points, and checking if either value was more than 25ä from its predicted value. I f either point was too rar away from the fitted line, they were both deleted. Before running the simulation we checked this procedure for deleting values by comparing it with simply using the least squares line obtained without deleting any small observations. We found that deleting values lead to estimates that were uniformly less biased and had smaller variances. Even using the deletion procedure there were still a few cases where the Weibull plot produced very large estimates for 6. We decided that when the estimate of 6 exceeded 8, to just use 8 as the estimate. The rational for this was that one could extrapolate from the scale on the Weibull paper to some degree, but at some point one would simply have to decide that 6 was "very large" (i.e., 8). The number of times this happened was recorded. Thus, there were three different situations where something in Weibull plot estimation procedure had to be modified. They are referred to as faults in Tables 3-6. The occurrences of these faults are recorded as ( f l , f 2 , f 3 ) , and Table 7 indicates which type of fault each count corresponds to. Table 7 Explanation of the fault counts in Tables 3-6 Fault count

Situation

f~ f2 f3

deleted X(1) and )((2) were deleted

ä-s.o X'(1 ) was

Examination ofTables 3 5 suggests the following. Out äoverestimates a and has a larger variance than the Weibull plot, which underestimates õ. In m a n y situations it would be preferable to overestimate a, thus inferring a larger hazard rate. The two estimators of fi are rauch closer. For a close to 1 our estimator appears better in terms of both bias and variance. This is particularly true for 6 = 0.5 and r3 _< 50, where the Weibull plot does a poor job in terms of both bias and variance. The smaller r3 is, the worse the results are. However, as ô increases from 1,

772

P. R. Nelson and K. B. Kulasekera

the situation reverses and the Weibull plot estimates have both smaller bias and variance. The Weibull plot estimate of 7 appears to have both smaller bias and smaller variance, p r o b a b l y due to it not depending on the estimates ~'and fi as ours does. Our estimate, however, does not appear unreasonable. Also, our estimators are easier to obtain since there is no trial and error plotting involved. 5.2.

U n c e n s o r e d data

Table 6 compares our procedure with both the Weibull probability plot and the procedure o f C o h e n et al. (1984) (referred to as C W D ) . The comparison of our procedure and the Weibull plot is similar to what was f o u n d in the censored comparison. Our procedure and that o f C W D behave similarly. In some circumstances one has smaller bias and variance, and for other circumstances it is reversed. Both procedures tend to overestimate c5. Our procedure is worse in this regard for large Ô and large n, while C W D is worse for ~ - 0.5 a n d / o r small n.

6. Conclusion We have presented estimates for the three parameters o f a Weibull distribution that are based on three order statistics and can be obtained graphically with relative ease. They work with either complete samples or with right-censored samples. F o r rightcensored data our procedure compares favorably with the Weibull probability plot and is simpler to implement. It does, however, tend to overestimate the shape parameter 6. F o r complete data it is competitive with the procedure o f C o h e n et al. (1984); but works best with small samples. F o r larger samples, the estimates o f C o h e n et al. (1984) have both less bias and smaller variance.

Appendix Table A1 Estimates and their variances when ~ t'/

t"3

0.5

?'2 = [r3/2j

r2 = it3/4j

x

,2(x)

x

s2(x)

20

20 15 13 10

0.5418 0.5408 0.5554 0.6313

0.1647 0.2219 0.2837 0.5392

0.5125 0.4933 0.5069 0.5093

0.I255 0.1962 0.2481 0.4411

30

30 23 20 15

0.5183 0.4781 0.5249 0.5533

0.1513 0.2044 0.1969 0.2930

0.5425 0.4450 0.4967 0.4979

0.1230 0.1628 0.1755 0.I984

50

50 38 33 25

0.5281 0.4787 0.5104 0.4984

0.1410 0.1584 0.1712 0.1422

0.5021 0.4684 0.4818 0.4855

0.1029 0.1140 0.1276 0.1292

Direct graphical estimation for the parameters in a three-parameter Weibull distribution Table A2 Estimates and their variances when c5 = 0.75 n

r3

r2 -

Lr3/2j

r2-

Lr3/4j

;-

,2l~

a-

,2(x)

20

20 15 13 10

0.8378 0.8525 0.7047 0.9672

0.3207 0.4664 0.4787 0.8213

0.7864 0.7608 0.7948 0.8134

0.2725 0.3297 0.4452 0.7201

30

30 23 20 15

0.5024 0.7717 0.8166 0.8429

0.3773 0.3032 0.3126 0.5271

0.6899 0.7370 0.7661 0.7540

0.3898 0.2297 0.2625 0.3541

50

50 38 33 25

0.7732 0.7525 0.7737 0.7596

0.1712 0.2105 0.2006 0.2389

0.7736 0.7608 0.7402 0.7451

0.1309 0.3297 0.1619 0.2100

Table A3 Estimates and their variances when 6 = 0.8 n

rs

r2 - Lrs/2J

r2 - Lrs/4J

20

20 15 13 10

0.8971 0.9174 0.9858 0.9994

0.3554 0.5194 0.7275 0.8695

0.8415 0.8197 0.8593 0.8842

0.3005 0.3735 0.4996 0.7919

30

30 23 20 15

0.9639 0.8280 0.8758 0.9057

0.4101 0.3290 0.3473 0.5712

0.7012 0.7939 0.8198 0.8118

0.3698 0.2458 0.2926 0.4049

50

50 38 33 25

0.8520 0.8161 0.8267 0.8120

0.1379 0.2125 0.2167 0.2488

0.8373 0.7468 0.7902 0.7960

0.1309 0.2036 0.1730 0.2391

773

774

P. R. Nelson and K. B. Kulasekera

Table A4 Estimates and their variances when 6 = 0.9 n

r3

r2

=

Lr3/2j

r2 = [ r 3 / 4 ]

s

,2(~-~

x

,2(x)

20

20 15 13 10

0.9918 1.0521 1.0121 1.1019

0.4348 0.6224 0.8054 0.9597

0.9514 0.9393 0.9941 1.0012

0.3441 0.4635 0.6269 0.8898

30

30 23 20 15

0.8636 0.9304 0.9981 1.0036

0.4580 0.4022 0.4271 0.6594

0.8723 0.9006 0.9296 0.9295

0.3077 0.2898 0.3646 0.5101

50

50 38 33 25

0.9434 0.9158 0.9307 0.9163

0.1723 0.2499 0.2556 0.2959

0.8447 0.8297 0.8872 0.8972

0.2618 0.2375 0.1992 0.2891

Table A5 Estimates and their variances when ~ = 1.1 ~

r 2 - [r3/4j

~ = L~/2j

3"

~2(g'/

3"

s2(;'t

20

20 15 13 10

1.2618 1.3133 1.3915 1.4851

0.5731 0.7737 0.9474 1.1372

1.1761 1.1901 1.2661 1.3031

0.4537 0.6696 0.8554 1.1351

30

30 23 20 15

1.1009 1.1671 1.2553 1.2886

0.4914 0.5542 0.6265 0.8183

1.0663 1.1177 1.1586 1.1809

0.3493 0.4166 0.5402 0.7168

50

50 38 33 25

1.1561 1.1351 1.1438 1.1295

0.2236 0.3070 0.3607 0.4266

1.0996 0.9989 1.0774 1.1024

0.1592 0.3202 0.2791 0.4340

Direct graphieal estimation for the parameters in a three-parameter Weibull distribution

775

Table A6 Estimates and their variances when 6 = 1.5 r2 = Lr3/4]

n

r3

ra -

Lr3/2]

20

20 15 13 10

1.7482 1.8234 1.8906 1.9704

0.8091 1.0303 1.1635 1.3296

1.6366 1.6834 1.7402 1.7531

0.6980 1.0055 1.1102 1.4122

30

30 23 20 15

1.5927 1.6735 1.7739 1.7865

0.5806 0.8187 0.9208 1.0732

1.5448 1.5754 1.6303 1.6985

0.4602 0.6897 0.7958 1.0805

50

50 38 33 25

1.5820 1.5811 1.5936 1.5852

0.3639 0.5267 0.5857 0.7104

1.4988 1.4397 1.4950 1.5468

0.2758 0.4815 0.5148 0.7043

Table A7 Estimates and their variances when ~ = 2 n

r3

r2 =

Lr3/2J

r2 - - [ r 3 / 4 J

x

s2(x~

x

s2(x)

20

20 15 13 10

2.3097 2.3531 2.3977 2.4159

1.0030 1.2075 1.3094 1.4364

2.1852 2.1880 1.2291 2.1040

0.9443 1.2203 1.2975 1.5402

30

30 23 20 15

2.0818 2.2684 2.3514 2.3125

0.8422 1.0662 1.1384 1.2427

2.1083 2.1524 2.2181 2.2318

0.6967 0.9755 1.0944 1.3572

50

50 38 33 25

2.1412 2.1583 2.1741 2.1411

0.5720 0.7913 0.8607 0.9733

2.0263 1.9898 2.0483 2.1150

0.4654 0.7325 0.8021 0.9989

References Cohen, A. C., B. J. Whitten and Y. Ding (1984). M odified m o m e n t estimation for the three-parameter Weibull distribution. J. Qual. Technol. 16, 159-167. Johnson, N. L., S. Kotz and N. Balakrishnan (1994). Continuous Univariate Distributions-1. Wiley, New York. Nelson, L. S. (1967). Weibull probability paper. Ind. Qual. Contro123 (11), 452-453. Nelson, W. (1972). Theory and applications of Hazard plotting for censored failure data. Technometrics 14, 945 966.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 EIsevier Science B.V. All rights reserved.

"~ ]

,)

Bayesian and Frequentist Methods in Change-Point Problems

Nader Ebrahimi and Sujit K. Ghosh

The change-point problem is one of the important problems of statistical inference in which one tries to detect abrupt change in a given sequence of random variables. This problem, which originally started with statistical control theory (see Page, 1955), has now been applied to different fields, including but not restricted to survival analysis and reliability studies. The literature about change-point problem, by now, is quite extensive. In this paper, our goal is to review recent developments in this area. In particular, statistical procedures to estimate discrete change point as well as continuous change point are reviewed.

1. Introduction

There are two types of change-point problems: (i) continuous change-point problem and (ii) discrete change-point problem. In the continuous change-point problem we assume that a continuous random variable T representing a survival time or a failure time has the hazard function

h(t) = ~ h°(t) i f O < t < t o ,

( hl(t)

ift>to

(1.1)

,

where ho(to) ¢ hl(t) as t --+ to. In other words h(.) has a discontinuity at to. The problem is to estimate the change point to. In the discrete change-point problem we assume that X 1 , . . , X n are observed as a sequence of independent random variables with an abrupt change at g0 E { 1 , .., n}. In other words, there exist two probability distribution functions F0 and FI such that e0 i~I Fl(xi) . P(X1 <_xl,... ,Xn <_xn) = HFo(xi) i=i

i=go+l

The problem is to estimate g0777

(1.2)

778

N. Ebrahimi and S. K. Ghosh

A rast amount of published literature on this area clearly signifies the importance of this topic. As a result, it is rather impossible to review this topic that would include all possible results and landmarks on change-point problems. However we have tried to bring in the recent developments in this area. The change-point problem, originally started with the question: is the sequence of observations in a sample obtained from the same identical distribution? If not, is it possible to detect several segments of the observed data that came from identical distributions. Parametric and non-parametric methods have been developed for discrete change-point problem. Also, parametric methods have been developed for continuous change-points problem. However, non-parametric methods are yet fully developed and the field is quite open. Section 2 of this paper presents various Frequentist and Bayesian approaches to estimate a continuous change point. Section 3, describes methods developed during recent years to estimate a discrete change point.

2. Continuous ehange-point problem In reliability theory, a widely accepted procedure is to apply "burn in" techniques to screen out defective items and improve the lifetimes of surviving items. More specifically, suppose the lifetimes of items are independent but that early failures appear to occur at one rate and late failures (after some threshold) appear to occur at another rate. Because survivors actually have greater resistance to fatal stresses than do newer ones, if the threshold parameter were known, the screening process could entail testing up to this threshold and selling only survivors. However, in practice the threshold parameter will not be known. One helpful tool is to model the aging process by model (1.1). Hefe the change point plays the role of the threshold parameter. Model (1.1) is also applicable in analyzing biomedical data. For example, suppose patients in the clinical trial receive a treatment at time 0. The event times may represent the time until undesirable side-effects occur, in which case we would have an initial hazard rate say ho(t) and expect a lower hazard rate say hl (t) after the treatment has been in place for some time. The problem gets quite involved if the patients get several treatments over time. In that case, there may exist more than one change point. Another application is in medical follow-up studies after a major operation, e.g. bone marrow transplantation. There is usually high initial risk and then risks settle down to a lower constant long-term risk. Both frequentist and Bayesian solutions have been proposed to detect a single change point for a parametric family. 2.1. Frequentist approach

The literature to date focuses primarily on examination of model (1.1) for specific forms of ho(x) and hl (x). Nguyen et al. (1984) consider the estimation of change

Bayesian and Frequentist methods in change-point problems

779

point to when ho(x) = fl0, hl(x) = 21,21 > fl0. Under their assumptions the density has the form

f(t) = (fl0 exp(-fl0t))/(0 _< t _< to) + (21 exp(-fl0t0 - ,~l(t - to)))I(t > to) ,

(2.1)

where

I(A)= {1

ifxcA,

0

otherwise .

Using the n-ordered observations t(1) < t(2) _< ..- _< t(n), they construct a kernel X~(t) such that the solution of Xn(t) provides a consistent estimate of to. The construction of X~(t)is as follows. Define

BI(T,t) = T,

B2(T,t) = I(T > t),

B4(T,t) --- T2I(T > t),

B3(T,t) = TI(T > t),

B = (B1,B2,B3,B4),

Il

Bj(t)=lZBj(t(i),t), n

j=1,2,3,4

,

i=1

and _

±

+ (1 -

. . . . /~3 IJ2[t)) ~ (t) +/~l(t) log/~2(t) •

The kernel Xn (t) is then x . ( o ) = o,

Xn(t(i)) = H@(t(i)),

i=l,...,n-1,

xn(t)

t _> t(n -

= x,,(t(n

- 1)),

1)

.

The construction of Xn(t) is ingenious but apart from providing a consistent estimate this estimator does not seem to have any attractive properties. Later, Yao (1986) and Nguyen and Pham (1987, 1989) propose an alternative method for estimating to. It is clear that for each n, the likelihood function is Ln(20, )«, to) =

!~ i=1

log f(t(i)) =

Jo~

(log y(t))dFù(t)

,

(2.2)

N. Ebrahimi and S. K. Ghosh

780

where Fn is the empirical distribution function and f(t) is given by Eq. (2.1). Now the procedure to get estimator of to will be in the following three steps: (i) For each fixed to < t(n), maximize Ln with respect to 20 and 21; (ii) Insert the values of 2o and )~1 from the step (i) into Ln. Denote the resulting function by L*~(to) and maximize it by varying to; (iii) Insert the value of to from step (ii) and maximize the function Ln to get the new values of 2o and 21. Repeat step (ii) with the new values of 2o and 21 to get new value of to. Return to step (i) and continue the remaining steps until the algorithm converges. It should be noted that as to tends to t(n) from below, Ln tends to infinity. Thus, we are led to restrict L* (to) to some random intervals JAn,Bnl depending on data alone, with 0 _< An < Bn < t(n). Examples of An and Bn are given by Nguyen and Pham (1989). Note also that the function L~(to) is not defined at data points t ( 1 ) , . . , t(n), only its limits at to tend to these data points, from below or above exist. Thus, even by restricting to to [A~,B~], we may not be able to achieve a maximum in step (ii). In order to solve this problem one can maximize L*~*(to) =L;(t0) if t(r) < to < t(r+ 1),r = 0, 1 , . . , n - 1 and = max{L~(tö) , L*(tö-)} if to = t(r),r = 1 , . . , n 1, where t(0) = 0 and L;(to) and L~(t +) are the left- and right-hand limits at to, respectively. Matthews et al. (1985) also considered the model ho(t) = 20 and hl (t) = )q, but they were interested in testing the hypothesis H0 : 21 = 2o against Ha : 2o > 2~. See also Loader (1991). Basu et al. (1988) extended the model proposed by Nguyen et al. (1984) to allow general ho(t) keeping h2(t)=)~1. They suggested two semi-parametric estimates for to based on the assumption that the proportion of population that dies or fails at time to or before is known. Let pl be such that po < pl < l, where po is the known number of items in the population that failed at time to or before. Also let k be the number of order statistics between T([ùpo]) and T([npl]), and

B, = 2(t(i) log F~(t(i) ) /k + 1) - (~~))(~-~~ log l~~(t(i) ) /k + 1)) t2(i) __ (~-~ t(i) ~2 k+l \z-~ k+lJ

and the summations range over i = [npol + 1 to i = estimators by Basu et al. (1988) of to are t0(1) = Inf{t

[np~]. Then the proposed

: y~(t + h~) - yn(t) <_ hnB1 +en}

and to(2) = Inr{t : log

F~(t) - log(1 - P o ) _< 2o(~.po - t) + e~} ,

where

hn ~-(Æ) -1/4,

Th(t) =--logffn(t)~

is p0th sample quantile and e~ -- c(log

n)n -1/2.

Irin(t) =

I(ti > t ) rt '~Po

Bayesian and Frequentist methods in change-point problems

781

Basu et al. (1988) proved consistency of their estimators. Later Ghosh and Joshi (1992) investigated asymptotic distribution of t0(2). Modeling the aging process by the change-point mean residual life function was initiated by Ebrahimi (1991). He considered the model

m(t) = mo(t)I(O < t < to) + mlI(t > to) , where ra(t) is the mean residual life function or the remaining life expectancy function at age t and is defined as { ' f ö # ( x ) d x if~e(t) > 0, m(t) = E ( X - t]X> «) = P~(t) 0 if Æ(t) = 0 . He proposed an estimator for to and also studied its asymptotic properties.

2.2. Bayesian methods None of the classical methods described in Section 2.1 is quite satisfactory and puts stringent restrictions in order to obtain asymptotic normality. Moreover, simulation studies show that asymptotics are poor for small-to-moderate sample sizes. Bayesian approach on the other hand avoids asymptotics and provides more reliable inference conditional only upon the data actually observed. Howerer, Bayesian methods are also susceptible to impropriety of posterior distribution of to if one is not careful in specifying the prior for to. In fact, Ebrahimi et al. (1999) showed that an improper prior on to, necessarily leads to an improper posterior for to. Achkar and Bolfarine (1989) consider the model ho(t) = 2o and hl(t) = 21 and avoid this problem by using a discrete uniform prior for to. However, such a choice could tremendously limit the scope of application. A nice review on problems that arises within Bayesian framework is presented in Ghosh et al. (1993) which concentrates on the case 2o _> ,~1. In fact Ghosh et al. (1993) shows that one needs the restriction 2o >_ -~-1 for some known 2o in order to make the posterior proper. Ebrahimi et al. (1999) gave general Bayesian formulation of the change-point problem and they discussed the case of h ( t ) = 2o > h l ( t ) = 21 which yields particularly simple fitting and the Weibull case. Returning to (1.1), Ebrahimi et al. (1999) adopt fully parametric modeling assuming ho(t)= ho(t;Oo) and hl (t) = hl (t; 01). Thus, h0 and hl are two, possibly distinct, parametric families of hazard functions indexed by 00 and 01, where 00 and 01 could be vector valued. To capture the order restriction on h0 and hl they ler S = {(00, 01) : h0(t; 00) _> hl(t;O1) for all t > 0}. The likelihood takes the form n

L(Oo, 01, to; t) = I-I h(tj) exp(-H(tj))

,

(2.3)

j=l

where t = ( q , . . . , th) denotes the observed values of the lifetimes. Let vj(to) = 1 if tj < to, = 0 if tj > to. Then (2.3) becomes

N. Ebrahimi and S. K. Ghosh

782

L( Oo, 01» to; t) = I~I ho( tj; Oo)V/t°) hl ( tj; 01) 1-vj(t°) j--1

× exp

- H(H0(min(tj, to); 00) + [Hl(t£ 01) - Hl(t0; 01)]+) I(S) . j=l " (2.4)

For their model they restrict the likelihood so that to _> t0). Certainty of a change-point during the period of observation would then add the further restriction to < t(n). In order to complete probability specifications, Ebrahimi et al. (1999) require a prior distribution for 00, 01 and to. They assume that it takes the general form

f(Oo, 01)-f(S). f(to)

(2.5)

and that it is proper which assures that the posterior f(Oo, 01,t0]t) is proper. The prior information on to places it on the interval (0, b) with b possibly oc. The actual support for to is truncated according to the restrictions imposed by the likelihood. When to is not bounded above they argue that iff(t0) is improper the posterior must necessarily be improper. Combining (2.4) and (2.5) provides the complete Bayesian specification and thus the posterior f(Oo, 01, tolt) which is proportional to

L(Oo, 01, to; t). f(Oo, O1)f(S) . f(to) •

(2.6)

The posteriors f(00[t) and f(01lt) enable us to learn about the pre- and postthreshold hazards. In fact for each t, since h0(t; 00), H0(t; 00), h l (t; 01),//1 (t; 01), h(t; 00, 01, to) and H(t; 00, 01, to) are all random variables, they all have posterior distributions which would be of interest as well. However, primary interest is in the posterior for to,f(tolt) and when a change is not certain, P(to > t(n)It). The expression in (2.6) is not analytically tractable so they turn to simulationbased approaches for fitting such a model and use Markov chain Monte Carlo techniques. For more details see Ebrahimi et al. (1999). Even if the proposed models are all parametric, they can be quite rich if one incorporates finite mixture distributions. The semi-parametric approach is proposed in Ebrahimi and Ghosh (1999) where the problem is formulated in terms of dynamic weight mixture model. Also, the methodologies described by Ebrahimi et al. (1999) could be extended to detect more than one change point. However, one must be careful in specifying the joint prior distribution of change points.

3. Discrete change-point problem Discrete change-points problem occurs as a result of non-homogeneity in a sample. In some cases we have a priori knowledge about the physical nature of the process that generates data and we propose some parametric family to characterize such knowledge. However, one must construct a statistical test based

Bayesian and Frequentist methods in change-point problems

783

on initial data to check the validity of the model. In this situation a vicious circle arises: in order to create such a model one must guarantee the statistical homogeneity of data, but it is just the same model which is used to check the statistical homogeneity hypothesis of the data obtained. Thus non-parametric methods are the only satisfactory way to tackle this kind of problem. Between opposing parametric and non-parametric methods there is a large area of semi-parametric methods of change-point detection. The broad applicability of discrete change-point problem in various areas makes this area an attractive field of research. In this section we give two applications. For more applications see Zacks (1983) and Broemeling (1972). Suppose we are monitoring the rate of occurrence of a rare health event, for example a specific congenital malformation. Since the number of malformed births is small, one can assume that the malformed births occur according to a realization of a Poisson process with parameter say 21. Suppose an epidemic occurs at an unknown instant of time and the normal rate is subject to increase. (Environmental risk factors such as toxic spills, contaminated drinking water and radiation may also increase the normal rate.) Let V be this change. Since the interarrival times for a Poisson process are i.i.d, exponential, X1,X2,.. ,Xv will be independent having common exponential distribution with parameter 21 and X v + l , X v + 2 , . . will be independent having common exponential distribution with parameter 22, where 22 >_ 21. The goal is to estimate V. Another important application is on-line quality control of a manufacturing process. Imagine a machine that produces some product. The machine might break down at some point. The purpose of an on-line quality control scheme is to determine, based on the observation of the manufacturing process, whether the machine is functioning properly or not. In this setting, it is assumed that the observations are independent and their common distribution function before the change is F0 and after the change is F1. Clearly, F0 = F1 implies that the machine is functioning properly. In this section we discuss several Frequentist as well as Bayesian approaches to the discrete change-point problem.

3.1. Frequentist approach Assume that there exists 1 < g0 ~< n - 1 such that the joint probability density function of X ~ , . . , X n is ~0

I-[ f(Xi;O1) ~ I f(x"02) i 1

'

i=e0+ 1

where f is known and 01 ~ 02. If we assume that both 01 and 02 are known, then the maximum likelihood estimator of g0 is r

80 = a r g

max ~ ~

l<_r
,

N. Ebrahirni and S. K. Ghosh

784

where

1 Ff(x~, 01)1 and any non-uniqueness in maximization is resolved by suitable convention. For unknown 01 and 02 the maximum likelihood estimator of (01,02, go) is the maximizer of

~0 g(01,02, g0) = ~

log f(xi, 01 ) +

i=1

~

log f(xi, 02) •

i=g0+l

If the conditional maximum likelihood estimator of (01,02), (Öle0, Ô2e0) given g0 is available in closed form, then the maximum likelihood estimator of g0, {0 is the maximizer of g(Ôleo,Ô2e0,g0). Asymptotic properties of the maximum likelihood estimator of g0 were derived by Bhattacharya (1987), see also Rukhin (1994). Worsley (1986) constructed a confidence interval of g0 in exponential families. Some other approaches to estimation as well as confidence intervals for g0 have been discussed by Siegmund (1988) and references therein. For more details see the book by Sinha et al. (1995). The non-parametric estimator of g0 was proposed by Darkovsky (1976). This estimator is based on the Mann-Whitney discrepancy measure. Later Carlstein (1988) generalized this estimator by considering different measures of discrepancy. Methods based on U-statistic was proposed by Ferger (1991). For more details about different non-parametric approaches we refer you to Brodsky and Darkhovsky (1993) and reference therein.

3.2. Bayesian approach First we discuss parametric modeling of change-point problem. Assume that F0 and F1 admit densities of f(x]Oo) and of f (x[01), respectively; where 00 and 01 could be vector valued but unknown parameters. As we mentioned in Section 1 the problem is to estimate the discrete-valued parameter g0 from the sampling distribution

p ( x , . . . ,xnleo, 00, ol) --

1-If(~~lOo) i=1

f(xilOl)

•

i=g0+I

In order to complete the full probability specification we consider prior distribution for g0,0o and 01. As 00 and 01 are treated as nuisance parameters we assume that g0 and (00, 01) are independent. In other words, p(e0, 00, 01) = p(eo)p(Oo, 01) • Then, from a Bayesian point of view orte can obtain the posterior distribution of g0 given the data x = (xl,... ,xù) as

Bayesian and Frequentist methods in change-pointproblems

p(6o Ix)

p(g0) / p(x_160,00,01)p(00,01)d00d01

.

785

(3.1)

Smith (1975) assumes 00 and 01 are independent, however such an assumption may not be valid even when the change point is known. Muliere and Scarsini (1984) extend the approach to hierarchial specifications. Typically the discrete uniform prior is used for p(60), however other parametric priors can also be used. If integration in (3.1) can be achieved analytically (e.g. if a conjugate family is used for p(Oo, 01)), then the posterior distribution (which is necessarily discrete) can be enumerated rather straightforwardly. Otherwise Markov Chain Monte Carlo (MCMC) methods are employed to obtain posterior distribution of 60. For simple parametric families (e.g. exponential, gamma, lognormal, etc.) a simple code is easily developed using a generic software called BUGS. We now turn to non-parametric approaches. Notice that given 6o,2(1, Xz,... ,Xe0 are independent and identically distributed random variables with the common distribution function/7o and Xeo+l,... ,Xn are independent and identically distributed random variables with the common distribution function Fa. Again assuming that the random measures F0 and F1 are independent of 60, nonparametric priors as developed in Ferguson (1973) can be considered for F0 and F1. Muliere and Scarsini (1985) consider Dirichlet process priors on F0 and F1. However such a prior process selects a discrete distribution with probability one. Such an undesirable feature can be removed by using mixture of Dirichlet process priors. Dykstra and Laud (1981) propose Gamma process prior and extended Gamma process prior on the hazard rate instead of directly priors on F0 and F1. Other possible priors on F0 and F1 include a Beta process prior (Hjort, 1990) or Levy process prior. Typically, for such non-parametric classes of priors analytical solutions are rarely available and hence one uses sampling-based method (e.g. Metropolis Hastigs, etc.) to obtain desired posterior distributions. We refer the reader to Damien et al. (1996) for sampling-based methods on non-parametric Bayesian models. In this part we have not considered a third approach to estimate 60 using sequential procedures. Brodsky and Darkhovsky (1993) present several nonparametric methods for change-point problems. A brief survey of such parametric and non-parametric problems can be found in Muliere and Scarsini (1993).

4. Concluding remarks In this paper we discussed both Frequentist and Bayesian approaches for different change-point problems. Discrete change-point problems are easily handled using sampling-based methods. However, when using improper priors one taust be careful in verifying the propriety of posterior distributions. On the other hand, continuous change-point problems are rauch harder to tackle. Even simple parametric models could lead to very unstable estimates. Bayesian approaches to this problems are flexible. However, they could lead to serious problem of iden-

786

N. Ebrahimi and S. K. Ghosh

t i f i a b i l i t y a n d i m p r o p r i e t y o f p o s t e r i o r d i s t r i b u t i o n s . M u c h w o r k is n e e d e d f o r t h e implementation of non-parametric Bayes methods for this problem. As a final r e m a r k , it m a y b e o f w o r t h t o e x t e n d t h e i d e a o f a single c h a n g e p o i n t t o s e v e r a l (countable) such points when necessary.

References Achcor, J. A. and H. Bolfarine (1989). Constant hazard against a change-point alternative: A Bayesian approach with censored data. Commun. Stat. Theory Meth. 18, 3801 3819. Basu, A. P., J. K. Ghosh and S. N. Joshi (1988). On estimating change point in a failure rate. Statistical Decision Theory and Related Topics IV, Vol. 2 Springer, New York, 239 252. Bhattancharya, P. K. (1987). Maximum likelihood estimation of a change point in the distribution of independent random variables: General multi-parameter case. J. Multivariate Anal. 23, 183 208. Brodsky, B. E. and B. S. Darkovsky (1993). Nonparametric Methods in Change-Point Problems. Kluwer Academic Publishers, Boston. Broemeling L. D. (1972). Bayesian procedures for detecting a change in a sequence of random variables. Metro 30, 214227. Carlstein, E. (1988). Non-parametric change point estimation. Ann. Statist. 16, 188-197. Damien, P., P. Laud and A. F. M. Smith (1996). Implementation of Bayesian nonparametric inference based on Beta processes. Scand. J. Stat. 23, 27 36. Darkovskiy, B. S. (1976). A non-parametric method for the a posteriori detection of the "disorder" time of a sequence of independent random variables. Theory Probab. Appl. 21, 178-183. Dykstra, R. J. and P. Laud (1981). A Bayesian nonparametric approach to reliability, Ann. Stat. 9, 356-367. Ebrahimi, N., A. E. Gelfand, S. K. Ghosh and M. Ghosh (1999). Bayesian Analysis of Change-Point Hazard Rate (submitted). Ebrahimi, N. and S. K. Ghosh (1999). Bayesian analysis of the mixing functions in a mixture of two exponential distributions (in preparation). Ebrahimi, N. (1991). On estimating change point in a mean residual life function. Sankhya, A. 53, 206-219. Ferger, D. (1991). Non-parametric change-point detection based on U-statistics. Ph.D. Dissertation, Giessen. Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Star. 1, 209 230. Ghosh, J. K., S. N. Joshi and C. Mukhopadhyay (1999). Inference about sharp changes in hazard rates. In Frontiers in Reliability (Eds. A. P. Basu, S. K. Basu and S. Mukhopadhyay). World Scientific, NJ. Ghosh, J. K., S. N. Joshi and C. Murhopadhyay (1993). A Bayesian approach to the estimation of change-point in a hazard rate. Advances in Reliability, pp. 141-170 (Ed. A. P. Basu). Ghosh, J. K. and S. N. Joshi (1992). On the asymptotic distribution of an estimate of the change point in a failure rate. Commun. Stat. Theory. Meth. 21, 3571-3588. Hjort, N. J. (1990). Nonparametric Bayes estimation based on beta processes in models for life history data. Ann. Stat. 18, 1259 1294. Loader, C. R. (1991). Inference for a hazard rate change point. Biometrika 78, 749-757. Matthews, D. E., V. T. Farewell and R. Pyke (1995). Asymptotic score statistics processes and tests for constant hazard against a change point alternative. Ann. Statist. 13, 583. Muliere, P. and M. Scarsini (1993). Some Aspeets of Change-Point Problems in Reliability and Deeision Making (Eds. R. E. Barlow, C. A. Clarotti and F. Spizzichino). Chapman & Hall, London. Muliere, P. and M. Scarsini (1985). Change-point problems: a Bayesian nonparametric approach. Appl. Math. 30, 397~402. Muliere, P. and M. Scarsini (1984). Bayesian inference for change-point problems. Riv. Stat. Appl. 17, 93-106.

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Nguyen, H. T. and H. T. Pham (1987). Maximum likelihood estimation in the change poiut hazard rate model. The IMS Bull. 16, 36. Nguyen, H. T. and T. D. Pham (1989). Strong consistency of maximum likelihood estimators in the change-point hazard rate model. Statistics 10, 150-158. Nguyen, H. T., G. S. Rogers and E. A. Walker (1984). Estimation in change point hazard rate models. Biometrika. 71, 229-304. Page, E. S. (1955). A test for a change in a parameter occuring at an unkuowu point. Biometrika 42, 523-526. Rukhin, A. L. (1994). Asymptotic minimaxity in the change-point problem. IMS Lecture Notes, Change-Point Problems, pp. 284-291 (Eds. E. Carlstein, H. G. Muller and D. Siegmund). Siegmund, D. (1988). Confidence sets in change-point problems. Internat. Statist. Rev. 56, 31~48. Sinha, B., A. Rukhin and M. Ahsanllah (1995). Applied Change- Point Problems in Statisties. Nova Science, New York. Smith, A. F. M. (1975). A Bayesian approach to inference about a change point in a sequence of random variables. Biometrika 62, 407~416. Worsley, K. J. (1986). Confidence regions and tests for a change-point in a sequence of exponential family random variables. Biometrika 73, 91 104. Yao, Y. C. (1986). Maximum likelihood estimation in hazard rate models with chauge point. Commun. Stat. 16, 2455-2466. Zacks, S. (1983). Survey of classical and Bayesian approaches to the change-point problem. Fixed sample and sequential procedures of testing and estimation. In Recent Advances in Statistics, pp. 245-269 (Eds. M. H. Rizvi, J. Rustagi and D. Siegmund). Papers in Honor of Herman Chernoff on his 60th Birthday, Academic Press, New York.

N. Balakrishnanand C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 200t ElsevierScienceB.V. All rights reserved.

'~ ~'~ ù.J

The Operating Characteristics of Sequential Procedures in Reliability

S. Z a c k s

1. Introduction

Sequential methods in reliability testing have been in use from the pioneering work of Dvoretsky et al. (1953), Epstein and Sobel (1955) and Kiefer and Wolfowitz (1956). Sequential life testing and sampling acceptance for the exponential distribution had been codified in the Military Standards 781C. Later, in the 1960s and 1970s many papers appeared on survival analysis, and among them also studies on sequential methods, like Epstein (1960), Basu (1971), Mukhopadhyay (1974), Bryant and Schmee (1979) and others. A comprehensive summary of these methods is given in the article of Basu (1991). In many of the studies on sequential methods, the evaluation of their operating characteristics is done via Wald's approximations, asymptotic analysis, or numerical techniques. The difficulty has been in finding the exact distributions of stopping times. Algorithms for the numerical determination of the operating characteristics of sequential procedures were developed by Aroian and Robinson (1969), Aroian (1976), Zacks (1980) and others. In the present article we present analytic methods for the determination of the distributions of stopping times, and from these, the evaluation of expected stopping times (AST), the operating characteristic function (OC) and coverage probabilities of test procedures and estimators. In Sections 2 and 3 we restrict attention to reliability experiments with exponential time till failure. In Section 4 we discuss general distributions of operating times and of repair times. In Zacks (1991, 1997), the exact distribution was derived analytically, for the first crossing times of a homogeneous Poisson counting process of two parallel straight line boundaries. These results can be immediately applied for evaluating the operating characteristics of sequential testing, as done in the present article. We start in Section 2 with the problem of estimating the reliability of a system whose life distribution is exponential. We discuss a sequential procedure for a prescribed proportional closeness probability (PPCP). This criterion is different from the fixed-width confidence interval, or the minimum risk estimators used in 789

790

s. Zacks

many studies (see Basu, 1971; Mukhopadahyay, 1974). It is straightforward to verify the asymptotic properties of the sequential PPCP estimator developed in Section 2 (see Simon and Zacks, 1967). We present, however, analytic equations, based on the sample path behavior of a homogeneous Poisson process, which yield the exact probabilities at which the process crosses a quadratic boundary. Based on this we develop formula for the exact (not asymptotic approximation) evaluation of the coverage probability, and of the expected values, variance and the first, second and third quartiles Q1, Me and Q3 of the stopping time. Generally, sequential experiments for PPCP, with high accuracy and high closeness probability, require long times till stopping. We develop therefore the formula for evaluating the procedures when m parallel experiments are conducted, with and without replacements. In Section 3 we provide similar analysis for the sequential testing. We restrict the presentation to a particular shape of the stopping boundaries. Also here we provide an m-parallel experiments version, with or without replacement, for the reduction of the expected length of the experiment. In Section 4 we discuss the problems of estimation after testing. We show that generally the PPCP of an estimator after testing is low. One cannot estimate the reliability after sequential testing, so that the estimator is, with high probability, within 10% of the true value, for all values of the parameter 0. We show, nevertheless, how one can determine upper confidence limits for the reliability, based on the crossing time and crossing level in a sequential testing. This problem of determining confidence intervals after sequential stopping is discussed in various papers. See in particular Siegmund (1978), and Bryant and Schmee (1979). In Section 5 we discuss the distribution of the total time, within a specified time period, that a repairable system is operating. This is done by utilizing recent results of Perry et al. (1999) on the distributions of stopping times for alternating renewal processes. We emphasize that also in this connection, the results presented are exact, and not asymptotic approximations. Sections 6 and 7 are devoted to some review of sequential methods in detection of the epoch of entrance into a wearout phase, and in software reliability. The material of Section 6 is based on the paper of Zacks (1984) and that of Section 7 on Zacks (1995). The article of Zacks (1995) provides a comprehensive review of sequential methods for software reliability. We do not reproduce this review here. We mention here that the problem of software reliability has been studied and discussed in many papers and books. In particular see Littlewood and Verall (1973), Musa et al. (1987), Dalal and Mallows (1988).

2. Sequential estimation of reliability with prescribed proportional closeness Consider a system whose time till failure is exponentially distributed, with an unknown mean time till failure (MTBF) 0, 0 < 0 < oc. We wish to estimate the reliability of the system at a specified time to, i.e., p(O) = e -t°/°. Without loss of generality, let to = 1. Let Pn be an estimator of p(O), based on n failure times

The operating characteristics of sequential procedures in reliability

791

,Am. Let 0 < 3,7 < 1 be specified values. We say that 15n has a prescribed proportional eloseness (3, 7) if

X1,X2,..

Po{Ißù - p(O)] < 3p(0)} > 7

(2.1)

for all 0 < 0 < c~. I f X 1 , . . , X, are i.i.d, random variables having an exponential distribution, E(O), then a maximum likelihood estimator (MLE) of p(0) is ~ù = exp{-1/Ä~ù }, where d{ù is the sample mean. The asymptotic distribution of ~ , as n -+ c~, is that of

Thus, for large values of n, the proportional closeness, which is the left-hand side of (2.1), is equal approximately to 2~(03v~ ) - 1. Thus, if n _> n(3), where n(6) - Z~[1]

(2.2)

0232

then the proportional closeness attains (approximately) the prescribed level 7- The term Z~[1] in (2.2) is the 7th quantile of the Z2 distribution, with one degree of freedom. Eq. (2.2) leads to the consideration of the stopping variable N(3) = least integer n , n > n*,

Z~[1] , such that n _> õ23~2

(2.3)

where n* is the size of an initial sample. Routine analysis (see Simons and Zacks, 1967) yields the following asymptotic results: (i) lim N!ó) = 1, a.s.; g40 n~o)

(ii) ~iöPo{[~N(õ ) (iii) lim E0{Æ(6)) ö~o

n(6)

--

p(0)l

< 6p(O)} Z 7 for all 0;

1 for all 0;

(iv) E o { N ( 3 ) ) < n(3) + O(1), as 3 ~ 0. In the present section we develop the exact distribution of the stopping variable N(ö), and the associated proportional closeness probability at stopping, namely

cP(3,o) = »0{l~N(~/-

p(0)l <

3p(o)}

m

(2.4)

Let {K(t), 0 _< t < ~ } be a regular, time-homogeneous, Poisson jump process (see Cinlar, 1975, p. 70), with intensity 2 = 1/0. This is a process of independent increments, such that: (i) K(0) _= 0, (il) for every 0 < tl < t2 < e~, K(t2) - K ( t l )

~ K(t: - tl) ~ Pols(il(t2 - tl)) ,

S. Zacks

792

where Pois(#) designates a random variable having a Poisson distribution, with mean #. We denote by p(j; #) and P(j; #) the p.d.f, and the c.d.f., respectively, of Pois(#). Define the jump random times ~n = inf{t : t

>_O,K(t) =

n} ,

(2.5)

n = 1 , 2 , . . . It is well known that ~ n - % - 1 , with z 0 - 0 , are i.i.d, random variables having the exponential distribution, with mean 0, i.e.,

zn-r~-l~E(O),

n> 1 .

(2.6)

If in out reliability design, we put systems on test sequentially orte by one, starting the operation (life) of the nth system immediately after the failure of the (n - 1)th system, then ~ù corresponds to the nth failure time, i.e., "c,, ~ ~i~1X/. In terms of the Poisson process {K(t), 0 < t < oo}, the stopping variable N(6) corresponds to the stopping time T(6) = inf t : K ( t ) =

max(n*,~T;7t2) \

Z~L1] ,/

(2.7)

and N(b) = K(T(8)).
t2

Z211]

(2.8)

'

Obviously, Pz{T(3) < ~ } = 1 for all 2, 0 < 2 < o« Indeed, by the law of the iterated-logarithm

P~~F-lm K(t)- 2t _

1}= 1

(2.9)

I t+oo ~/22t log 2(t) for all 2. Since K(t) may cross B~(t; b) only at levels of non-negative integers, T(Ô) is a discrete random variable assuming the values in {t~),t ~ , . . } where t~, k _> 0, is the root t of BT(t ;3) = k + n * , or t~-~Z~[1]~,

k=0,1,...

We develop now the probability distribution of T(3). Let 2" = 2~/7~7211]/ô, and let

O(vS(k) =P2{r(3)

= tl} ,

k = 0,1,...

(2.10)

Since the Poisson process is strongly Markovian (see Cinlar, 1975, p. 117), the probability function {~(8 (k), k >_ O} can be determined recursively according to the equations ~~~) (0) = P (n*; 2* x/~7)

(2.11)

The operating characteristics of sequential procedures in reliability

793

and, for k _> 1,

#/*

l=0 k-1 j=l

(2.12) A non-iterative solution of (2.11) and (2.12) is not available. However, the computation of {tp(r;) (k), k _> 0} is very fast. Once the values of ~~) are determined one can immediately determine the moments of N(6) and of T(cS) and their quantiles. The results depend, however, on the selected initial sample size n*. We illustrate this dependence in Table 1, in which c5 = 0.1,7 = 0.9. We denote by Q1,Me and Q3 the first quartile, the median, and the third quartile of N(6), respectively. Generally, as expected, Eo{N(6)} is slightly increasing with n*, while //õ{N(cS)} is slightly decreasing. The quantiles Q1,Me, Q3 are quite robust against changes of n*. Furthermore, as 0 increases, the distribution ofN(~5) becomes more symmetric. Notice also that, according to Table 1, if

Os <

(02~ 2 02,

E < { N ( 6 ) } / E o 2 { N ( 6 ) } ~- \ÕlJ

"

This is in accordance with Eq. (2.2). The proportional closeness interval estimator of

p(O) at stopping is

PCI(N(6)) = (fiN(a) ÆN(a)']

(2.13)

\TT;' 7 --$)

Table 1 The expected values, variance Q1, Me, Q3 of N(c~), ~

0.1, ~ = 0.9

0

n*

E{N(6)}

V{N(cS)}

Q~

Me

Q3

2

20 40 10 15 3 6 1 3 1 3

64.01 64.38 29.02 29.43 15.64 16.03 9.28 10.04 6.38 7.26

335.82 314.17 119.64 106.82 69.09 60.27 44.60 37.23 26.67 21.02

55 55 21 21 9 10 3 5 2 3

66 66 29 29 15 15 9 9 5 3

78 78 35 36 21 21 14 14 10 10

3 4 5 6

S. Zacks

794

where

~N/~,=exp

~

(2.14)

J

The question is, what is the closeness probability (2.4)? Since

Ä~N(a)-~(5)ZI{N(6)=n* r(a) ~ k=O

+ k } 1 /Z•[1]

(2.15)

aVT7~

the proportional closeness probability at stopping is OO

(2.16)

cP(a, 0) = Z 0~l/°/(k)~{k*(0) < ~ < k**(0)} , k=O

where

k*(o) : ~z~[1] [max(O, 1 - Olog(1 + 5))]2-n *

(2.17)

z~[1] k**(O)= 0B5(1

(2.18)

and - 0log(1 - Ô))2-n* .

In Table 2 we present the closeness probabilities CP(5, 0) at stopping. We see in Table 2 that the CP(cS, 0) is generally very close to the nominal value 7- It depends somehow on n*. If n* is too small there might be cases of early stopping, which lead to decrease in the coverage probability. If n* is too large, the coverage probability might be higher than the desired one. It is a conservative strategy to take n* high, especially since the values of are not affected rauch. Finally, the expected length of the experiment is

Eo{N(6)}

Table 2 The closeness probabilities at stopping CP(6, 0), 6 = 0.1, 7 = 0.9

0

,*

cP(a, 0)

2

20 40 10 15 3 6 1 3 I 3

0.8919 0.8943 0.8780 0.9783 0.8498 0.9816 0.8247 0.9843 0.9879 0.9876

3 4 5 6

The operating characteristics of sequential procedures in reliability

795

O(3

Eo{T(fi)} = Z~[1] Z ( n * + k)U2o(rl/°)(k) . &

(2.19)

k=0

In a similar fashion we can compute the variance of T(&). In Table 3 we present these values. As seen in Table 3, the expected duration of this sequential estimation is quite large, especially for 0 close to 1. One can reduce the expected duration by constructing an experiment in which m independent systems operate in parallel. Whenever a system fails it is immediately replaced. Let {Kz(t),t >_ 0} be the Poisson counting process for the /th system, i = 1 , . . , m , and let K(m)(t) = ~im_l Kz(t). K(m)(t) is the total number of failures in the time interval (0, tI. We define now the boundary m2~52

Bm(t)=--t

Z211]

2,

t>0 -

(2.20)

and the stopping time

rm(ô)

-- max ( tm,mf {~*:

K(m)(*)<_ Bm(t)}),

(2.21)

where

t*m = t ~ / m -

;

~

.

With this stopping rule, one has the relationship Tm(&) = T~(&)/m. Moreover, if Nm(6) = max{K(m)(Tm(&)), n*} then at stopping the estimator of p(O) is fiN.,(a) = exp

{

Nm(~)'~

(2.22)

mT~(a)J "

One can immediately verify that @~) T,m\fk~) = P)~{Nm(6) = n* + k} = P~{N1 (&) = n* + k} =~r,~(k) (~)

for a l l k > O

.

Table 3 Expected value and variance of T(6), & = 0.1, y = 0.9

0

n*

Eo{T(6))

Vo{T(6)}

2 3 4 5 O

40 10 3 2 1

133.58 86.93 62.08 48.38 37.89

257.95 293.13 338.53 294.70 290.34

S. Zacks

796

Moreover, the coverage probability of

777'7 --X) is the same as that of

l÷5'l:B)

"

Thus, with such an experiment we can estimate p(O) sequentially with the desired proportional closeness probability but reduce the expected duration by a factor of m, since

Eo{ Tm(6) } = 1Eo{ TI (6) }

.

(2.23)

It is often the case that m independent systems are put on test, but failed systems are not replaced. In this case the interfailure times are independent but not identically distributed, and all formulae should be changed. We will still use the stopping boundary Bm(t) given by (2.20). Thus, 7],ù(6) defined as in (2.21) has the same possible stopping points, but the probability distribution of Nm(6) has to be changed. Let b(j'; n,p) and B(j; n,p) denote, respectively, the p.d.f, and the c.d.f, of the binomial distribution with parameters (;0 (n,p). Let Ob,m(k),k >_O, denote the probability function of Nm(6) when failed systems are not replaced. We obtain the following recursive equations: @(;0<m,(05,= B(n*; m, 1 - e where 2~, =

2*/m,

<,~) ,

<2.24)

and for k _> 1,

b,mk ] 11"

- Z b(l;m, l -exp{-2mV%7} ) /=0

xb(n*+k-l;m-l,l-exp{-2*m('~*+k-W~7)}) k-1

- l{k > 2} ~ ~~;'~(]') j=i

×b(k-j;m-ù*-j,e-exp{-2*m(V/nT+k-~)})

. (2.25)

Notice that n* must be smaller than m, otherwise ~b.~( (~) 0 ) = 1. Moreover, g%m(k) > 0 only if k _< m - n*. In Table 4 we present the expected duration and the coverage probability when failed systems are not replaced.

The operating characteristics of sequential procedures in reliability

797

Table 4 The expected value of Tm(6) and the coverage probability 6 = 0.1,7 = 0.9 0

m

n*

EO{Tm(Ô)}

CP(c~, 0)

2

300 200 100 50 40 30 10 5

20 20 10 10 3 3 2 1

0.401 0.576 0.767 1.395 1.323 1.690 3.512 5.579

0.8080 0.7204 0.8356 0.7342 0.8292 0.8133 1.0000 0.7893

5 3

1 1

5.165 7.563

1.000 1.000

3 4 5

6

We see in Tables 3 and 4 that by running m parallel systems, without replacement, one reduces the expected duration of the experiment by order of magnitude, but one has to use large values of m, when 0 are small, in order to attain CP values over 0.7, when the nominal value is 7 = 0.9. To achieve high proportional closeness one should pay either by long durations, or running independent systems in parallel with replacement. For example, if 0 = 2 and we wish to reduce the expected duration of the experiment to 2.58, we need to run, with replacement m = 100 systems. One hundred systems without replacement yield a coverage probability of 0.3457.

3. Sequential testing Sequential testing is an important procedure both in system development and in reliability demonstration, when one attempts to demonstrate that a given system meets the reliability standards. In the present Section we focus attention on systems whose time till failure are exponentially distributed with MTBF, 0. The objective is to test systems sequentially in order to demonstrate that 0 _> 01. Suppose that systems with 0 < 00 should be rejected, while systems with 0 >_ 01 are acceptable, where 00 < 01, and 00, 01 are specified values. We consider here the sequential testing of the hypotheses H0 : 0 < 00 versus//1 : 0 _> 01, with level of significance c~and power at least 1 - / ~ if H1 is true. As in the previous section, we start with a sequential testing in which the tested system is replaced immediately after each failure. Since the interfailure times are exponentially distributed, we consider a homogeneous, regular Poisson counting process {K(t),t _> 0} and terminate testing as soon as {K(t)} crosses a stopping boundary. The properties of such a test were studied in m a n y papers. In particular see Dvoretsky et al. (1953), and Zacks (1991, 1997). As explained in Zacks (1992, p. 188), the Wald Sequential probability ratio test (SPRT) terminates as soon as K ( t ) crosses either one of the two parallel boundaries B u ( t ) =- he +sC ,

(3.1)

s. Zacks

798

or

B L ( t ) = - h 1 4;- st »

(3.2)

where

(~,),

(3.3)

hl = !°g (LÜ) log

hz = . l ° g ( L ~ )

(3.4)

log(~) 21 - 20

'-,og(~)

t~»/

Here 20 = 1/01 and 21 = 1/00. ~ and/~ are the error probabilities. If {K(t)} hits the lower b o u n d a r y BL(t), H~ is accepted, otherwise H~ is rejected. In the present formulation, H~ : 2 _< 2o corresponds to H1 : 0 > 01, and H{ corresponds to H0 : 0 < 00. F o r example, if 00 - 4481 and 01 = 6153, c~ = 0.05 and/3 = 0.95 the two parallel boundaries are Bu(t) = 9.107 + 0.00019125t and BL(t) = --7.093 + 0.00019125t. Since K(t) is a unit jump process, BL(t) can be crossed only every 1Is = 5228.758 time units. By making the transformation to # = 2Is and t' = ts we can consider two parallel boundaries Bu(t') = h2 + t' and BÆ(t') = - h l + t'. Thus, without loss of generality, we can consider two parallel boundaries with slope s = 1. We will also consider hl = k l and h2 --k2, where kl and k2 are positive integers. Closed form formula for the distribution of the stopping time is given in Zacks (1997). In the presentation here we considered a stopping rule in which the lower boundary is, as before BL(t) = - k l + t, while the upper b o u n d a r y is Bu(t) = k» This corresponds to a frequency truncation of the SPRT. Thus, we define the stopping time :r = i n f ( t

: K(t)

= -kl

+ t,

or K(t)

-- k 2 }

.

(3.6)

At stopping, if K(T) -- - k l + T we accept H0; if K(T) = ka, we reject H0 (accept //1). Obviously, T _< kl + kz - 1, with probability one. Since we use a boundary with slope s = 1, we consider {K(t)} with intensity # = 2Is, where s is given by Eq. (3.5). Let ~2~)(j) = P~{K(T) = j}, j = 0, ù " , k2 - 1; i.e., ~7kl ,/,(")~;~ kJ ) is the probability that K(t) crosses the lower boundary at level K(T) = j. The values of t, at which K(T) crosses the lower b o u n d a r y are {to,..., tk2 1), where tj = k l q - j ,

j = 0,..

,k2 -

1

(3.7)

The operating characteristics of sequential procedures in reliability

799

(~)(i5 ¢k~ ~, can be found recursively. According to the equations ¢(~) /q (0~ v ~ = exp{-ffki}

(3.8)

and for j _> 1, j-1

¢(#)

¢(~)(1)p(j

k, Ü') = P(J; Iztj) - Z

- l; # ( j - l))

(3.9)

.

I=0

Since the second term on the right-hand side of (3.9) is a convolution one can obtain an explicit solution to (3.8) and (3.9) by using generating functions (see Zacks, 1991). Explicitly, J O(ff)(J) = ~q(#)PÜ"l; ]~(kl • j kz

l))

(3.10)

1=0

where q'(#) =

{1, (-1)¢det{Al(#)},

l=0, l_> 1

(3.11)

and det{Az(/~)} is the determinant of the l × l matrix

p(11~) ......

p(1;t~)

p(1;/*)

...

".

"..

A,(#) =

p(l-1;(/-1)/~)

(3.12)

"

p(1;#) The determinants det{Al(#)} can be computed recursively by det{A0 (#)} = 1 ,

(3.13)

j+l

det{Aj+l(#)} = Z ( - 1 )

i-1 x p(i; @)det{Aj+l-i(#)},

j = 0,

i=1

(3.14) The c.d.f, of the stopping time T is

[t <]

+I{t _>kl} Z

v,<'l'(~)(J)P(k2- 1 -j;t*(t- k l - j ) )

(3.15)

j=0

The quantiles of this distribution can be obtained directly by tabulating the values of ~!Y),_2(t)., " " Moreover, the expected value of T is

S. Zacks

800

k2 [1 - P(k2; ~(k~ + k2 - 1))]

e~{T} = 7

k2-1

+ ~ ( k l +J)t)k," (~)(j)P(k2 - j j-O lk2-I -Z(k2

B j=0

1;#(k2 - j ) )

-- j)~(k~) (j) [1 - P ( k 2 - j ; # ( k

2 -j))]

.

(3.16)

Finally, in addition to the expected stopping time one should characterize the procedure by computing the operating characteristic (OC) function ~z(#) = Pv{accepting H0}. This OC-function is given by k2-1

r~(#) =

k2-i

k2-1-j

/~) Ü) • = ~-~~«l(#) Z Z Ok~ j-0

I=0

p(j;#(kl +j)) •

(3.17)

j=o

In Table 5 we present some of these values numerically. T0.5 denotes the median. We see in this table that ifwe wish to test H0 : # _< 0.8 versus//l : # _> 1.6 with kl = 3 and k2 = 42, then the attained c~ level is ~z(0.8)= 1 - 0.9406 = 0.0594, while the attained fi level is ~(1.6) = 0.0459. The expected value of T, when H0 is true does not exceed 11.65, while if//1 is true it might be up to 24.63. As in estimation, if one wishes to reduce the expected time till termination, m parallel tests can be tun independently, with immediate replacement of failed systems. Let {K,.(t), t > 0}, i = 1 , . . , m, be the Poisson process corresponding to t h e / t h trial and K(m)(t) = ~'i~=1Ki(t). {K(m)(t), t >__0} is a Poisson counting process with intensity m#. We consider then the boundary B~ )(t) = -kl + rot, and the upper boundary B(Ü) (t) = k2. In this case the true points at which B ~ ) (t) can be crossed Table 5 Values of 2(#), E~{T}, and T05 for kl = 3, k2

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.9999 0.9989 0.9885 0.9406 0.8237 0.6456 0.4552 0.2974 0.1867 0.1159 0.0725 0.0459 0.0294 0.0192 0.0126 0.0084

42

6.00 7.42 9.31 11.65 14.87 t9.10 23.19 25.89 26.95 26.85 26.12 25.11 24.01 22.90 21.83 20.82

4.5 5.5 6.5 7.5 12.5 20.5 29.5 30.0 28.9 27.60 26.09 24.63 23.26 22.00 20.83 18.80

The operating characteristics of sequential procedures in reliability

are (t}m),j = 0 , . . ,k2 - 1}, where t!m) = with the stopping boundaries B(m)(t) and

Ora.kl (~), ~;~ • J = Pu{Km(Tm) = j}

=

801

tj/m. If Tm denotes the stopping time B(m)(t)= k2, then

~,/.(~) l,kt

Ü)

for a l l j = 0 , . . ,kz - 1. Thus, the OC function is the same as in the case o f m = 1, but E~{Tm} = E~{T~}/m. If such an experiment can be performed a considerable saving can be attained in termination time. We consider now the case of m parallel systems, without replacement of failed systems. Let Jm,b(t) be the total number of failures in (0, t]. Consider the stopping time Tm,b = inf{t : Jm,b(t) = - k l 4-

rot,

or

Jm,b(t)

=

k2}

•

(3.18)

We define the crossing probabilities

(~)

O~l,m»(J)

"

=

Pu{-kl

+

mTm,b= j } ,

j = 0, 1 , . . , k2 - 1. One can determine recursively, t)(v) kt ,m,b ;0' \ ) = b(0;m, 1 -

e -k'~/m)

(3.19)

and for j _> 1,

(~)

~'k,,~»(J)

"

b(j;m,

=

1 -

e -~(k~+')/m)

j-I -

V/~a~Pkl,m,bk ' , (u) ;l'b(j-1; ]

-

m

-l,l-e

-~(j-O/m)

(3.20)

/=0

Also here the second term on the right-hand side of (3.20) is a convolution, and one can obtain explicit solutions to (3.19) and (3.20). However, for computational purposes, one can use Eqs. (3.19) and (3.20). The OC function is k2-1

~b,.,(/~)

(~) " • = Z O<,m,ó(J)

(3.21)

j-O

The expected value of

Tm is 1 k2-1

1

/1

Ern'l~{Tm}=;j~_o-m7 ~ ~ - / x p ( - # ~ ) 1 k2 1 B

l~=_ó_o'l'(~) fl) 7*kl,m,b\J

(m - j , j ÷ 1))

k2-1-1

j~=O

m - l -1j ( 1 -

In(m -- l -- j , j + 1)) (3.22)

S. Zacks

802

where ( (k2 - 1 - l ) ) t / = exp - # m and [p(Vl, •2) is the incomplete-beta-function ratio. It is clear that m should be greater or equal to kz. In Table 6 we present the characteristics of the procedures, for kl = 3, k2 = 10 and m = 15. Comparing Tables 5 and 6 we see that with m = 15 parallel systems, we can reduce kz from 42 to 10. The values of the OC functions are quite close for # _< 0.9. For values o f # _> 1.0, rC»,m(#) are greater than re(#). Thus, ifH1 : # _> 1.5 the one-system with replacement experiment yields an error probability/~ _< 0.07, while the m-systems without replacement has error probabilities/~ _< 0.33. On the other hand, the expected duration of the m-system experiments, with m = 15, is about 0.30-0.59, while for one-system with replacement the expected duration is from 6.00 to 26.95 time units.

4. Reliability estimation after testing Consider again the sequential testing studied in the previous section. We will discuss here the testing with stopping boundaries BL(t) = - k l + t and Bu(t) = kz, with the stopping time given by (3.6). We also consider a one-system experiment with replacement of failed systems. After stopping H0 is accepted if K(T) < k2. People are. often interested in estimating p(O) at stopping. An estimator of # = 1/0, at stopping, is ~(T) = K ( T ) / T , and the corresponding estimator of the

Table 6 Values of 7Zb,m(/l),Em,~{Tm} and the median, Tm.o.5 for kl = 3, k2 = 10 and m = 15

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.9947 0.9831 0.9590 0.9183 0.8593 0.7837 0.6959 0.6015 0.5066 0.4162 0.3341 0.2626 0.2023 0.1531 0.1139 0.0835

0.352 0.395 0.439 0.481 0.519 0.549 0.572 0.586 0.591 0.590 0.582 0.570 0.555 0.537 0.519 0.500

0.288 0.330 0.383 0.442 0.501 0.549 0.580 0.597 0.602 0.599 0.589 0.574 0.555 0.534 0.513 0.491

The operating characteristicsof sequentialprocedures in reliability

803

reliability p(O) is/3(T) - e x p { - K ( T ) / T } . Generally, sequential estimation for a prescribed proportional closeness and prescribed coverage probability, requires longer experimental times than sequential testing. This can be seen by comparing Tables 3 and 5. From Table 3, if # = 0.5(0 = 2), 6 = 0.1 and 7 = 0.9, the expected stopping time is 133.58 (time units), while for testing, the expected stopping time is 6 (time units). This shows that the estimator/5(T) at stopping after sequential testing, cannot be sufficiently precise. To evaluate the proportional closeness coverage probability of f3(T) we write

CPv(#; 6) = P~{lfi(T) - e-~l < 6e ~} = P # ( I ~ ( T ) - e ~1 < ae L K ( T ) < k2}

÷Pl,{Ijb(T) -- e-Ul < a e - " , x ( r ) For K ( T ) = j , j = O , . . . , k 2 - 1 , P ~ { K ( T ) = j } (3.9). Thus, P , { [ ~ ( ~ ) - e "1 < a e % K ( T )

: ~2}.

(4.1)

=0~~)(j), as given by (3.8) and < k2}

k2-1

= Z =

j-0 k2 1

0~~)(J)[{J : e ~(1 - 6) < e j/(kL+j) < e-~(1 + 6)} ,/,(/~)

{

klal(#)

2~=o,zk~ Ü)I 1-~ ä l ~ )

<j < --

kla2(#)

"~

-- 1 -- a 2 ( ~ ) J

,

(4.2)

where al(#) = (# - log(1 + 6)) +, a2(#) = ( # - log(1 - 6)) .

(4.3)

Notice that if al(#) > 1 then (4.2) equals zero. If al(/~) < 1 and a2(#) _> 1 then (4.2) is equal to

,I,(~) klal (#) j~o ~,k~ (j)I i ~ ~~ (~) -< j < - k2 - 1

.

If K(T) -- k2 then/3(T) = exp{-k2/T}. We need, however, the distribution of T, conditional on {K(T) = k2}. Let TL denote the first time the Poisson process {K(t); t > 0} crosses the lower boundary BL(t). Thus,

P~{T <_ t,K(T) = kz} = P~{T _< t, TL < t} .

(4.4)

From (3.15) we ger P#{T ~. t,K(T) = k2} = 1 - P ( k 2 - 1;#t) - I { t > kl} It-<] × ~ ,//#)(j)(1 - P(k2 - 1 - j ; #(t - k, - j ) ) ) ',"kl j--0

(4.5)

S. Zacks

804

Finally, since

{,~(T)-e-I*, < Õe-~,K(T)=k2} = { ~

< T <&,K(T)=k2} (4.6)

we obtain that

< be U,K(T) = k2} { T < a l ~k2 K(T) = k 2 } - P ~ ~,

P~{Ifi(T) - e -~1 =P~

{

k2 'K(T)=k2}

T_
(4.7)

Thus, the coverage probability of the estimator ~(T), can be computed by adding (4.2) and (4.7). In Table 7 we present some values of the proportional closeness coverage probabilities. We show now a method of determining upper confidence limits for the reliability p(O) at stopping. There are two situations at stopping, one is that K(T) < k2, i.e., H0 is accepted; the other is that K(T) = k» IfH0 is accepted, we determine an upper confidence limit for p(O)in the following manner. For each # such that ~(#) _> 1 - c~, let jl-«(#) = least

j,j

J

_> 0

such that ~

~'k~'/'(~)(l)_> 1 --c~ .

(4.8)

/=0

Given K(T) = j, j = 0, 1 , . . , k2 - 1, a lower confidence limit for #, with level of confidence (l - ~ ) is

BI~~(K(T)) =

inf{# : jl-~(#) = K(T)} .

(4.9)

The corresponding upper confidence limit for the reliability p is

BI~~(K(T)) = exp{-#1~~(K(T)) } . (4.10) If K(T) = k2, i.e., H0 is rejected, we determine the ~-quantile of P~{T <_t,K(T) = kz} for all # such that ~(#) < c« Let Tv#(#) = inf{t : P~{T <_t,K(T) = k2} _> ~}. Then, the lower confidence limit for #, given (T, K(T) = kz), at confidence level 1 - c~ is /~(u)«~, l _ ~ k ~ ~ k2)

=

inf{#: Tu,s(#)) <_ T}

(4.11)

•

The (1 - ~)-upper confidence limit for p is then (4.12)

B(u/~r kz) = exp{-glu)~(T,k=)} Tabie 7 P r o p o r t i o n a l closeness c o v e r a g e p r o b a b i l i t i e s o f iS(T); ~ = 0.1, kl = 3, k2 = 42 #

0.5

0.75

1.00

1.25

1.50

2.00

CP

0.4054

0.3942

0.2011

0.3271

0.3185

0.2528

The operatingcharacteristicsof sequentialprocedures in reliability

805

W e r e m a r k t h a t in the case o f K ( T ) < kz, we b a s e d the confidence limit o n K(T). Recall that, in this case, T = k] + K(T). I n the case o f K(T) = k2, we base the confidence limit on T. I n T a b l e 8 we p r e s e n t the values o f j ] _ ~ ( # ) a n d o f Tu,a(#), for the case o f kl = 3, k2 = 42, d = 0.05. Thus, for example, if the s t o p p i n g b o u n d a r i e s have p a r a m e t e r s kl = 3, kz = 42 a n d at s t o p p i n g H0 is accepted, with K(T) = 10, (T = 13), the 0.95-upper confidence limit for p is e - ° 5 = 0.6065. If, on the o t h e r h a n d , K(T) = kz = 42, a n d T = 13, the 0.95-upper confidence limit for p is e -24 = 0.0907.

5. The total operating time of repairable systems C o n s i d e r a system which can be r e p a i r e d after each failure, a n d o p e r a t e a g a i n after being repaired. Let us d e n o t e b y & , $2, • • • the r a n d o m o p e r a t i n g times o f the system, a n d b y R1,R2,.. the r a n d o m repair times. Let C I = S I + R I , C2 = $2 + R 2 , . . , C~ (n >_ 1) be the length o f the nth cycle o f o p e r a t i o n followed b y repair. W h e n the system o p e r a t e s we say t h a t it is O N a n d when it is r e p a i r e d we say t h a t it is O F F . Let t, 0 < t < oc, be a fixed time. The n u m b e r o f system failures in the time interval (0, t] is

NF(t)=

{° / max

n:n

if & >_ t,

n] } _>l,Z©+&
if SI < t ,

j=O where Co - 0. The t o t a l o p e r a t i n g time d u r i n g the interval (0, t I is a r a n d o m variable N~(t)

Wt = I { & > t } t + I { & < t} Z S n

.

(5.1)

n=l

I n the present section we present results, recently p u b l i s h e d by P e r r y et al. (1999), c o n c e r n i n g the d i s t r i b u t i o n o f Wt, when {Sn}n_>l are i.i.d., {Rn}~>l are i.i.d. Table 8 Quantiles of K(T) and of Tu, for kl

3, k2

=

42, c~= 0.05

#

jl-~(Y)

~

Tu#(~)

#

Tu,~(p)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2 3 4 7 10 15 24 40

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

42 36 32 29 26 24 22 21

1.0 1.7 1.8 1.9 2.0 2.2 2.4 2.6

19 18 17 16 15 14 13 12

S. Zacks

806 and

{Œ} is independent

of {Rn}. Under

this assumption,

the sequence

{SI, R1, ,5'2,R 2 , . . } is called an alternating renewal process. We mention here that results concerning the distribution o f ~ are k n o w n in the literature only for the asymptotic case o f t ~ oc. The results we present here are valid for all t, 0 < t < ec. Consider the c o m p o u n d process NF(t)

x(t) = t + Z

Rù,

t »_ o ,

(5.2)

n--O

where R0 = 0. With respect to {X(t), t _> 0}, the variable o f interest W~, for any fixed time t = a, is a stopping time defined as W~ = inf{t : X ( t ) >_ a} .

(5.3)

Notice that the process {X(t); t _> 0} crosses the level a either as X(Wa) = a when the system is ON, or as X(W~) > a when the system is O F F . The distribution o f W~ has an a t o m at t = a. In other words, the c.d.f, o f Wa is absolutely continuous on (0, a). At t = a it has a j u m p o f size P{S1 > a}. W h e n the distributions o f the O N times and o f the O F F times are both exponentials, with parameters 2 and #, respectively, where 2 1 _ E { S I } and g-1 = E{R1 }, the p.d.f, o f W~, on (0, a), is oo

Oa(t) = Z p O " ; g(a - t) ) ( g p ( j + 1; 2t) + 2p0'; 2t) ) .

(5.4)

j-0

Generally, i f F and G a r e the distribution functions o f S and R, respectively, and if F *(") and G *(~) denote their n-fold convolutions, with F *(°) = G *(°) = 1(0,~), then for the alternating renewal process, oo

/

N

P{W~ > t} = ~ ( F * ( " ) ( t ) - F*("+l)(t))G*(")((a - t ) - )

.

(5.5)

/

n--0

If, for example,

F(t) = 1 - e -zt,

G(t; v + j , g )

t >_ 0 and G(t) = e -~ j=0

,

•

where G(t; v + j , g ) is the g a m m a c.d.f, with shape parameter v + j , and scale parameter l / p , then, from (5.5) we obtain, OQ

P{W~ > t}

OQ

e ~t+ZZp(n;2t)pO';n~)G(a-t;n(v+j),g

) .

(5.6)

n=l j=0

Setting c~ = 0 and v = 1 yields the special case o f exponential distributions, with F(t) = 1 - e -xt ,t >_O;G(t) = 1 - e ut, t > O, for w h i c h oo

P{Wa > t}

~p(j; j-0

g(a - t))P(j; At) .

(5.7)

The operating characteristics of sequential procedures in reliability

807

Differentiation of (5.7) yields the p.d.f. (5.4). We conclude the present section with the following result. Let M(a) = E{W~}. From (5.5) we get for absolutely continuous F and G,

M(a)

= a(1 -

F(a)) +

/0~(1 - F(t))dt

+ ~~_lfoa (F*(n)(t) -F*(n+l)(t))G*(n)(a-t)dt .

(5.8)

Let f(t) and g(t) be the densities of F and G, respectively, and let fr(s) and be their LT's. Then, the LT of M(a) is

M*(s) =

g*(s)

e ~aM(a)da

(

1

B(1 - f*(s)) \1 + 1 -

1

"~ 1 d * s ,

f*(s)g*(s)/ + -s" dssf ( )

s>0

.

(5.9) In particular, in the double exponential case

1( M*(«)-«(Ä+,)

1+ (~+s)(~_+_s).~

~

(5.10)

Inversion of this LT yields

M(a)

(~2 + # )

~ - R + # + a e Xa

_ _+e -~)2 (;.+U)a (~

(5.11)

Recall that M(a)/a is the proportion of total time in (0, a] the system is ON. From (5.11) we obtain the well-known formula lim M(a) a--+oo a

_

#

2+#

(5.12)

Eq. (5.9) can yield the LT of M(a) for any distributions F and G. Inversion of this transform yields the formula of M(a).

6. Sequential detection of wearout

Generally, systems operate for long periods with an almost constant hazard rate function. This phase in the life of the system is called the "mature phase". After the mature phase comes the "wearout phase", in which the hazard rate is increasing. For purposes of controlling the operation of a system, the inventory of spare parts, etc., it is important to detect, as early as possible, the shift epoch from the mature phase to the wearout phase. Zacks (1984) modeled the problem as that of estimating the epoch of shift to the wearout phase, which is a parameter

s. Zacks

808

%0 < z < oo, intrinsic to the system. The hazard function of the system is modeled as

h(t;fl, Œ,z)=

{~ ~,+fl~Œ(t-z) ~-1

if t _< z, ift>~ ,

(6.1)

where (2, c~,z) are all positive parameters, and ct > 1. The shift of h(t; .) from a hazard function of an exponential to that of a Weibull distribution with ~ > 1 (increasing), is at the time point r. Moreover, the hazard at t > -c is greater than Ä. The life distribution of such a system has a density function

f(t;)~,c~,~) = fle-~t(1 + ~ ( 2 ( t - ~)+)~-1). e x p { _ 2 ~ ( t _ r)+} ,

(6.2)

where (t - z)+ = max(0, t - z). When z is unknown, one should be able to estimate it from the available data on failure times of such systems. We remark here that usually we consider this problem with repairable systems. Thus, time measurements on such systems are cumulative operation time. Zacks (1984) studied systems which are repaired immediately after failure, or after A units of operation time, for preventive maintenance. The data were the history of sequential random failure or maintenance times Tl, T 2 , . . , where 0 < T1 < T2 < ' ' " < Tn < " ' '. A Bayesian estimator of z, was derived. For the first time the estimate of z is smaller than the chronological time, the shift is detected. We do not present here the formula of the Bayesian estimator of r, which is not of a particular interest in a general discussion. Moreover, the detection rule mentioned is not necessarily optimal. For optimization one needs to formulate a risk function which depends on various loss components, like the loss, per time unit, of late or of early detection. Generally, the derivation of the optimal detection rule is difficult. A few exceptions are in cases of Markov decision processes, see Zacks (1991b) for details.

7. Sequential methods in software reliability Software reliability studies have special models, since the software is a finite collection of units (modules, functions, etc.), which does not change if a fault is not found and corrected. Several models of software reliability and sequential testing procedures are discussed in Zacks (1995). In the present section we will present some of the stopping rules, which are of special interest. Since the article of Zacks (1995) is a comprehensive chapter in a book, we will not give here all the details. Software might contain a certain unknown number, N, of faulty units. As long as no faulty unit is chosen for execution, the software fulfills its requirements. What is software reliability? The reliability function R(z) is the probability that the computer running with the given software will survive z time units without crashing or showing faults. There are two main software reliability models: time-domain models, and data-domain models. A common time-domain

The operating characteristics of sequential procedures in reliability

809

model can be described in the following terms. Suppose there are N faults in the system where N is unknown. The model assumes that the random time required until a fault is detected by the requirement of a random customer is exponentially distributed. The times to detect different faults are i.i.d. Thus, the observed times of the first n detected faults are distributed like the first n-order statistics, T(I:N) < T(2:N) < "'" < T(n:N), from a sample of N i.i.d, r a n d o m variables. If we denote by E(fi) the exponential distribution with mean/~, 0
OO,

T(n:N) --

T(1,N )

,',-'E(fl~ T(2,N) --T(1,N ) ~ E ( N ~ ) \NJ'

r(n_l:N/ ~ E

'

...

'

N - n + 1

The problem is, after observing n failures (faults), should one continue or stop searching (testing). If N = n no more faults can be found, and one should stop sooner or later if a good estimator of/~ is available. Sometimes the risk function is such that it is optimal to stop even before observing all the N faults. A sequential stopping rule is given in Zacks (1995) which, after each detected fault specifies how long testing should continue if no additional fault is detected. We show now some of the results concerning this question, which are proven in Zacks (1995). Let U(1) < • • • < U(N) denote the order statistics of a random sample of size N from an exponential distribution, E ( 1 / # ) . Define N

n

J(u)=ZI{U(i)_
0_
W~=ZU(i)

i=1

with U0-=0 .

i=0

Define the stochastic process N-1

W(ii) = Z I { U ( n )

~ Il <

U(n+l)}Wn -~

WN-[{U(N) ~ Il},

0 ~ Il < OO .

n=0

The likelihood function of (#, N) at time u is

L(#,N; u,J(Ii), W(u) ) _

ArT

-uJ(~) e x p { - # W ( u ) - #(N - J(u))il}

(7.1)

(N - J(il))! --

Thus, {(J(u), W(u); 0 _< u} is a minimal sufficient process. In a Bayesian framework we ascribe (#, N) a joint prior distribution. If we assume that # and N are priorly independent, N ~ Pois(~) and

i.e., g a m m a with mean «~, then the posterior of (#,N) at time Il, given

(J(Il), W(u) ),

is

s. Zacks

810

h(#, N[u, J(u), W(u))

r(~ + J(ù//(x- J(ù//! v'~~m=0cm,[w(ù/+++ mù]/~+Jl")/ The n u m b e r o f remaining faults at time u is M ( u ) = N - J ( u ) , O < u < oc. Let ù(u,J(u), W(u)) = E{M(u)[u,J(u), W(u)} be the posterior expectation. One can prove (see Zacks, 1995) that q(u, J(u), W(u)) is a decreasing function o f u in intervals between failure points. At failure times q(u,J(u), W(u)) m a y j u m p upwards. Let K[$1 be the cost (penalty) of finding a fault in the field (after release o f the software). Let c[$] be the cost o f testing per unit time. The posterior risk if testing is stopped at time u is R(u;J(u), W(u)) = «u + K ~ ( u , J ( u ) , W(u)) .

(7.3)

One could consider the following stopping rule: S = i n f { u : argmtBnR(t,J(u), W ( u ) ) = u } .

(7.4)

It is shown in Zacks (1995) that as long as S does not stop it is optimal to continue. Data-based models consider various sampling strategies f r o m the finite population of units which represents the software. In each sample, all or some faulty units are detected and corrected. In a second sample one continues to detect and correct units. The question is, h o w long should one continue sampling? Two sequential testing procedures for d a t a - d o m a i n models are studied in Zacks (1995). The first one is sequential testing using quantal prediction models, and the second one is a sequential capture-recapture procedure. The interested reader is referred to this article for details.

References

Aroian, L. A. (1976). Application of the direct method in sequential analysis. Technometrics 18, 301 306. Aroian, L. A. and D. E. Robinson (1969). Direct methods for exact truncated sequential tests of the mean of a normal distribution. Technometrics 11, 661 675. Basu, A. P. (1971). On a sequential rule for estimating the location parameter of an exponential distribution. Nav. Res. Logist. Q. 18, 329-337. Basu, A. P. (1991). Sequential methods in reliability and life testing. In Handbook of Sequential Analysis (Eds., B. K. Ghosh and P. K. Sen), Chapter 25. Marcel Dekker, New York. Bryant, C. M. and J. Sehmee (1979). Confidence limits of MTBF for sequential test plans of MIL-STD 781. Technometrics 21, 33M-2. Cinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ. Dvoretsky, A., J. Kiefer and J. Wolfowitz (1953). Sequential decision problems for processes with continuous time parameter. Ann. Math. Stat. 24, 254-264.

The operating characteristics of sequential procedures in reliability

811

Dalal, S. R. and C. L. Mallows (1988). When should one stop testing software? J. Amer. Stat. Assoc. 83, 872 879. Epstein, B. (1960). Statistical life tests acceptance procednres. Technometrics 2, 435446. Epstein, B. and M. Sobel (1955). Life tests in exponential case. Arm. Math. Stat. 26, 8243. Kiefer, J. and J. Wolfowitz (1956). Sequential tests of hypotheses abont mean occurrence time of a continuous parameter Poisson process. Naval Res. Logis. Q. 3, 205-219. Littlewood, B. and J. L. Verall (1973). A Bayesian reliability growth model for computer software. J. R. Star. Soc. B 22, 332-346. Mukhopadhyay, N. (1974). Sequential estimation of location parameter in exponential distribution. Calcutta Stat. Assoc. Bull. 23, 85 95. Musa, J. D., A. Iannino and K. Okumoto (1987). Software Reliability Measurement, Prediction, Applications. McGraw-Hilt, New York. Perry, D., W. Stadje and S. Zacks (1999). First exit times for increasing compound Poisson processes. Stochastic Models 15, 977-992. Siegmund, D. (1978). Estimation following sequential tests. Biometrika 65, 341-349. Simons, G. and S. Zacks (1967). A seqnential estimation of the tail probabilities in exponentiai distributions with a prescribed proportional closeness. Technical Report, Department of Statistics, Stanford University. Stadje, W. (1993). Distribution of first-exit times for empirical counting and Poisson processes with moving boundaries. Stochastic Models 9, 91 103. Zacks, S. (1980). Numerical determination of the distributions of stopping variables associated with sequential procedures for detecting epochs of shifts in distributions of discrete random variables. Commun. S t a t . - Simul. Comput. 9, 1 18. Zacks, S. (1984). Estimating the shift to wear-out of systems having exponential-Weibull life distributions. Oper. Res. 32, 741-749. Zacks, S. (1991a). Distributions of stopping times for Poisson processes with linear boundaries. Stochastic Models 7, 233-242. Zacks, S. (1991b). Detection and change-point problem. In Handbook ofSequential Analysis (Eds. B. K. Ghosh and P. K. Sen). Chapter 23. Marcel Dekker, New York. Zacks, S. (1992). Introduction to Reliability Analysis. Springer, New York. Zacks, S. (1995). Sequential procedures in software reliability. In Recent Advances in Life-Testing and Reliability (Ed., N. Balakrishman), Chapter 6. CRC Press, Boca Raton. Zacks, S. (1997). Distributions of first exit times for Poisson processes with lower and upper linear boundaries. In Advances in the Theory and Practice ofStatistics (Eds. N. L. Johnson and N. Balakrishman). Wiley, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook ofStatistics, Vol. 20 © 2001 ElsevierScienceB.V. All rights reserved.

"4 J J

Simultaneous Selection of Extreme Populations from a Set of Two-Parameter Exponential Populations

Khaled Hussein and S. Panchapakesan

1. Introduction

Let [ I 1 , . . , F [ k be k independent two-parameter exponential populations E(#i, ffi), i = 1 , . . ,k, where the density associated with F[i is f ( x ; # i , ai) = _1 e (x-~i)/o,, ai

X > #i,

- - 0 0 < # i < OO,

(7i > 0

.

(1.1)

Although technically the Pi could be any real numbers, in the context of lifelength distributions they are positive and are referred to as the guaranteed lifetimes. To describe the basic goals and the background of the selection and ranking, we will first consider ranking the populations according to the values of the location parameters (guaranteed lifetimes), assuming that al . . . . . crk = « (known). It is also assumed that there is no prior knowledge about the correspondence between the ordered and the unordered #i. Most of the investigations in the literature are concerned with the goal of selecting the population associated with the largest #~ or the one associated with the smallest #» the target population being called the best population. Two classical formulations of these problems are known as the indifference zone (IZ) formulation due to Bechhofer (1954) and the subset selection (SS) formulation due mainly to Gupta (1956). Let us consider selecting the population associated with the largest #i. In the IZ approach, one of the k populations is selected as the best. A correct selection (CS) occurs if the selected population is a best population. Let P(CSIR) denote the probability of a correct selection (PCS) using the rule R. It is required that a valid rule R satisfies: P(CSIR ) _> P* whenever #[k] -

#[k-I] ~ 3" »

(1.2)

where the positive constant c5" and the guaranteed PCS P* (k -1 < P* < 1) are specified in advance by the experimenter. For a rule R based on a single sample of size n from each population, one has to determine the minimum sample size n for which (1.2) holds. 813

814

K. Hussein and S. Panchapakesan

The part of the parameter space defined by the restriction: #[k] - ~[k-1] ~ ~* is called the preference zone (PZ) and its complement is the so-called IZ. For selecting the population associated with the smallest #» the PZ is characterized by: BI2] -- //[1] --~ b*.

In the SS approach for selecting the best population, the goal is to select a nonempty subset of the k populations so that the selected subset includes the best population (which results in CS) with a guaranteed minimum probability P* whatever be the configuration of the unknown/~i- It is assumed that, in the case of a tie for the best population, one of the contenders is tagged as the best. The size S of the selected subset is not specified in advance but is determined by the data. The usual measures of the performance of a valid rule are the expected subset size E(S) and the expected number of non-best populations included in the selected subset (which is equal to E(S) - PCS). In order to meet the guaranteed minimum PCS under either formulation, one needs to evaluate the infimum of the PCS over the appropriate set of configurations of/~ = (#l,---,#k). Any configuration of # for which this infimum is attained is called a least favorable configuration (LFC). For selecting the best two-parameter exponential population where the best population is either the one associated with the largest value or the one with the smallest value of the parameter of interest, several procedures in the literature have been reviewed by Panchapakesan (1995) who has also considered variations and modifications of the basic selection goals. For detailed discussions of various aspects of the theory of selection and ranking and related problems, the reader is referred to Gibbons et al. (1977), Gupta and Panchapakesan (1979), and Bechhofer et al. (1995). Now, one can consider the goal of simultaneously selecting the extreme populations, namely, the populations associated with the largest and the smallest values of the parameter of interest. Mishra (1986) considered this goal under the IZ formulation for selecting fi'om a set of populations belonging to a one-parameter family where the parameter of interest is of location or scale. Mishra and Dudewicz (1987) have considered this goal under the SS formulation for selecting from normal populations in terms of means. Dhariyal and Misra (1994) have considered a Bayesian approach to this problem while Misra and Dhariyal (1994) have considered Bayes-P* and minimax rules. In the present paper, we consider selecting the extreme populations from a set of k two-parameter exponential populations with densities f(x; #t, ai) given by (1.1) for i = 1 , . . ,k. Section 2 discusses selection in terms of the location parameter under the IZ approach whereas Section 3 deals with the problem under the SS approach. Selection with respect to the scale parameter is addressed in Sections 4 and 5 under the IZ and SS approaches, respectively. The procedures discussed in Sections 2-5 are based on complete sample observations. Section 6 considers all the above selection problems based on Type-II censored samples in the context of life-testing experiments. Some remarks regarding future investigations are made in Section 6.

Simultaneous selection of extreme populations

815

2. Selection in terms of the loeation parameter; IZ approaeh Our goal is to select the two populations that are associated with the largest and the smallest #~, both identified correctly. We assume that al . . . . . «k = a. A CS occurs when a rule selects the two extreme populations correctly identified. Any valid rule R is required to satisfy P(CSIR ) _> P* whenever #[21 - #[1] ~

fT

and #N - il[k-l]

~ f~ '

(2.1)

where the positive constants f~ and f~ and the probability level P* are specified in advance by the experimenter. F o r a meaningful problem, we take (1/k(k - 1)) < P* < 1. The set O1 = {_~= ( # 1 , . . , # k ) : #[2]--#[11--> fiT, #[k]--#[k 1] ~ Ô~} is the PZ. We consider below two cases: a known and a unknown.

2.1. Known a case Let X~-I,...,Xi, be a r a n d o m sample of size n from Hi, i = 1 , . . , k . De#ne Y/= minl<j<,X/j, i = 1 , . . , k . Let Y[1]-< " " - < Y[g] denote the ordered Y/. We propose the following rule RI:

Select the populations yielding Y[1] and YN as the populations associated with #[I] and #[k], respectively .

(2.2)

2.1.1. P C S and its infimum Let Y(i) denote the statistic from the population associated with #[i], i = 1 , .., k. De#ne Ui = (n/cr)(Y(o- #[i]), i = 1 , . . , k . Then the U~ are i.i.d each having a standard exponential distribution E(1), with mean one. Then letting #~j = (n/c)(#['l - #~l)' P(CSIR1) = Pr[Y(1) _< Y(i) _< Y(k),i = 2 , . . , k -

1]

= Pr[U1 - #il < Ui < Uk + # v , i = 2 , . . , k -

11

k-1

=

[F(t + #ki) - F(s - #il)]f(s)f(t)ds dt , 30

J0

i=2

where F(.) and f ( . ) are, respectively, the c.d.f and the p.d.f of the standard exponential r a n d o m variable. It is easy to see that the infimum of P(CSIR1) over f21 occurs when BI1] + f~ = #[2] . . . . .

#[k 11 = #[k] - f~ ,

(2.3)

which is therefore, the LFC. Thus inf P(CSIRI )

[o~ ft+(n/o)(~*l+~~) x f ( s ) f ( t ) d s dt .

_n [F(t + a g : ) - F ( s - n

~] f * k2 (2.4)

816

K. Hussein and S. Panchapakesan

When 6~ = c5~ = c5" (say), letting A = n6*/a, Eq. (2.4) becomes Boe ft+2A

inf P(CS[R1) = Jo ]o

[F(t + A ) - F ( s - A)]k 2

x f(s)f(t)ds dt . By straightforward integration, this gives e-A inf al P(CS[R1) - k ( k 1- 1) {1 - (1 _ )k

1

{l_(l_e-3)

k 1}{1_ e 4} .

-~ ( k - 1)e -~ The minimum sample size n needed to satisfy the probability requirement (2.1) can be obtained by first equating the right-hand side in (2.5) to P*. This gives

where @} denotes the smallest integer greater than or equal to y, A=-log(1-u),

0
1 ,

and

h(u)-u k+l-(k+l)u k+(k-1)(l+kP*)u-k(k-1)P*+l

=0 . (2.6)

It is easy to check that h"(u) < 0 for k > 2 and 0 < u < 1. Thus h(u) is concave on (0, 1). Further, limù_~0h(u)< 0 and lim~+l h(u)= 0. Therefore, there exists a unique solution for Eq. (2.6) for u C (0, 1). This leads to the determination of the minimum sample size n. Table 1 gives values of A = A(k,P*) for k = 2 ( 1 ) 1 0 , 15, 20, 25, 30 and P*= 0.90, 0.95 and 0.99.

2.2. Unknown « case When Œis known, the determination of the minimum sample size n required for the procedure R1 depends on the knowledge of «. When a is not known, we cannot determine the minimum sample size needed to satisfy the probability requirement (2.1). In fact, there does not exist a single sample procedure that can satisfy (2.1). We propose a two-stage procedure R2 described below. R2: (1) First, take n independent observations X/l,... ,X/n from ~Ii, i = 1 , . . , k, where n is arbitrarily chosen. Let

1 ä - k (1n ~

k ~'~ ~ ( X ~ j - Yi), i=1 j=l

where Yi = minl_<j
817

Simultaneous selection of extreme populations Table 1 Value of A for which P(CS[R1) = P*

k•*

0.90

0.95

0.99

2 3 4 5 6 7 8 9 10 15 20 25 30

0.80471 2.30090 2.78197 3.07235 3.28549 3.45630 3.59969 3.72371 3.83325 4.24700 4.53667 4.76042 4.94274

1.15129 2.99531 3.49120 3.78649 4.00254 4.17514 4.31984 4.44492 4.55523 4.97131 5.26217 5.48657 5.66967

1.95601 4.60517 5.11300 5.41258 5.63043 5.80448 5.94993 6.07571 6.18650 6,60469 6,89681 7,12094 7,30525

(2) Let N = max{n, ( a ä ) } , where (y) denotes the smallest integer _> y, a is the unique solution of

(2.7)

x f ( s ) i ( t ) 9~ (w)d« dt dw = P* ,

v = 2k(n 1) and 9u(-) is the density function of a chi-square variable with v degrees of freedom. (3) N o w take a second sample of size N - n from each population. Let Y~be the smallest of the N observations from IIi, i = 1 , . . , k, and let Y[1] _< "'" _< Y[kl denote the ordered Y/. Select the populations that yield Y3] and Y[k] as the populations associated with #[1] and #[k], respectively.

2.2.1. P C S and its infimum Letting, as before, Y(i) to be the statistic from the sample corresponding to the

population a s s o c i a t e d

with

#[i], i ---- 1 , . . ,

k, w e h a v e

P(CSIR2) = Pr[Y0)-< Y(i) -< Y(~),i = 2 , . . , k -

11

(3Q

= ZPr[N

= m]PrIY(1) < Y(i) <_ Y(k),i = 2 , . . , k - IIN = m]

m=F/ oQ

=~Pr[N=m]Pr q- rna (kt[k] _ kt[i]) '

[U s - ä m (#[i] _ i = 2,..,k-

_ Ui < _ Uk #[1]) c5~ and #[k]-#[k-1]->@ it is easy to see that P(CSIR:) is minimized when #[11 + 31 = #[2] . . . . . #[k-1] = #N -- c52' So we get P(CSIR2 ) _> Z

-rr, Pr[N = m]Pr [U1 - m~)*l ~r <_ Ui <_ Uk + -m62

m~n

i = 2,..,k=

IIN = m]

N~~ ~ <_U~<_u~ + - -a,

Pr[N = m]Pr U1 m~t/

i = 2,..,k_>

I[N = m]

aa6~ <_ Ui <_ Uk + - - ,

P r [ N = m ] P r U1

ry

m~n

ff

i =2,..,k-l[N=m]

P[WCBm]P [UI

=

aW6~ v <_ Ui <_ Uk +aW6: v '

m=n

i = 2, . . , k - l]W C Bm] , where W = vä/a has a chi-square distribution with v = 2k(n- 1) degrees of freedom and is independent of G, and Bm = {N = m}, Bm being the appropriate subset in the sample space of W. We note that the sets Bm are disjoint and UmBm = (0, eX)). Let Dm(a) = {w[w = vä/G w ŒBm}. Then P(CSIR2) > ~ ß m=n

Pr[U1 ra(a)

aW3~GUiGUk+aW6~ ~

i = 2,..,k-

V

l[gv(w) dw

= fo~ fo°° fo~('~+~;)w+t[F(t+aWv6:)_F(s_a;fi~)l k-2 x f(s)f(t)gv(w)dsdtdw = P* by (2.7)

.

When c~~ = 6~ = ~5" (say), by letting A = a6*/v, Eq. (2.7) becomes

Jo~ f ~ f2~w+t[F(t Aw) - F(s- Aw)]k-z x f(s)f(t) gu(w) dsdtdw = P* .

(2.8)

For given k, P*, 3", and n, orte can solve for A, which gives the second stage sample size. There are no tables available for A at this time.

Simultaneous seleetion of extreme populations

819

3. Seleetion in terms of the loeation parameter; SS approach O u r goal in this section is to select two n o n - e m p t y subsets of the k t w o - p a r a m e t e r exponential populations, one (call it SG) containing the p o p u l a t i o n associated with largest #i and the other (call it SB) containing the p o p u l a t i o n associated with the smallest #i, simultaneously. In case of a tie for the largest #~ or the smallest #;, we assume that one of the contenders is tagged as that extreme population. A CS occurs if Sc includes the p o p u l a t i o n associated with #[< and SB includes the p o p u l a t i o n associated with #[1]- We assume that al . . . . . ak = a.

3.1. Known cr case As in Section 2.1, we take a r a n d o m sample of size n from each population. Let Y~ denote the smallest order statistic of the sample f r o m Ili, i = 1 , . . , k , and let Y[[1] ~ "'" ~ Y[k] be the ordered Y/. We p r o p o s e the following procedure R3 :

Put IIi in SB iff Y/_< Y[1] -~- 0"c~l and, n ~C2 put FIi in S~ iff I7//_> Y[k] - - ,

(3.1)

H

where C1, C2 are hOrt-negative constants to be chosen so that R3 satisfies the requirement: P(CSIR3 ) > P * where 1 / k ( k -

for a l l #

(#1,--.,#k)

,

(3.2)

1) < P * < 1.

3.1.1. P C S and its infimum Let Ui = (n/«)(Y(i) - #[il), i = 1 , . . ,k, where Y(i) is the statistic o f the sample f r o m the p o p u l a t i o n associated with #[i]. Further, let bqt = (n/a)(#[i] - #[JI) + cl, for l = 1, 2, and bkl = (n/a)(# N -- #[11) + min(c1, c2). The U,. are i.i.d E(1). N o w , °'c1 o'c2] Y(k) -> Y[< n n J = Pr[U1 -bi11 <_ Ui « U k - - b k i 2 , i - 2 , ..,k-

P(CSIR3 ) = Pr

[y(

1) < Y[1] + - - ,

1,

u1 _< u~ + b~l]

foof,+ó~lk-1

H [F( t + bki2) - F(s - bi11 )] f (s) f ( t) ds dt .

J0

J0

i=2

It is easy to see that the infimum ofP(CSIR3 ) over t~ -- {# = ( # 1 , - - - , & ) } occurs when #1 . . . . . #k. Thus inf P(CSIR3) = foo°° f t+min(cl'ca)[F(t + «2) - F(s - «, )]k-2 O

J0

× f ( s ) f ( s ) ds dt .

(3.3)

K. Hussein and S. Panchapakesan

820

In order to satisfy the requirement (3.1), we want cl and C 2 such that the righthand side of (3.3) equals P*. W h e n cl = c2 = c (say), Eq. (3.3) yields info P ( C S I R 3 ) :

~0oo J0pt+c r[e-(,

c)_ e-(t+«)]k-2e-(,+,)dsdt

- ( k - - l l ) e c [ k e ß {l -- ( 1 - - e-«)k} -- ~ { 1 - - e-«}k-I 1 1

-~ ( k - 1)e « {1 - (1 - e «)k 1}{ 1 _ e-C}

(3.4)

by straightforward integration. The smallest constant c = c(k,P*) satisfying the probability requirement (3.2) can now be obtained by equating the last expression in (3.4) to P* and solving for c. This gives « ---- - log(1 - u) ,

(3.5)

where u E (0, 1) satisfies

h(u) =_ (1 - k)u k - u~-1 + ( k - 1)(1 ÷ k P * ) u - k ( k - 1)P* + 1 = 0 .

(3.6) It is easy to check that h"(u) < 0 for k _> 2 and 0 < u < 1. Thus h(u) is concave on (0, 1). Further, l i m ù ~ 0 h ( u ) < 0, and l i m ù ~ l h ( u ) = 0. Therefore, there exists a unique solution for Eq. (3.6) for u E (0, 1). This yields the constant c by (3.5). Table 2 gives values o f c = c(k,P*) for k = 2(1)10, 15, 20, 25, 30, and P* = 0.90, 0.95 and 0.99.

3.1.2. Expected sizes of the selected subsets As p e r f o r m a n c e characteristics of procedure R3, we need to evaluate the suprema of E(B) and E(G), the expected sizes of the selected subsets B and G, respectively. Table 2 Value of c for which P(CSIR3 ) = P*

k•*

0.90

0.95

0.99

2 3 4 5 6 7 8 9 10 15 20 25 30

1.60943 2.43129 2.82533 3.09206 3.29626 3.46228 3.60395 3.72666 3.83533 4.24763 4.53695 4.76053 4.94288

2.30258 3.13741 3.53729 3.80739 4.01383 4.18199 4.32428 4.44791 4.55742 4.97203 5.26256 5.48681 5.66967

3.91203 4.75681 5.16116 5.43390 5.64193 5.81147 5.95455 6.07876 6.18894 6.60543 6.89681 7.12094 7.30525

Simultaneous selection of extreme populations

821

We first consider E(G). Let Qi denote the probability that the p o p u l a t i o n associated with #[i] is included in the selected subset G. Then it is easily seen that E(G) = ~~=1 Qi, where _

-

-

C1(7]

Qi = Pr [Y(i) > Y[k] ~ - j =Pr

[YÜ) <- Y(o + c](7 -- , J = n

]

l,'",k,j ¢ i

= PrIUj <_ Ui + n_(#[i] - #7]) + Cl, j = 1 , . . ,k, j ¢ i ] (7

=

J=' F ( , + ä ( # [ / ] -

#~.]) + c , )

f(,)dt.

j¢i The expression for E(G) is the same as that for a procedure of Ofosu (1972) for a different selection goal. Ofosu (1972) has claimed without giving the p r o o f that the s u p r e m u m o f E ( G ) over the p a r a m e t e r space O is attained when #1 . . . . . #kWe conjecture that is true for k _> 3. It can be shown to be true for k = 2. In this case, we can write E(G) in the f o r m

E(G) =

/max{0,~-cl ~ }

[1 - e-(t-~+c~)]e -t dt +

= {2-1e-C'[e~+e 1 +

-~1

½e-~[e C' - e «~]

/0 ~

[1 - e-(«+«+Cl)]e -t dt

ifc~<«, otherwise ,

where ~ = (n/(7)(#[2]- #Eil)" Since (d/dc~)E(G)<_ 0 for c~ _> 0, E(G) attains its m a x i m u m when #[1] = #E2I" We can have a parallel discussion of the case of E(B). The c o m m e n t s m a d e a b o u t the s u p r e m u m of E(G) applies to the s u p r e m u m of E(B) in toto.

3.1.3. Unknown G case W h e n a is u n k n o w n , unlike in the I Z approach, we can p r o p o s e a single-stage procedure R4 by replacing « in R3 defined in (3.1) by its unbiased estimator 1

k

ù

~-~(n-l~ z/=1 z(x~j~)' j=l where the X/j, j = 1 , . . , n , are independent observations f r o m IIi and B = minl_<j_
Put IIi in S, iff I1//< Y[1] -- bi(} and, n putIIiinSGiffY~_>

b2ôYN--, n

(3.7)

K. Hussein and S. Panchapakesan

822

where bi and b2 are non-negative integers to be chosen so that R4 satisfies the requirement: P(CSIR4 ) _> P* where 1 / k ( k -

for all # = ( # l , . . . , # k ) a n d a

(3.8)

1) -- Y[k] --

bl Wa Y(k) >-- Y[k] vn

Pr Y(1) < Y[I] + blwa -

vn

' Y(kl >

b2 W«] vä , where W = - vn j a ~~]

b2wu 9v(w) dw , vn j

where gv(.) is the density of a chi-square variable with v degrees of freedom. Based on the arguments in the k n o w n a case,

[

blwa vn

Pr 17(1) -< Y [ [ 1 ] - + - - - , Y(k) >-- YN

-->[~[(t+l/v,)1m0in(b"b2)W[jo

b2wa] vn j

[_F(t-7~) -F(s -- ~W-)] k-2

× f ( s ) f ( t ) ds dt . Thus, in order to satisfy (3.8), the constants bl and b2 should be chosen to satisfy

fO°°~O°°~ot+(1/v)min(bl'b2)W[F(t@~) _F(s bS_)]k-2 × f ( s ) f ( t ) Ov(w)ds dt dw = P*

(3.9)

W h e n bi = b2 = b, Eq. (3.9) becomes

× f(«) f ( t ) gv (w) d« dt dw = P* .

(3.10)

A table of values of b satisfying (3.10) for selected values of k, n and P* is not available at present.

4. Selection in terms of the scale parameter; IZ approach O u r goal now is to select the two populations that are asso¢iated with the largest and the smallest ui, both identified corre¢tly. Let 0 < «[11 -~ "'" -~ aN denote the

823

Simultaneous selection of extreme populations

ordered a» It is assumed that there is no prior information about the correct pairing of the ordered and the unordered O-i. A CS occurs when a rule selects the extreme populations correctly identified. Let

B2 = ~'O-= (O-1,.. ,O'k) : 0"[2] > ~~ > 1, O-[h] _> 61 > 1 ; , O-[1] O-[k-Il J t where 6~ and 6~ are specified in advance. Any valid rule R is required to satisfy: P(CSIR ) _> P*

whenever a E ~ 2 ,

(4.1)

where 1/k(k- 1) < P* < 1. We assume that the Pi are all known or all unknown. Let X~j, j = 1 , . . , n, be independent observations from Ili, i = 1 , . . , k. Define n

Si =

v

Pz)

[1i~/~~-~/

if

#i is

known, (4.2)

~~~,i~ ~nknown,

where Y,- = minl<j<_n Xij. Let S[1] _< ... _< S[k] be the ordered We propose the following procedure Rs:

Si.

Select the populations that yield S[1] and S[k] as those associated with O-[1] and

O-[k],respectively

.

(4.3)

4.1. PCS and its infimum Letting S(i) as the statistic arising from the sample drawn from the population associated with O-?], i = 1 , . . , k, we have P(CS]Rs) = Pr[S(1) <

S(i) <_S(k), i = 2,.. ,k -

1]

=pr[a[1]Tl<Ti<_a[klTk, i=2, . . , k - l l Lo-[i]

O-[i]

,

where Ti = vS(i)/o[i] with v = 2n if Pi is known and v = 2(n - 1) if #i is unknown. The T~ are i.i.d, each having a chi-square distribution with v degrees of freedom. We thus get P(CS]Rs) =

fo °~ fa0 (~Ekll~I'l)t~~_~ [ Gk v (o-[k]t'l-G~(~s)]9~(s)9~(t)dsdt ko-[i] /

where Gv(.) and gv(.) are the c.d.f, and the density function of a chi-square r.v. having v degrees of freedom. F o r _a E ~'~2, (a[k]/a[i]) ~ (~~ and (a[1]/o[i]) _< 6~. It follows easily that the L F C in ~22 for P(CSIR5 ) is given by

,

K. Husseinand S. Panchapakesan

824

310"[1] : 0"[2]

" = 0"[k-l]

0"[kl 32

Hence k-2

G~(3~t) - G~ s

infP(CSIR5 ) = ~2 dO dO

g~(s) g~(t)ds dt . (4.4)

I n p a r t i c u l a r , when 6~ = 3~ = 3* (say), (4.4) becomes

inf P(CSIRs ) =

fo~fo õ% [G~(3*t) - Gu(~,)]k-2 gv(S)g~(t)dsdt. (4.5)

The smallest v for which (4.1) is satisfied can n o w be o b t a i n e d b y e q u a t i n g the r i g h t - h a n d side o f (4.5) to P* a n d solving for v. T a b l e 3 gives the smallest v for 3" = 2(0.5)4, k = 2(1)10 a n d P* = 0.90.

5. Selection in terms of the scale parameter; SS approaeh O u r goal is to select two n o n - e m p t y subsets o f the k p o p u l a t i o n s , n a m e l y , Se which includes the p o p u l a t i o n a s s o c i a t e d with a[1] a n d Sc which includes the p o p u l a t i o n associated with a N ' I n case o f a tie for either extreme p o p u l a t i o n , we assume t h a t one o f the c o n t e n d e r s is tagged as t h a t extreme p o p u l a t i o n . A CS occurs when a n y two subsets So a n d SB are selected consistent with the goal. A n y valid p r o c e d u r e R is r e q u i r e d to satisfy the c o n d i t i o n t h a t P(CSIR6 ) > P*

for all a E ~2 ,

(5.1)

where ~2 = { a : _a = ( a l , . . . ,ffk), a > O,i = 1 , . . ,k} a n d

1 / k ( k - 1) < P* < 1.

Table 3 Value of v for which P(CSIR5) = 0.90

k•

2

2.5

3

3.5

4

2 3 4 5 6 7 8 9 10

3 12 16 18 20 21 22 23 23

2 7 9 11 12 13 13 14 14

1 5 7 8 8 9 9 10 10

1 4 5 6 7 7 8 9 9

1 3 4 5 5 6 6 6 7

Simultaneousselectionof extremepopulations

825

5.1. Procedure R6 Based on sample of size n from each population, we define Si as in (4.2). As before, S[1] _< .-. _< S N are the ordered Si. We now propose procedure R6:

Put IIi in SG iffSi > clS N and,

(5.2)

put IIi in SB iff Si <_c~S[1] ,

where cl and c2 constants in the interval (0, 1) to be chosen so that the probability requirement (5.1) is satisfied.

5.2. PCS and its infimum P(CSIRo)

Pr[S(k) >_clS N, S ( 1 ) < I s [ l l ]

Il

S(1)~< min(ca

sk

c2) ( )1

= Pr[«2«Ill Tl _< ~ _<-1 «[k~rk, i = 2 , . . , k Cl 0"[i]

L a[i]

1,

T1 <_m i n ( 1 , 1 " ~ a[k~]TkI ,

k,Cl C2f O'[X]

where the T~. = v(S(i)/a[ij) are i.i.d, having a chi-square distribution with v degrees of freedom. Here v = 2(n - 1) when the #i are unknown and v = 2n when the #i are known. Thus

P(CSlN6) z ~0°°min(1/cl'l/c2)(aN/~I'])tk~lIGv(LO' [il_ k,C[k]t' 1 0"[i] ]J00/ -Gv \(c2°'S~]o' [1] /[i] x gv(S)9u(t)ds dt . It is easy to see that the infimum of P(CSIR6 ) over O is attained when a[l] . . . . . a[k]. Thus infP(CSIR6) = Q

fo°° fomin(l/cl'l/c2)t[Gv( t ) _ Gv(c2s)]k-2 × g~(s)g(t)dsdt .

(5.3)

The constants cj and c 2 should be chosen so that the right-hand side of (5.3) equals P*. When c~ = c2 = c (say), then (5.3) becomes

oo t/c infP(CSIR6)=f0

f0

t k-2 [Gv(c)-Gv(cs)] g~(s)g(t)d, d t .

(5.4)

K. Hussein and S. Panchapakesan

826

Table 4 Value of c for which P(CS[R6) = 0.90 v

2

3

2 3 4 5 6 7 8 9 10

0.337894 0.145901 0.347621 0.223001 0.387095 0.297640 0.391154 0.310126 0.449899 0.345762 0.489621 0.375892 0.495401 0.412057 0.512398 0.420378 0.552579 0.446297

4

5

6

7

8

9

10

0.117621 0.160109 0.273561 0.279839 0.310145 0.350222 0.362197 0.385589 0.410275

0.092791 0.159208 0.229762 0.249804 0.280118 0.330198 0.335914 0.364027 0.384994

0.089001 0.139973 0.190138 0.230986 0.271098 0.297981 0.324075 0.345776 0.371213

0.070972 0.130120 0.182232 0.220017 0.259953 0.285421 0.303376 0.327791 0.347912

0.069971 0.121193 0.169871 0.211091 0.241101 0.271389 0.292754 0.301972 0.319781

0.060778 0.097821 0.160001 0.209011 0.230579 0.265470 0.277651 0.293496 0.300192

0.060012 0.082165 0.149809 0.200359 0.221239 0.257011 0.260012 0.275429 0.297312

F o r given v, k a n d P*, we solve for c b y e q u a t i n g the r i g h t - h a n d side o f (5.4) to P*. Table 4 gives the values o f c = c ( k , v , P * ) for k = 2 ( 1 ) 1 0 , v = 2 ( 1 ) 1 0 a n d P* = 0.9O.

5.3. Expected sizes of the selected subsets As p e r f o r m a n c e characteristics o f p r o c e d u r e R6, we need to evaluate the s u p r e m a o f E(B) a n d E(G), the expected sizes o f the selected subsets B a n d G, respectively. W e first consider E(G). T h e n k

E(<

Pr[the p o p u l a t i o n associated with ~r[i] is included in G]

= i--I k

--E PrIS(i) _> eiS(I), j

= 1 , . . , k~ j vk i]

i--I

k

l Gj < [ i _l - - - - T ~ , j = l , . . . , k , j ~ i Pr [T Z cl a[j] i=1

i=1

[~'~"

1

k.C1 0"[/] /

k j¢i The expression in (5.5) is the same as t h a t o f the expected size o f the selected subset for the p r o c e d u r e o f G u p t a (1963) whose selection goal is different. By T h e o r e m 4 a n d C o r o l l a r y 1 o f G u p t a (1963), the s u p r e m u m o f E(G) is a t t a i n e d when «[1] . . . . . a3.J. T h u s supE(G) = k /2

:o~ Gk-1 (~1) gv(t)dt

.

Similarly, it can be s h o w n t h a t supE(B) = k g2

/0~ [1 -

Gv(c2t)]k-lg~(t)dt .

Simultaneous selection of extreme populations

827

6. Selection based on Type-Il censored samplc In this section, we will consider selection from k two-parameter exponential populations where the samples arise out of life-testing experiments under Type-Il censoring. We put n items from each population on test and terminate the experiment at the rth (1 _< r < n) failure. When r = n, we have the single-stage procedures based on complete samples which have been discussed in the preceding sections. Let X~[1] < --. _< X/[d denote the r observed failure times for the experiment associated with IIi, i = 1 , . . , k . Let Y//=Xi[1] and Y[1] ~ ' ' " ~ Y[k l denote the ordered Yi. We consider selection in terms of the guaranteed life (location parameter) as well as the scale parameter.

6.1. Selection in terms o f the guaranteed life: I Z approach We assume that al . . . . . erk = ~r (known). Our proposed procedure R'1 is same as R1 defined in (2.2). In other words, the sampling rule is different but the decision rule is the same. We note that the derivation of the PCS and its infimum over •I(PZ) are still the same. Thus the minimum sample size (the number of units to be put on test) is the same as for R1. This does not depend on the value of r. This is intuitively clear because we need only the smallest observation in each sample for the procedure. We may as well take r = 1.

6.2. Selection in terms o f the guaranteed life: S S approach We again assume that ~rl . . . . . ak = a (say). When a is known, out procedure R~ uses the same decision rule as of procedure R3 defined in (3.1). The derivation of the PCS and its infimum carry over. Thus the equation for finding the constants cl and c2 is the same. When cl = c2 = c, we get the same solution for c. Again, the solution does not depend on r and we may as well take r = 1. Also, the discussion regarding the expected size of the selected subsets carries over. When a is unknown, we use its unbiased estimator = -r

~/-] -- X/[1] ) ~- (n -- F)(X/[r] -- X/[1]

It is known that v(r/a with v = 2k(r - 1) has a chi-square distribution with v degrees of freedom. Also, 6 is independent of the Y/. Our procedure R~ has the same decision rule a s R4 defined in (3.7). The equation for determining the constants bi and b2 is (3.9) with v = 2k(r - 1) instead of 2k(n - 1). When bs = b2 = b, we get the value of b by solving (3.10). As pointed out in the case of R4, a table of values of b for selected values of k, v and P* is not available at this time. We note that the constant b depends on v only through k and r. Of course, we need r _> 2 because of estimating a.

K. Hussein and S. Panchapakesan

828

6.3. Selection in terms of the scale parameter: I Z approach We assume that the #i are all k n o w n or all unknown, not necessarily equal. F o r the T y p e - I I censored sample f r o m IIi (1 < i < k), define ~] - Xi[i]) + (n - r) (Xi[r] - X/[1]

if #i is unknown,

It is k n o w n that T~ = v6i/«[i], i = 1 , . . ,k, are i.i.d, having a chi-square distribution with v degrees of freedom, where v = 2r when the #i are k n o w n and v = 2(r - 1) when the #i are u n k n o w n . Also, the di are independent of the ~[1]. The P Z is (22 defined in Section 4 for the complete sample case. Our procedure R~ is same as R» defined in (4.3) with S i replaced by di. Let ~[I] ~--- "'" ~ Ô-[k] denote the ordered ~[i]. Then the p r o p o s e d procedure is R~:

Select the populations that yield ä[1] and ô-N as those associated with «[1] and «[k], respectively .

The derivation of the PCS and its infimum over ~2 are the same as for procedure Rs. The m i n i m u m value of v (therefore, of r) is determined such that the righthand side of (4.4) [(4.5) when 6~ = 6~ = 6*] is at least P*. We note that we can have any n _> r to start with. In the case of u n k n o w n #i, we need r > 2.

6.4. Selection in terms of the scale parameter." SS approach In this case, out procedure R~ is same as R6 defined in (5.2) with Si replaced by äi. In other words, we p r o p o s e R~:

Put IIi in SG iff äi >_ e~ä[k] and 1 ^ put IIi in SB iff äi _< ~«[1] ,

where Cl and e2 are constants in the interval (0,1) to be chosen so that the probability requirement (5.1) is satisfied. With derivations parallel to the case of R 6 , the infimum of P(CSIR~) is given by (5.3) with v equal 2r or 2 ( r - 1) depending on whether the #i are all k n o w n or u n k n o w n , respectively. W h e n cl = c2 = c, then the values o f c for selected values o f k and v, and P* -- 0.90 are given in Table 4. We again note that the constant c is the same for any n > r and we need r > 2 when the #i are unknown.

Simultaneous selection of extreme populations

829

7. Coneluding remarks I n o u r discussion o f several selection p r o c e d u r e s in the preceding sections, we have n o t c o n s i d e r e d some aspects o f these procedures. F o r example, SS p r o c e d u r e R3 involves two c o n s t a n t s Cl a n d c » W h e n we t a k e cl = c2 = c, the s o l u t i o n is unique. H o w e v e r , w h e n cl a n d c2 are n o t necessarily equal, there could be m a n y pairs o f (Cl, c2) satisfying the p r o b a b i l i t y requirement; in which case, one has to investigate the o p t i m a l s o l u t i o n b y considering, for example, E ( G ) + E ( B ) . These c o m m e n t s also a p p l y to other SS p r o c e d u r e s discussed earlier. In o u r f o r m u l a t i o n o f SS, o u r goal is to select two n o n - e m p t y subsets, B c o n t a i n i n g the lower extreme p o p u l a t i o n a n d G c o n t a i n i n g the other. It is n o t r e q u i r e d t h a t these two subsets should be n o n - o v e r l a p p i n g . The p r o b l e m to be investigated is p a r t i t i o n i n g o f the k p o p u l a t i o n s into three subsets B, M, G so t h a t B a n d G include the lower a n d the u p p e r extreme p o p u l a t i o n s , respectively, a n d M, the m i d d l e subset, c o u l d p o s s i b l y be empty. Some efforts have been m a d e in this direction b y the a u t h o r s b u t w i t h o u t m u c h success at this time. O n e can consider progressive T y p e - I I censored s a m p l i n g which is a generalization o f the s t a n d a r d T y p e - I I censoring c o n s i d e r e d here; see A g g a r w a l a a n d B a l a k r i s h n a n (1998) for some p r o p e r t i e s o f progressively c e n s o r e d - o r d e r statistics a n d related references. W e do n o t see a n y technical difficulties in extending o u r results u n d e r progressive censoring.

References Aggarwala, R. and N. Balakrishnan (1998). Some properties of progressive censored order statistics from arbitrary and uniform distributions with applications to inference and simulation. J. Stat. Plan. Inf 70, 35 49. Bechhofer, R. E. (1954). A single-sample multiple decision procedure for ranking means of normal populations with known variances. Arm. Math. Stat. 25, 16 39. Bechhofer, R. E., T. J. Santner and D. M. Goldsman (1995). Design and Analysis of Experimentsfor Statistical Selection, Screening and Multiple Comparisons. Wiley, New York. Dhariyal, I. D. and N. Misra (1994). Simultaneous selection of extreme populations: a Bayesian approach. Comm. Stat. Theor. Meth. 23, 1993~027. Gibbons, J. D., I. Olkin and M. Sobel (t977). Selecting and Ordering Populations: A New Statistical Methodology. Wiley, New York; Reprint: SIAM, Phlladelphia, 1999. Gupta, S. S. (1956). On a decision rule for a problem in ranking means. Ph.D. Dissertation, Institute of Statistics, University of North Carolina, Chapel Hi11, North Carolina. Gupta, S. S. (1963). On a selection and ranking procedure for gamma populations. Ann. Inst. Statist. Math. 14, 199-216. Gupta, S. S. and S. Panchapakesan (1979). Multiple Deeision Proeedures: Theory and Methodology of Selecting and Ranking Populations. Wiley, New York. Mishra, S. N. (1986). Simultaneous selection of extreme population means: indifference zone formulation. Am. J. Math. Management Sci. 6, 131 142. Mishra, S. N. and E. J. Dudewicz (1987). Simultaneous selection of extreme populations: A subset selection approach. Biomet. J. 4, 471-483. Misra, N. and I. D. Dhariyal (1994). Simultaneous selection of extreme populations: Bayes-P* and minimax rules. Comm. Star. - Theor. Meth. 23, 1963-1992.

830

K. Hussein and S. Panchapakesan

Ofosu, J. B. (1972). On selection procedures for exponential distribution. Bull. Math. Stat. 16, 1-9. Panchapakesan, S. (1995). Selection and ranking procedures. In The Exponential Distribution." Theory Methods and Applications, pp. 259378 (Eds. N. Balakrishnan and A. P. Basu). Gordon and Breach, New York, New Jersey.

Subject Index

A consecutive k-out-of-n: F(G) 239 A priori distribution 487 Accelerated life test 623 lifetime tests 623 test 611 tests 458 Acceleration factor 626 function 625 Acceptance constant 463 number 458 sampling plans 415 Action space 486 Active redundancy 302 Adaptive rejection sampling 532 Aging 69 properties 307, 361 Alternating renewal process 806 Approximate estimators 431 Approximations 394 Arbitrary continuons distributions 386 Arrhenius function 612 law 626 Asymptotic covariance matrix 462 Attribute sampling 458 Audit laboratory testing 693 Availability 2, 7 Bartlett's test 322 Bathtub failure rate functions Bathtub-shaped 69 Bayes 529 decision function 487 risk 487 Bayesian 523, 527 530, 538 analysis 634, 733 models 708, 724 robustness 496

Best linear invariant estimation 400 Best linear unbiased estimates 432 estimation 336, 396 estimators 465 Beta 140 process 785 BFR 73 Blas 431 Binary multistate system 7 Bivariate normal distribution 641 Weibull distribution 652 Blackwell renewal theorem 17 BLUE 336, 344 Bounded relative error property 35 Burn-in 70, 185, 573 Burr 307 Cause-of-failure 529 Cause-specific 534 hazards 520 probability density function 520 Censored data 685 Weibull data 518 Censoring 669 Central limit theorem 18 Change point 74 Chapman-Kolmogorov equation 3 Characteristic Iife 612 Characterization 105, 106, 117, 121 Circular connected X-out-of-(n, m): F(G) lattice 240 Circular consecutive k-out-of-n: F systems 250 Closure properties 115, 124, 135 Coherent system 216 Common location parameter 351 Competing failure modes 604

189

831

832 risk 77 risk model 005 risks 523, 535 Component-level tests 659 Compound Gompertz 140 Weibull 140 Computer algebra systems 283 Conditional probability generating functions 282 Conditioning methods 28 Confidence intervals 790, 804 limits 616 Connected (r,s)-or-(s, r)-out-of-(n, m): F(6) 241 lattice system 241 Consecutive k-out-of-n systems 238 k-out-of-n: F 238 system 281 Consistency 781 Constant-stress modei 612 Construction techniques 81 Consumer's risk 460 Continuous time Markov chain 2 Correct selection (CS) 813 Correlation methods 26 Countable mixture 144 Counting process 43 Cox model 626 proportional hazard model 729 Cumulative exposure 614 exposure model 623, 627 TTT-statistic 476 Cutset 292 Decision procedures 485 Degradation data 458 Degrees of censoring 460 Demonstration testing 682 Dependent competing risks 519 Detection ofwearont 807 Deterioration 44 DFR 361 Diagnostic probability 500 Directed Markov distribution 284 tree 283 Dirichlet process 785

Subject index Discrete bathtub shape 95 distribution theory 282 time Markov chains 2 Z2 distribution 791 Distributions of stopping times 789, 792, 798 Dominance relations 105-107, 114, 117 Double proofload design 646 Doubly truncated exponential distribution 389 Duane model 715 Dynamic importance sampling measure 33 stresses 611 Embedded Markov chain 15 method 38 Equilibrium distribution in higher dimension 106, 124, 135 Ergodic Markov chain 6 Expected Fisher information 438 Expected stopping time 800, 803 times 789 ExponentiaI 630 distribution 303, 377, 460, 783, 789, 792 distributions 307, 330 power 78 Exponentiated Weibull 79 Extreme-value distribution 462 distributions 383 Eyring law 627 Eyring-Weibull model 613 Failure intensity function 709, 711,722 mode 538 modes 523 process 708 rate 69 rate ordering 365 truncated 49 Failure-step stress test 624 Field life 611 reliability 585 Finite mixture 144 Finite-state Markov chain 245 Fisher information 438,642 matrix 463, 710 Follow-up survey 587

Subject index Freund's bivariate exponential distribution 328 Full conditional 531 Functional central limit theorem

19

Gamma distribution 196, 319 mixture 82 Gauss hypergeometric distribution 359 Generalized beta distribution 356, 358 Eyring relationship 612 order statistics 311, 314 Pareto distribution 307 Generating function 296 functions 267, 799 Gibbs sampler 530 Goel-Okumoto model 711 Gompertz 140 Goodness of fit 538 tests 57 Graphical interpretation 716, 717 Hartley's test 323 Hazard function 200, 207, 310, 777, 781 Hazard measure 199201, 209 211 order 203 ordering 199, 200, 203206 Hazard rate 199, 208210, 212 function 142 functions 189 order 202 ordering 199, 203,210 Heterogeneous populations 140, 185 Hierarchical structures 227 Higher order equilibrium distribution 111, 125, 132 Hm ordering 206, 208 Homogeneous Markov chain 3 populations 140 Hr ordering 205 Hypergeometric function 356 Identifiability 146 Identification probability 499 IFR 361 IFRA 362 Importance sampling 2, 29, 32 Indicator of the event 266 Indifference zone 813 Inference 373

833

Intensity function 43 Inter-event times 474 Interval censored 538 censoring 533 Inverse Gaussian-Weibull mixed model (IG-W) 168 power function 612 Power Law 626 transform method 23 Inverted gamma distribution 318 Jeffrey's prior 528, 530 Jelinski 721 )th component in the mixture

I44

k-out-of-n system 215 systems 302 k-out-of-n: G systems 238 ( k - 1)-step Markov dependence kth record values 314

247

Laplac~Stieltjes transform 14 Laplace's trend test 478 Latent variables 530, 533 Law of large numbers 17 Least squares median ranks estimator 407 Lieberman and Resnikoff (1955) procedure 459 Life distribution 611 distributions 140 testing 139, 373 Lifetime 292 Likelihood function 779 Likelihood ratio 26 confidence limits 617 ordering 366 Linear connected X-out-of-(n, m): F(G) lattice system 240 Linear consecutive k-out-of-n: F systems 241 Linear-hazard-rate 196 Location-scale 530, 534 distribution 459 family 306, 326 Log-concave 533 Log-concavity 531 Logistic distribution 431 Lognormal distribution 620 Log-power model 717

834 Lomax distributions 307, 333 Loss of utility 486 Maintainability 7 Majorization 216 Marginal count data 595 Markov chain Monte Carlo (MCMC) 529, 733 Monte Carlo techniques 782 of first-order 249 Markov model 708, 721 Markov renewal equation 16 process 15 Markov trees 284 Masked 523, 524, 532, 535 cause-of-failure 524, 536, 538 group 499, 500 Masking 524, 529, 534 group g 499 probability 500 Mathematical programming 661 Maximum likelihood 651,709 fitting 615 estimation 333, 336, 395, 630 estimator 783, 784 estimators (MLEs) 306, 317, 431,460 (ML) estimates 681 MCMC 530-533 Mean cumulative function (MCF) 693, 694 down time 8 Mean residual life 70, 222 function 199, 200, 206 ordering 199, 200 Mean squared error 431 Mean time between failure 8 to failure 8 to repair 8 Mean up time 8 Mean value function 709 Measure Specific Dynamic Importance Sampling 34 Method of competing risk 36 Miner's rule 615, 620 Minimal cut 266 Minimal repair 46, 471 scheme 316 Minimum random subset 525 Mission time 92 Mixing proportion 144 Mixture 143 of distributions 186

Subject index Mixtures of exponential components 157 Gompertz components 162 Inverse Gaussian components 153 lognormal components 149 normal components 147 Rayleigh components 159 Weibull components 160 ML estimates 616 MLE 306, 336, 344, 354 Model of additive accumulation of damages 626 Modified semi-Poisson model 172 Modular decomposition 220 Modulated power law process 66 Moments 373 Monte Carlo methods 2 MRL function 222 order 202 ordering 205 MRS 533 Multiple step-stress test 624 systems 60 Musa-Okumoto model 714 New products 545 No-data problem 489 Node criticality 217 Non-homogeneous discrete time Markov chain 9 Poisson process 315, 316, 471, 708 Non-identifiability 530, 535 Non-monotonic aging classes 91 Nonparametric 653 Normal distribution 409 Norma~exponential mixed model 166 Observed Fisher information 438 OC curve 459~461 One-dimensional case 239 One-parameter models 396 One-sided sampling plans 420 Operating characteristic 800 characteristics 789 Optimal censoring schemes 373 test plans 630, 637 Optimization 663 Order statistic 303

Subject index statistics 200, 206,212, 295, 311,314, 750, 754, 772 Parallel system 13 systems 670 Parametric empirical Bayes models 65 Pareto distribution 140, 307, 333, 382 Partial masking 525 ordering 199, 200, 203, 307 Partially accelerated life test 624 Pearson distributions 332 I distributions 307 Perfect repair 46 Permutation equivalent 217 Pfeifer's records 311,314 Phase-type distributions 13 Piecewise exponential distributions 190 Pivotal quantities 431 quantity 690 Poincaré equations 228 Poisson binomiaI variables 216 process 43, 783, 790, 792 Population cdf 617 reliability 611 Posterior distribution 781,784 Power-law process 43, 475, 480 Preference zone 814 Prior distribution 782, 784 Probability coverages 431 generating functions 282 plot 750, 772 Producers risk 459 Progressive censoring 373 Type II censored order statistics 311 Type II censoring 314, 432 Type-I right censoring 432 Progressively Type-II censored samples 431 Proof load 641 Proportional closeness 790, 791,793, 803 failure rate 221 MRL 224 Proportional hazard model 195, 626 models 512 Pseudo-likelihood 599

Quality control

835 569

Rare event estimation 2 Reconstruction of reliability polynomial 233 Record valnes 200, 311,314 Recursive computation 388 Redundancy 568 Redundant cold standby system 13 systems 302 Regenerative simulation 33 Regrets 487 Relative error 29 Relayed bipolar consecutive k-out-of-n: F system 239 consecutive k-out-of-n: F 239 unipolar consecutive k-out-of-n: F system 239 Release time modeling 730 Relevation transform 316 Reliability 2, 7, 45, 139, 227, 281, 373,457 improvement 44, 568 testing 789, 802 tests 659 Reliability polynomial 227 of monotone structures 228 Renewal process 46 repair 46 Repair 570 Repairable system 131, 132, 790 systems 45, 105, 698, 699, 805 Residual lifetime 69 Reverse hazard measure 210 rate 210 Reverse time hazard function 594 Roller-coaster 99 Sample size 463 determination 679 Sampling acceptance 789 Scale mixtures 196 Scaled TTT-transform 470, 472, 473 Schick 723 Sectional models 74, 77 Selection and ranking 813 Semi-Markov model 2 processes 2 Sequential

836

Subject index

k-out-of-n systems 303,304 estimation 790, 795, 803 probability ratio test 797 Sequential-order statistics 303, 309, 311 Series system 13 systems 666 Shape parameter 612 Shock models 315 Simple step-stress test 624 Simulation 373, 682 Simultaneous selection of extreme populations 813 Software reliability 707, 733, 808 growth model 709, 720 models 707 Specification limits 459 S-shaped models 713 St ordering 206 Stabile stress distribution 619 State of nature 485 Stationary Markov chain 246 Step-stress 611,623 testing 623 Stochastic order 202 ordering 200, 204, 365 Stopping variable 791 Stratified sampling 28 Stress corrosion crack initiation 680 function 625 profiles 611,614 Stres~strength models 354 Strong Markov property 792 Subset selection (SS) 813 Sums ofindependent random variable 200,206 Supermajorized 216 Survival analysis 789 Symmetric proof load design 649 Symmetry assumption 501,526 System reliability 693, 694, 704 System-based component test plans 659 System-level tests 659 Taboo probability 245 Tail-ordering 366 Taylor approximation 463 Test plans 621 Tests of exponentiality 83 Time stress pattern 624 truncated 49

Time-step stress test 624 Time-varying stresses 611 TTT-plot 85, 469, 472 Two-dimensional consecutive k-out-of-n: F systems 240, 268 Two-parameter exponential distribution 462 models 401 Two-sided sampling plans 424 Two-stage optimization 662 Type I censoring 432, 457, 624 Type-II censoring 432, 458,463,024 progressive censoring 458 UMVUE 336, 344, 354 Uniform distribution 307, 379 Uniformly minimum variance unbiased estimation 336 estimators (UMVUEs) 306 Upside-down bathtub 88 Used products 549 U-shape 70 Variables-sampling plans 458,461 Variance reduction methods 2, 25 Warrant policies combination 547 cumulative 548 extended 551 FRW 546 PRW 547 RIW 549 Warranty cost basis 556 costs 551, 593 data 577, 585 reserves 575 servicing 574 Weak submajorization 216 Wearout 72, 807 Weibull 140 distribution 195, 307, 336, 383, 462, 600, 612, 681,715, 749, 750, 753, 772, 781 distributions 331 lifetime 631 mixture 82 nonhomogeneous Poisson process 44 process 43, 715 Weibull-exponential mixed model 172 Weight function 213, 214

Subject index Weighted least squares 431 estimators 433 Weighted random variable 213, 214

Weinman multivariate exponentiaI distribution 328 "Worth" of the experiment or test 487

837

Handbook of Statistics Contents of Previous Volumes

Volume 1. Analysis of Variance Edited by P. R. Krishnaiah 1980 xviii + 1002 pp.

1. Estimation of Variance Components by C. R. Rao and J. Kleffe 2. Multivariate Analysis of Variance of Repeated Measurements by N. H. Timm 3. Growth Curve Analysis by S. Geisser 4. Bayesian Inference in MANOVA by S. J. Press 5. Graphical Methods for Internal Comparisons in ANOVA and MANOVA by R. Gnanadesikan 6. Monotonicity and Unbiasedness Properties of ANOVA and MANOVA Tests by S. Das Gupta 7. Robustness of ANOVA and MANOVA Test Procedures by P. K. Ito 8. Analysis of Variance and Problem under Time Series Models by D. R. Brillinger 9. Tests of Univariate and Multivariate Normality by K. V. Mardia 10. Transformations to Normality by G. Kaskey, B. Kolman, P. R. Krishnaiah and L. Steinberg 11. ANOVA and MANOVA: Models for Categorical Data by V. P. Bhapkar 12. Inference and the Structural Model for ANOVA and MANOVA by D. A. S. Fraser 13. Inference Based on Conditionally Specified ANOVA Models Incorporating Preliminary Testing by T. A. Bancroft and C. -P. Han 14. Quadratic Forms in Normal Variables by C. G. Khatri 15. Generalized Inverse of Matrices and Applications to Linear Models by S. K. Mitra 16. Likelihood Ratio Tests for Mean Vectors and Covariance Matrices by P. R. Krishnaiah and J. C. Lee

839

840 17. 18. 19. 20. 21. 22. 23. 24. 25.

Contents of previous volumes

Assessing Dimensionality in Multivariate Regression by A. J. Izenman Parameter Estimation in Nonlinear Regression Models by H. Bunke Early History of Multiple Comparison Tests by H. L. Harter Representations of Simultaneous Pairwise Comparisons by A. R. Sampson Simultaneous Test Procedures for Mean Vectors and Covariance Matrices by P. R. Krishnaiah, G. S. Mudholkar and P. Subbiah Nonparametric Simultaneous Inference for Some MANOVA Models by P. K. Sen Comparison of Some Computer Programs for Univariate and Multivariate Analysis of Variance by R. D. Bock and D. Brandt Computations of Some Multivariate Distributions by P. R. Krishnaiah Inference on the Structure of Interaction in Two-Way Classification Model by P. R. Krishnaiah and M. Yochmowitz

Volume 2. Classification, Pattern Recognition and Reduction of Dimensionality Edited by P. R. Krishnaiah and L. N. Kanal 1982 xxii + 903 pp.

1. Discriminant Analysis for Time Series by R. H. Shumway 2. Optimum Rules for Classification into Two Multivariate Normal Populations with the Same Covariance Matrix by S. Das Gupta 3. Large Sample Approximations and Asymptotic Expansions of Classification Statistics by M. Siotani 4. Bayesian Discrimination by S. Geisser 5. Classification of Growth Curves by J. C. Lee 6. Nonparametric Classification by J. D. Broffitt 7. Logistic Discrimination by J. A. Anderson 8. Nearest Neighbor Methods in Discrimination by L. Devroye and T. J. Wagner 9. The Classification and Mixture Maximum Likelihood Approaches to Cluster Analysis by G. J. McLachlan 10. Graphical Techniques for Multivariate Data and for Clustering by J. M. Chambers and B. Kleiner 11. Cluster Analysis Software by R. K. Blashfield, M. S. Aldenderfer and L. C. Morey 12. Single-link Clustering Algorithms by F. J. Rohlf 13. Theory of Multidimensional Scaling by J. de Leeuw and W. Heiser 14. Multidimensional Scaling and its Application by M. Wish and J. D. Carroll 15. Intrinsic Dimensionality Extraction by K. Fukunaga

Contents of previous voßmes

841

16. Structural Methods in Image Analysis and Recognition by L. N. Kanal, B. A. Lambird and D. Lavine 17. Image Models by N. Ahuja and A. Rosenfeld 18. Image Texture Survey by R. M. Haralick 19. Applications of Stochastic Languages by K. S. Fu 20. A Unifying Viewpoint on Pattern Recognition by J. C. Simon, E. Backer and J. Sallentin 21. Logical Functions in the Problems of Empirical Prediction by G. S. Lbov 22. Inference and Data Tables and Missing Values by N. G. Zagoruiko and V. N. Yolkina 23. Recognition of Electrocardiographic Patterns by J. H. van Bemmel 24. Waveform Parsing Systems by G. C. Stockman 25. Continuous Speech Recognition: Statistical Methods by F. Jelinek, R. L. Mercer and L. R. Bahl 26. Applications of Pattern Recognition in Radar by A. A. Grometstein and W. H. Schoendorf 27. White Blood Cell Recognition by E. S. Gelsema and G. H. Landweerd 28. Pattern Recognition Techniques for Remote Sensing Applications by P. H. Swain 29. Optical Character Recognition Theory and Practice by G. Nagy 30. Computer and Statistical Considerations for Oil Spill Identification by Y. T. Chinen and T. J. Killeen 31. Pattern Recognition in Chemistry by B. R. Kowalski and S. Wold 32. Covariance Matrix Representation and Object-Predicate Symmetry by T. Kaminuma, S. Tomita and S. Watanabe 33. Multivariate Morphometrics by R. A. Reyment 34. Multivariate Analysis with Latent Variables by P. M. Bentler and D. G. Weeks 35. Use of Distance Measures, Information Measures and Error Bounds in Feature Evaluation by M. Ben-Bassat 36. Topics in Measurement Selection by J. M. Van Campenhout 37. Selection of Variables Under Univariate Regression Models by P. R. Krishnaiah 38. On the Selection of Variables Under Regression Models Using Krishnaiah's Finite Intersection Tests by J. L Schmidhammer 39. Dimensionality and Sample Size Considerations in Pattern Recognition Practice by A. K. Jain and B. Chandrasekaran 40. Selecting Variables in Discriminant Analysis for Improving upon Classical Procedures by W. Schaafsma 41. Selection of Variables in Discriminant Analysis by P. R. Krishnaiah

842

Contents of previous volumes

Volume 3. Time Series in the Frequency D o m a i n Edited by D. R. Brillinger and P. R. Krishnaiah 1983 xiv + 485 pp.

1. Wiener Filtering (with emphasis on frequency-domain approaches) by R. J. Bhansali and D. Karavellas 2. The Finite Fourier Transform of a Stationary Process by D. R. Brillinger 3. Seasonal and Calender Adjustment by W. S. Cleveland 4. Optimal Inference in the Frequency Domain by R. B. Davies 5. Applications of Spectral Analysis in Econometrics by C. W. J. Granger and R. Engle 6. Signal Estimation by E. J. Hannan 7. Complex Demodulation: Some Theory and Applications by T. Hasan 8. Estimating the Gain of a Linear Filter from Noisy Data by M. J. Hinich 9. A Spectral Analysis Primer by L. H. Koopmans 10. Robust-Resistant Spectral Analysis by R. D. Martin 11. Autoregressive Spectral Estimation by E. Parzen 12. Threshold Autoregression and Some Frequency-Domain Characteristics by J. Pemberton and H. Tong 13. The Frequency-Domain Approach to the Analysis of Closed-Loop Systems by M. B. Priestley 14. The Bispectral Analysis of Nonlinear Stationary Time Series with Reference to Bilinear Time-Series Models by T. Subba Rao 15. Frequency-Domain Analysis of Multidimensional Time-Series Data by E. A. Robinson 16. Review of Various Approaches to Power Spectrum Estimation by P. M. Robinson 17. Cumulants and Cumulant Spectral Spectra by M. Rosenblatt 18. Replicated Time-Series Regression: An Approach to Signal Estimation and Detection by R. H. Shumway 19. Computer Programming of Spectrum Estimation by T. Thrall 20. Likelihood Ratio Tests on Covariance Matrices and Mean Vectors of Complex Multivariate Normal Populations and their Applications in Time Series by P. R. Krishnaiah, J. C. Lee and T. C. Chang

Contents of previous volumes

843

Volume 4. Nonparametric Methods Edited by P. R. Krishnaiah and P. K. Sen 1984 xx + 968 pp.

1. Randomization Procedures by C. B. Bell and P. K. Sen 2. Univariate and Multivariate Mutisample Location and Scale Tests by V. P. Bhapkar 3. Hypothesis of Symmetry by M. Hugkovä 4. Measures of Dependence by K. Joag-Dev 5. Tests of Randomness against Trend or Serial Correlations by G. K. Bhattacharyya 6. Combination of Independent Tests by J. L. Folks 7. Combinatorics by L. Takäcs 8. Rank Statistics and Limit Theorems by M. Ghosh 9. Asymptotic Comparison of Tests A Review by K. Singh 10. Nonparametric Methods in Two-Way Layouts by D. Quade 11. Rank Tests in Linear Models by J. N. Adichie 12. On the Use of Rank Tests and Estimates in the Linear Model by J. C. Aubuchon and T. P. Hettmansperger 13. Nonparametric Preliminary Test Inference by A. K. Md. E. Saleh and P. K. Sen 14. Paired Comparisons: Some Basic Procedures and Examples by R. A. Bradley 15. Restricted Alternatives by S. K. Chatterjee 16. Adaptive Methods by M. Hugkovä 17. Order Statistics by J. Galambos 18. Induced Order Statistics: Theory and Applications by P. K. Bhattacharya 19. Empirical Distribution Function by E. Csäki 20. Invariance Principles for Empirical Processes by M. Csörgö 21. M-, L- and R-estimators by J. Jureökovä 22. Nonparametric Sequantial Estimation by P. K. Sen 23. Stochastic Approximation by V. Dupa6 24. Density Estimation by P. Révész 25. Censored Data by A. P. Basu 26. Tests for Exponentiality by K. A. Doksum and B. S. Yandell 27. Nonparametric Concepts and Methods in Reliability by M. Hollander and F. Proschan 28. Sequential Nonparametric Tests by U. Müller-Funk 29. Nonparametric Procedures for some Miscellaneous Problems by P. K. Sen 30. Minimum Distance Procedures by R. Beran 31. Nonparametric Methods in Directional Data Analysis by S. R. Jammalamadaka 32. Application of Nonparametric Statistics to Cancer Data by H. S. Wieand

844

Contents of previous volumes

33. Nonparametric Frequentist Proposals for Monitoring Comparative Survival Studies by M. Gail 34. Meterological Applications of Permutation Techniques based on Distance Functions by P. W. Mielke, Jr. 35. Categorical Data Problems Using Information Theoretic Approach by S. Kullback and J. C. Keegel 36. Tables for Order Statistics by P. R. Krishnaiah and P. K. Sen 37. Selected Tables for Nonparametric Statistics by P. K. Sen and P. R. Krishnaiah

Volume 5. Time Series in the Time D o m a i n Edited by E. J. Hannan, P. R. Krishnaiah and M. M. R a o 1985 xiv + 490 pp.

1. Nonstationary Autoregressive Time Series by W. A. Fuller 2. Non-Linear Time Series Models and Dynamical Systems by T. Ozaki 3. Autoregressive Moving Average Models, Intervention Problems and Outlier Detection in Time Series by G. C. Tiao 4. Robustness in Time Series and Estimating ARMA Models by R. D. Martin and V. J. Yohai 5. Time Series Analysis with Unequally Spaced Data by R. H. Jones 6. Various Model Selection Techniques in Time Series Analysis by R. Shibata 7. Estimation of Parameters in Dynamical Systems by L. Ljung 8. Recursive Identification, Estimation and Control by P. Young 9. General Structure and Parametrization of ARMA and State-Space Systems and its Relation to Statistical Problems by M. Deistler 10. Harmonizable, Cramér, and Karhunen Classes of Processes by M. M. Rao 11. On Non-Stationary Time Series by C. S. K. Bhagavan 12. Harmonizable Filtering and Sampling of Time Series by D. K. Chang 13. Sampling Designs for Time Series by S. Cambanis 14. Measuring Attenuation by M. A. Cameron and P. J. Thomson 15. Speech Recognition Using LPC Distance Measures by P. J. Thomson and P. de Souza 16. Varying Coefficient Regression by D. F. Nicholls and A. R. Pagan 17. Small Samples and Large Equation Systems by H. Theil and D. G. Fiebig

Contents of previous volumes

845

Volume 6. Sampling Edited by P. R. Krishnaiah and C. R. R a o 1988 xvi + 594 pp.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22. 23. 24.

A Brief History of Random Sampling Methods by D. R. Bellhouse A First Course in Survey Sampling by T. Dalenius Optimality of Sampling Strategies by A. Chaudhuri Simple Random Sampling by P. K. Pathak On Single Stage Unequal Probability Sampling by V. P. Godambe and M. E. Thompson Systematic Sampling by D. R. Bellhouse Systematic Sampling with Illustrative Examples by M. N. Murthy and T. J. Rao Sampling in Time by D. A. Binder and M. A. Hidiroglou Bayesian Inference in Finite Populations by W. A. Ericson Inference Based on Data from Complex Sample Designs by G. Nathan Inference for Finite Population Quantiles by J. Sedransk and P. J. Smith Asymptotics in Finite Population Sampling by P. K. Sen The Technique of Replicated or Interpenetrating Samples by J. C. Koop On the Use of Models in Sampling from Finite Populations by I. Thomsen and D. Tesfu The Prediction Approach to Sampling theory by R. M. Royall Sample Survey Analysis: Analysis of Variance and Contingency Tables by D. H. Freeman, Jr. Variance Estimation in Sample Surveys by J. N. K. Rao Ratio and Regression Estimators by P. S. R. S. Rao Role and Use of Composite Sampling and Capture-Recapture Sampling in Ecological Studies by M. T. Boswell, K. P. Burnham and G. P. Patil Data-based Sampling and Model-based Estimation for Environmental Resources by G. P. Patil, G. J. Babu, R. c. Hennemuth, W. L. Meyers, M. B. Rajarshi and C. Taillie On Transect Sampling to Assess Wildlife Populations and Marine Resources by F. L. Ramsey, C. E. Gares, G. P. Patil and C. Taillie A Review of Current Survey Sampling Methods in Marketing Research (Telephone, Mall Intercept and Panel Surveys) by R. Velu and G. M. Naidu Observational Errors in Behavioural Traits of Man and their Implications for Genetics by P. V. Sukhatme Designs in Survey Sampling Avoiding Contiguous Units by A. S. Hedayat, C. R. Rao and J. Stufken

846

Contents of previous volumes

Volume 7. Quality Control and Reliability Edited by P. R. Krishnaiah and C. R. R a o 1988 xiv + 503 pp.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

Transformation of Western Style of Management by W. Edwards Deming Software Reliability by F. B. Bastani and C. V. Ramamoorthy Stress-Strength Models for Reliability by R. A. Johnson Approximate Computation of Power Generating System Reliability Indexes by M. Mazumdar Software Reliability Models by T. A. Mazzuchi and N. D. Singpurwalla Dependence Notions in Reliability Theory by N. R. Chaganty and K. Joag-dev Application of Goodness-of-Fit Tests in Reliability by H. W. Block and A. H. Moore Multivariate Nonparametric Classes in Reliability by H. W. Block and T. H. Savits Selection and Ranking Procedures in Reliability Models by S. S. Gupta and S. Panchapakesan The Impact of Reliability Theory on Some Branches of Mathematics and Statistics by P. J. Boland and F. Proschan Reliability Ideas and Applications in Economics and Social Sciences by M. C. Bhattacharjee Mean Residual Life: Theory and Applications by F. Guess and F. Proschan Life Distribution Models and Incomplete Data by R. E. Barlow and F. Proschan Piecewise Geometric Estimation of a Survival Function by G. M. Mimmack and F. Proschan Applications of Pattern Recognition in Failure Diagnosis and Quality Control by L. F. Pau Nonparametric Estimation of Density and Hazard Rate Functions when Samples are Censored by W. J. Padgett Multivariate Process Control by F. B. Alt and N. D. Smith QMP/USP-A Modern Approach to Statistical Quality Auditing by B. Hoadley Review About Estimation of Change Points by P. R. Krishnaiah and B. Q. Miao Nonparametric Methods for Changepoint Problems by M. Csögö and L. Horväth Optimal Allocation of Multistate Components by E. E1-Neweihi, F. Proschan and J. Sethuraman Weibull, Log-Weibull and Gamma Order Statistics by H. L. Harter Multivariate Exponential Distributions and their Applications in Reliability by A. P. Basu

Contents of previous volumes

847

24. Recent Developments in the Inverse Gaussian Distribution by S. Iyengar and G. Patwardhan

Volume 8. Statistical Methods in Biological and Medical Sciences Edited by C. R. R a o and R. C h a k r a b o r t y 1991 xvi + 554 pp.

1. Methods for the Inheritance of Qualitative Traits by J. Rice, R. Neuman and S. O. Moldin 2. Ascertainment Biases and their Resolution in Biological Surveys by W. J. Ewens 3. Statistical Considerations in Applications of Path Analytical in Genetic Epidemiology by D. C. Rao 4. Statistical Methods for Linkage Analysis by G. M. Lathrop and J. M. Lalouel 5. Statistical Design and Analysis of Epidemiologic Studies: Some Directions of Current Research by N. Breslow 6. Robust Classification Procedures and Their Applications to Anthropometry by N. Balakrishnan and R. S. Ambagaspitiya 7. Analysis of Population Structure: A Comparative Analysis of Different Estimators of Wright's Fixation Indices by R. Chakraborty and H. DankerHopfe 8. Estimation of Relationships from Genetic Data by E. A. Thompson 9. Measurement of Genetic Variation for Evolutionary Studies by R. Chakraborty and C. R. Rao 10. Statistical Methods for Phylogenetic Tree Reconstruction by N. Saitou 11. Statistical Models for Sex-Ratio Evolution by S. Lessard 12. Stochastic Models of Carcinogenesis by S. H. Moolgavkar 13. An Application of Score Methodology: Confidence Intervals and Tests of Fit for One-Hit-Curves by J. J. Gart 14. Kidney-Survival Analysis of IgA Nephropathy Patients: A Case Study by O. J. W. F. Kardaun 15. Confidence Bands and the Relation with Decision Analysis: Theory by O. J. W. F. Kardaun 16. Sample Size Determination in Clinical Research by J. Bock and H. Toutenburg

848

Contents of previous volumes

Volume 9. Computational Statistics Edited by C. R. R a o 1993 xix + 1045 pp.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

Algorithms by B. Kalyanasundaram Steady State Analysis of Stochastic Systems by K. Kant Parallel Computer Architectures by R. Krishnamurti and B. Narahari Database Systems by S. Lanka and S. Pal Programming Languages and Systems by S. Purushothaman and J. Seaman Algorithms and Complexity for Markov Processes by R. Varadarajan Mathematical Programming: A Computational Perspective by W. W. Hager, R. Horst and P. M. Pardalos Integer Programming by P. M. Pardalos and Y. Li Numerical Aspects of Solving Linear Lease Squares Problems by J. L. Barlow The Total Least Squares Problem by S. Van Huffel and H. Zha Construction of Reliable Maximum-Likelihood-Algorithms with Applications to Logistic and Cox Regression by D. Böhning Nonparametric Function Estimation by T. Gasser, J. Engel and B. Seifert Computation Using the QR Decomposition by C. R. Goodall The EM Algorithm by N. Laird Analysis of Ordered Categorial Data through Appropriate Scaling by C. R. Rao and P. M. Caligiuri Statistical Applications of Artificial Intelligence by W. A. Gale, D. J. Hand and A. E. Kelly Some Aspects of Natural Language Processes by A. K. Joshi Gibbs Sampling by S. F. Arnold Bootstrap Methodology by G. J. Babu and C. R. Rao The Art of Computer Generation of Random Variables by M. T. Boswell, S. D. Gore, G. P. Patil and C. Taillie Jackkuife Variance Estimation and Bias Reduction by S. Das Peddada Designing Effective Statistical Graphs by D. A. Burn Graphical Methods for Linear Models by A. S. Hadi Graphics for Time Series Analysis by H. J. Newton Graphics as Visual Language by T. Selker and A. Appel Statistical Graphics and Visualization by E. J. Wegman and D. B. Carr Multivariate Statistical Visualization by F. W. Young, R. A. Faldowski and M. M. McFarlane Graphical Methods for Process Control by T. L. Ziemer

Contents of previous volumes

849

Volume 10. Signal Processing and its Applications Edited by N. K. Bose and C. R. R a o 1993 xvii + 992 pp.

1. Signal Processing for Linear Instrumental Systems with Noise: A General Theory with Illustrations for Optical Imaging and Light Scattering Problems by M. Bertero and E. R. Pike 2. Boundary Implication Rights in Parameter Space by N. K. Bose 3. Sampling of Bandlimited Signals: Fundamental Results and Some Extensions by J. L. Brown, Jr. 4. Localization of Sources in a Sector: Algorithms and Statistical Analysis by K. Buckley and X.-L. Xu 5. The Signal Subspace Direction-of-Arrival Algorithm by J. A. Cadzow 6. Digital Differentiators by S. C. Dutta Roy and B. Kumar 7. Orthogonal Decompositions of 2D Random Fields and their Applications for 2D Spectral Estimation by J. M. Francos 8. VLSI in Signal Processing by A. Ghouse 9. Constrained Beamforming and Adaptive Algorithms by L. C. Godara 10. Bispectral Speckle Interferometry to Reconstruct Extended Objects from Turbulence-Degraded Telescope Images by D. M. Goodman, T. W. Lawrence, E. M. Johansson and J. P. Fitch 11. Multi-Dimensional Signal Processing by K. Hirano and T. Nomura 12. On the Assessment of Visual Communication by F. O. Huck, C. L. Fales, R. Alter-Gartenberg and Z. Rahman 13. VLSI Implementations of Number Theoretic Concepts with Applications in Signal Processing by G. A. Jullien, N. M. Wigley and J. Reilly 14. Decision-level Neural Net Sensor Fusion by R. Y. Levine and T. S. Khuon 15. Statistical Algorithms for Noncausal Gauss Markov Fields by J. M. F. Moura and N. Balram 16. Subspace Methods for Directions-of-Arrival Estimation by A. Paulraj, B. Ottersten, R. Roy, A. Swindlehurst, G. Xu and T. Kailath 17. Closed Form Solution to the Estimates of Directions of Arrival Using Data from an Array of Sensors by C. R. Rao and B. Zhou 18. High-Resolution Direction Finding by S. V. Schell and W. A. Gardner 19. Multiscale Signal Processing Techniques: A Review by A. H. Tewfik, M. Kim and M. Deriche 20. Sampling Theorems and Wavelets by G. G. Walter 21. Image and Video Coding Research by J. W. Woods 22. Fast Algorithms for Structured Matrices in Signal Processing by A. E. Yagle

850

Contents of previous volumes

Volume 11. Econometrics Edited by G. S. Maddala, C. R. Rao and H. D. Vinod 1993 xx + 783 pp.

1. Estimation from Endogenously Stratified Samples by S. R. Cosslett 2. Semiparametric and Nonparametric Estimation of Quantal Response Models by J. L. Horowitz 3. The Selection Problem in Econometrics and Statistics by C. F. Manski 4. General Nonparametric Regression Estimation and Testing in Econometrics by A. Ullah and H. D. Vinod 5. Simultaneous Microeconometric Models with Censored or Qualitative Dependent Variables by R. Blundell and R. J. Smith 6. Multivariate Tobit Models in Econometrics by L.-F. Lee 7. Estimation of Limited Dependent Variable Models under Rational Expectations by G. S. Maddala 8. Nonlinear Time Series and Macroeconometrics by W. A. Brock and S. M. Potter 9. Estimation, Inference and Forecasting of Time Series Subject to Changes in Time by J. D. Hamilton 10. Structural Time Series Models by A. C. Harvey and N. Shephard 11. Bayesian Testing and Testing Bayesians by J. -P. Florens and M. Mouchart 12. Pseudo-Likelihood Methods by C. Gourieroux and A. Monfort 13. Rao's Score Test: Recent Asymptotic Results by R. Mukerjee 14. On the Strong Consistency of M-Estimates in Linear Models under a General Discrepancy Function by Z. D. Bai, Z. J. Liu and C. R. Rao 15. Some Aspects of Generalized Method of Moments Estimation by A. Hall 16. Efficient Estimation of Models with Conditional Moment Restrictions by W. K. Newey 17. Generalized Method of Moments: Econometric Applications by M. Ogaki 18. Testing for Heteroskedasticity by A. R. Pagan and Y. Pak 19. Simulation Estimation Methods for Limited Dependent Variable Models by V. A. Hajivassiliou 20. Simulation Estimation for Panel Data Models with Limited Dependent Variable by M. P. Keane 21. A Perspective on Application of Bootstrap methods in Econometrics by J. Jeong and G. S. Maddala 22. Stochastic Simulations for Inference in Nonlinear Errors-in-Variables Models by R. S. Mariano and B. W. Brown 23. Bootstrap Methods: Applications in Econometrics by H. D. Vinod 24. Identifying outliers and Influential Observations in Econometric Models by S. G. Donald and G. S. Maddala 25. Statistical Aspects of Calibration in Macroeconomics by A. W. Gregory and G. W. Smith

Contents of previous volumes

851

26. Panel Data Models with Rational Expectations by K. Lahiri 27. Continuous Time Financial Models: Statistical Applications of Stochastic Processes by K. R. Sawyer

Volume 12. Environmental Statistics Edited by G. P. Patil and C. R. R a o 1994 xix + 927 pp.

1. Environmetrics: An Emerging Science by J. S. Hunter 2. A National Center for Statistical Ecology and Environmental Statistics: A Center Without Walls by G. P. Patil 3. Replicate Measurements for Data Quality and Environmental Modeling by W. Liggett 4. Design and Analysis of Composite Sampling Procedures: A Review by G. Lovison, S. D. Gore and G. P. Patil 5. Ranked Set Sampling by G. P. Patil, A. K. Sinha and C. Taillie 6. Environmental Adaptive Sampling by G. A. F. Seber and S. K. Thompson 7. Statistical Analysis of Censored Environmental Data by M. Akritas, T. Ruscitti and G. P. Patil 8. Biological Monitoring: Statistical Issues and Models by E. P. Smith 9. Environmental Sampling and Monitoring by S. V. Stehman and W. Scott Overton 10. Ecological Statistics by B. F. J. Manly 11. Forest Biometrics by H. E. Burkhart and T. G. Gregoire 12. Ecological Diversity and Forest Management by J. H. Gove, G. P. Patil, B. F. Swindel and C. Taillie 13. Ornithological Statistics by P. M. North 14. Statistical Methods in Developmental Toxicology by P. J. Catalano and L. M. Ryan 15. Environmental Biometry: Assessing Impacts of Environmental Stimuli Via Animal and Microbial Laboratory Studies by W. W. Piegorsch 16. Stochasticity in Deterministic Models by J. J. M. Bedaux and S. A. L. M. Kooijman 17. Compartmental Models of Ecological and Environmental Systems by J. H. Matis and T. E. Wehrly 18. Environmental Remote Sensing and Geographic Information Systems-Based Modeling by W. L. Myers 19. Regression Analysis of Spatially Correlated Data: The Kanawha County Health Study by C. A. Donnelly, J. H. Ware and N. M. Laird 20. Methods for Estimating Heterogeneous Spatial Covariance Functions with Environmental Applications by P. Guttorp and P. D. Sampson

852

Contents of previous volumes

21. Meta-analysis in Environmental Statistics by V. Hasselblad 22. Statistical Methods in Atmospheric Science by A. R. Solow 23. Statistics with Agricultural Pests and Environmental Impacts by L. J. Young and J. H. Young 24. A Crystal Cube for Coastal and Estuarine Degradation: Selection of Endpoints and Development of Indices for Use in Decision Making by M. T. Boswell, J. S. O'Connor and G. P. Patil 25. How Does Scientific Information in General and Statistical Information in Particular Input to the Environmental Regulatory Process? by C. R. Cothern 26. Environmental Regulatory Statistics by C. B. Davis 27. An Overview of Statistical Issues Related to Environmental Cleanup by R. Gilbert 28. Environmental Risk Estimation and Policy Decisions by H. Lacayo Jr.

Volume 13. Design and Analysis of Experiments Edited by S. G h o s h and C. R. R a o 1996 xviii + 1230 pp.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

The Design and Analysis of Clinical Trials by P. Armitage Clinical Trials in Drug Development: Some Statistical Issues by H. I. Patel Optimal Crossover Designs by J. Stufken Design and Analysis of Experiments: Nonparametric Methods with Applications to Clinical Trials by P. K. Sen Adaptive Designs for Parametric Models by S. Zacks Observational Studies and Nonrandomized Experiments by P. R. Rosenbaum Robust Design: Experiments for Improving Quality by D. M. Steinberg Analysis of Location and Dispersion Effects from Factorial Experiments with a Circular Response by C. M. Anderson Computer Experiments by J. R. Koehler and A. B. Owen A Critique of Some Aspects of Experimental Design by J. N. Srivastava Response Surface Designs by N. R. Draper and D. K. J. Lin Multiresponse Surface Methodology by A. I. Khuri Sequential Assembly of Fractions in Factorial Experiments by S. Ghosh Designs for Nonlinear and Generalized Linear Models by A. C. Atkinson and L. M. Haines Spatial Experimental Design by R. J. Martin Design of Spatial Experiments: Model Fitting and Prediction by V. V. Fedorov Design of Experiments with Selection and Ranking Goals by S. S. Gupta and S. Panchapakesan

Contents of previous volumes

853

18. Multiple Comparisons by A. C. Tamhane 19. Nonparametric Methods in Design and Analysis of Experiments by E. Brunner and M. L. Puri 20. Nonparametric Analysis of Experiments by A. M. Dean and D. A. Wolfe 21. Block and Other Designs in Agriculture by D. J. Street 22. Block Designs: Their Combinatorial and Statistical Properties by T. Calinski and S. Kageyama 23. Developments in Incomplete Block Designs for Parallel Line Bioassays by S. Gupta and R. Mukerjee 24. Row-Column Designs by K. R. Shah and B. K. Sinha 25. Nested Designs by J. P. Morgan 26. Optimal Design: Exact Theory by C. S. Cheng 27. Optimal and Efficient Treatment - Control Designs by D. Majumdar 28. Model Robust Designs by Y-J. Chang and W. I. Notz 29. Review of Optimal Bayes Designs by A. DasGupta 30. Approximate Designs for Polynomial Regression: Invariance, Admissibility, and Optimality by N. Gaffke and B. Heiligers

Volume 14. Statistical Methods in Finance Edited by G. S. M a d d a l a and C. R. R a o 1996 xvi + 733 pp.

1. Econometric Evaluation of Asset Pricing Models by W. E. Ferson and R. Jegannathan 2. Instrumental Variables Estimation of Conditional Beta Pricing Models by C. R. Harvey and C. M. Kirby 3. Semiparametric Methods for Asset Pricing Models by B. N. Lehmann 4. Modeling the Term Structure by A. R. Pagan, A. D. Hall, and V. Martin 5. Stochastic Volatility by E. Ghysels, A. C. Harvey and E. Renault 6. Stock Price Volatility by S. F. LeRoy 7. GARCH Models of Volatility by F. C. Palm 8. Forecast Evaluation and Combination by F. X. Diebold and J. A. Lopez 9. Predictable Components in Stock Returns by G. Kaul 10. Interset Rate Spreads as Predictors of Business Cycles by K. Lahiri and J. G. Wang 11. Nonlinear Time Series, Complexity Theory, and Finance by W. A. Brock and P. J. F. deLima 12. Count Data Models for Financial Data by A. C. Cameron and P. K. Trivedi 13. Financial Applications of Stable Distributions by J. H. McCulloch 14. Probability Distributions for Financial Models by J. B. McDonald 15. Bootstrap Based Tests in Financial Models by G. S. Maddala and H. Li

854

Contents of previous volumes

16. Principal Component and Factor Analyses by C. R. Rao 17. Errors in Variables Problems in Finance by G. S. Maddala and M. Nimalendran 18. Financial Applications of Artificial Neural Networks by M. Qi 19. Applications of Limited Dependent Variable Models in Finance by G. S. Maddala 20. Testing Option Pricing Models by D. S. Bates 21. Peso Problems: Their Theoretical and Empirical Implications by M. D. D. Evans 22. Modeling Market Microstructure Time Series by J. Hasbrouck 23. Statistical Methods in Tests of Portfolio Efficiency: A Synthesis by J. Shanken

Volume 15. Robust Inference Edited by G. S. M a d d a l a and C. R. Rao 1997 xviii + 698 pp.

1. Robust Inference in Multivariate Linear Regression Using Difference of Two Convex Functions as the Discrepancy Measure by Z. D. Bai, C. R. Rao and Y. H. Wu 2. Minimum Distance Estimation: The Approach Using Density-Based Distances by A. Basu, I. R. Harris and S. Basu 3. Robust Inference: The Approach Based on Inftuence Functions by M. Markatou and E. Ronchetti 4. Practical Applications of Bounded-Influence Tests by S. Heritier and M-P. Victoria-Feser 5. Introduction to Positive-Breakdown Methods by P. J. Rousseeuw 6. Outlier Identification and Robust Methods by U. Gather and C. Becker 7. Rank-Based Analysis of Linear Models by T. P. Hettmansperger, J. W. McKean and S. J. Sheather 8. Rank Tests for Linear Models by R. Koenker 9. Some Extensions in the Robust Estimation of Parameters of Exponential and Double Exponential Distributions in the Presence of Multiple Outliers by A. Childs and N. Balakrishnan 10. Outliers, Unit Roots and Robust Estimation of Nonstationary Time Series by G. S. Maddala and Y. Yin 11. Autocorrelation-Robust Inference by P. M. Robinson and C. Velasco 12. A Practitioner's Guide to Robust Covariance Matrix Estimation by W. J. den Haan and A. Levin 13. Approaches to the Robust Estimation of Mixed Models by A. H. Welsh and A. M. Richardson

Contents of previous volumes

855

14. Nonparametric Maximum Likelihood Methods by S. R. Cosslett 15. A Guide to Censored Quantile Regressions by B. Fitzenberger 16. What Can Be Learned About Population Parameters When the Data Are Contaminated by J. L. Horowitz and C. F. Manski 17. Asymptotic Representations and Interrelations of Robust Estimators and Their Applications by J. Jureökovä and P. K. Sen 18. Small Sample Asymptotics: Applications in Robustness by C. A. Field and M. A. Tingley 19. On the Fundamentals of Data Robustness by G. Maguluri and K. Singh 20. Statistical Analysis With Incomplete Data: A Selective Review by M. G. Akritas and M. P. LaValley 21. On Contamination Level and Sensitivity of Robust Tests by J. Ä. Visgek 22. Finite Sample Robustness of Tests: An Overview by T. Kariya and P. Kim 23. Future Directions by G. S. Maddala and C. R. Rao

Volume 16. Order Statistics - Theory and Methods Edited by N. Balakrishnan and C. R. Rao 1997 xix + 688 pp.

1. Order Statistics: An Introduction by N. Balakrishnan and C. R. Rao 2. Order Statistics: A Historical Perspective by H. Leon Harter and N. Balakrishnan 3. Computer Simulation of Order Statistics by Pandu R. Tadikamalla and N. Balakrishnan 4. Lorenz Ordering of Order Statistics and Record Values by Barry C. Arnold and Jose A. Villasenor 5. Stochastic Ordering of Order Statistics by Philip J. Boland, Moshe Shaked and J. George Shanthikumar 6. Bounds for Expectations of L-Estimates by Tomasz Rychlik 7. Recurrence Relations and Identities for Moments of Order Statistics by N. Balakrishnan and K. S. Sultan 8. Recent Approaches to Characterizations Based on Order Statistics and Record Values by C. R. Rao and D. N. Shanbhag 9. Characterizations of Distributions via Identically Distributed Functions of Order Statistics by Ursula Gather, Udo Kamps and Nicole Schweitzer 10. Characterizations of Distributions by Recurrence Relations and Identities for Moments of Order Statistics by Udo Kamps 11. Univariate Extreme Value Theory and Applications by Janos Galambos 12. Order Statistics: Asymptotics in Applications by Pranab Kumar Sen 13. Zero-One Laws for Large Order Statistics by R. J. Tomkins and Hong Wang 14. Some Exact Properties Of Cook's DI by D. R. Jensen and D. E. Ramirez

856

Contents of previous volumes

15. Generalized Recurrence Relations for Moments of Order Statistics from Non-Identical Pareto and Truncated Pareto Random Variables with Applications to Robustness by Aaron Childs and N. Balakrishnan 16. A Semiparametric Bootstrap for Simulating Extreme Order Statistics by Robert L. Strawderman and Daniel Zelterman 17. Approximations to Distributions of Sample Quantiles by Chunsheng Ma and John Robinson 18. Concomitants of Order Statistics by H. A. David and H. N. Nagaraja 19. A Record of Records by Valery B. Nevzorov and N. Balakrishnan 20. Weighted Sequential Empirical Type Processes with Applications to ChangePoint Problems by Barbara Szyszkowicz 21. Sequential Quantile and Bahadur-Kiefer Processes by Miklós Csörgö and Barbara Szyszkowicz

Volume 17. Order Statistics: Applications Edited by N. Balakrishnan and C. R. Rao 1998 xviii + 712 pp.

1. Order Statistics in Exponential Distribution by Asit P. Basu and Bahadur Singh 2. Higher Order Moments of Order Statistics from Exponential and Righttruncated Exponential Distributions and Applications to Life-testing Problems by N. Balakrishnan and Shanti S. Gupta 3. Log-gamma Order Statistics and Linear Estimation of Parameters by N. Balakrishnan and P. S. Chan 4. Recurrence Relations for Single and Product Moments of Order Statistics from a Generalized Logistic Distribution with Applications to Inference and Generalizations to Double Truncation by N. Balakrishnan and Rita Aggarwala 5. Order Statistics from the Type III Generalized Logistic Distribution and Applications by N. Balakrishnan and S. K. Lee 6. Estimation of Scale Parameter Based on a Fixed Set of Order Statistics by Sanat K. Sarkar and Wenjin Wang 7. Optimal Linear Inference Using Selected Order Statistics in Location-Scale Models by M. Masoom Ali and Dale Umbach 8. L-Estimation by J. R. M. Hosking 9. On Some L-estimation in Linear Regression Models by Soroush Alimoradi and A. K. Md. Ehsanes Saleh 10. The Role of Order Statistics in Estimating Threshold Parameters by A. Clifford Cohen 11. Parameter Estimation under Multiply Type-II Censoring by Fanhui Kong

Contents of previous volumes

857

12. On Some Aspects of Ranked Set Sampling in Parametric Estimation by Nora Ni Chuiv and Bimal K. Sinha 13. Some Uses of Order Statistics in Bayesian Analysis by Seymour Geisser 14. Inverse Sampling Procedures to Test for Homogeneity in a Multinomial Distribution by S. Panchapakesan, Aaron Childs, B. H. Humphrey and N. Balakrishnan 15. Prediction of Order Statistics by Kenneth S. Kaminsky and Paul I. Nelson 16. The Probability Plot: Tests of Fit Based on the Correlation Coefficient by R. A. Lockhart and M. A. Stephens 17. Distribution Assessment by Samuel Shapiro 18. Application of Order Statistics to Sampling Plans for Inspection by Variables by Helmut Schneider and Frances Barbera 19. Linear Combinations of Ordered Symmetric Observations with Applications to Visual Acuity by Marlos Viana 20. Order-Statistic Filtering and Smoothing of Time-Series: Part I by Gonzalo R. Arce, Yeong-Taeg Kim and Kenneth E. Barner 21. Order-Statistic Filtering and Smoothing of Time-Series: Part II by Kenneth E. Barner and Gonzalo R. Arce 22. Order Statistics in Image Processing by Scott T. Acton and Alan C. Bovik 23. Order Statistics Application to CFAR Radar Target Detection by R. Viswanathan

Volume 18. Bioenvironmental and Public Health Statistics Edited by P. K. Sen and C. R. R a o 2000 xxiv + 1105 pp.

1. Bioenvironment and Public Health: Statistical Perspectives by Pranab K. Sen 2. Some Examples of Random Process Environmental Data Analysis by David R. Brillinger 3. Modeling Infectious Diseases - Aids by L. Billard 4. On Some Multiplicity Problems and Multiple Comparison Procedures in Biostatistics by Yosef Hochberg and Peter H. Westfall 5. Analysis of Longitudinal Data by Julio M. Singer and Dalton F. Andrade 6. Regression Models for Survival Data by Richard A. Johnson and John P. Klein 7. Generalised Linear Models for Independent and Dependent Responses by Bahjat F. Qaqish and John S. Preisser 8. Hierarchial and Empirical Bayes Methods for Environmental Risk Assessment by Gauri Datta, Malay Ghosh and Lance A. Waller 9. Non-parametrics in Bioenvironmental and Public Health Statistics by Pranab Kumar Sen

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Contents of previous volumes

10. Estimation and Comparison of Growth and Dose-Response Curves in the Presence of Purposeful Censoring by Paul W. Stewart 11. Spatial Statistical Methods for Environmental Epidemiology by Andrew B. Lawson and Noel Cressie 12. Evaluating Diagnostic Tests in Public Health by Margaret Pepe, Wendy Leisenring and Carolyn Rutter 13. Statistical Issues in Inhalation Toxicology by E. Weller, L. Ryan and D. Dockery 14. Quantitative Potency Estimation to Measure Risk with Bioenvironmental Hazards by A. John Bailer and Walter W. Piegorsch 15. The Analysis of Case-Control Data: Epidemiologic Studies of Familial Aggregation by Nah M. Laird, Garrett M. Fitzmaurice and Ann G. Schwartz 16. Cochran-Mantel-Haenszel Techniques: Applications Involving Epidemiologic Survey Data by Daniel B. Hall, Robert F. Woolson, William R. Clarke and Martha F. Jones 17. Measurement Error Models for Environmental and Occupational Health Applications by Robert H. Lyles and Lawrence L. Kupper 18. Statistical Perspectives in Clinical Epidemiology by Shrikant I. Bangdiwala and Sergio R. Mufioz 19. ANOVA and ANOCOVA for Two-Period Crossover Trial Data: New vs. Standard by Subir Ghosh and Lisa D. Fairchild 20. Statistical Methods for Crossover Designs in Bioenvironmental and Public Health Studies by Gail E. Tudor, Gary G. Koch and Diane Catellier 21. Statistical Models for Human Reproduction by C. M. Suchindran and Helen P. Koo 22. Statistical Methods for Reproductive Risk Assessment by Sati Mazumdar, Yikang Xu, Donald R. Mattison, Nancy B. Sussman and Vincent C. Arena 23. Selection Biases of Samples and their Resolutions by Ranajit Chakraborty and C. Radhakrishna Rao 24. Genomic Sequences and Quasi-Multivariate CATANOVA by Hildete Prisco Pinheiro, Frangoise Seillier-Moiseiwitsch, Pranab Kumar Sen and Joseph Eron Jr 25. Statistical Methods for Multivariate Failure Time Data and Competing Risks by Ralph A. DeMasi 26. Bounds on Joint Survival Probabilities with Positively Dependent Competing Risks by Sanat K. Sarkar and Kalyan Ghosh 27. Modeling Multivariate Failure Time Data by Limin X. Clegg, Jianwen Cai and Pranab K. Sen 28. The Cost-Effectiveness Ratio in the Analysis of Health Care Programs by Joseph C. Gardiner, Cathy J. Bradley and Marianne Huebner 29. Quality-of-Life: Statistical Validation and Analysis An Example from a Clinical Trial by Balakrishna Hosmane, Clement Maurath and Richard Manski 30. Carcinogenic Potency: Statistical Perspectives by Anup Dewanji

Contents of previous volumes

859

31. Statistical Applications in Cardiovascular Disease by Elizabeth R. DeLong and David M. DeLong 32. Medical Informatics and Health Care Systems: Biostatistical and Epidemiologic Perspectives by J. Zvärovä 33. Methods of Establishing In Vitro-In Vivo Relationships for Modified Release Drug Products by David T. Mauger and Vernon M. Chinchilli 34. Statistics in Psychiatric Research by Sati Mazumdar, Patricia R. Houck and Charles F. Reynolds III 35. Bridging the Biostatistics-Epidemiology Gap by Lloyd J. Edwards 36. Biodiversity Measurement and Analysis by S. P. Mukherjee

Volume 19. Stochastic Processes: Theory and Methods Edited by D. N. Shanbhag and C. R. R a o 2001 xiv + 967 pp.

1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16.

Pareto Processes by Barry C. Arnold Branching Processes by K. B. Athreya and A. N. Vidyashankar Inference in Stochastic Processes by I. V. Basawa Topics in Poisson Approximation by A. D. Barbour Some Elements on Lévy Processes by Jean Bertoin Iterated Random Maps and Some Classes of Markov Processes by Rabi Bhattacharya and Edward C. Waymire Random Walk and Fluctuation Theory by N. H. Bingham A Semigroup Representation and Asymptotic Behavior of Certain Statistics of the Fisher Wrigh~Moran Coalescent by Adam Bobrowski, Marek Kimmel, Ovide Arino and Ranajit Chakraborty Continuous-Time ARMA Processes by P. J. Brockwell Record Sequences and their Applications by John Bunge and Charles M. Goldie Stochastic Networks with Product Form Equilibrium by Hans Daduna Stochastic Processes in Insurance and Finance by Paul Embrechts, Rüdiger Frey and Hansjörg Furrer Renewal Theory by D. R. Grey The Kolmogorov Isomorphism Theorem and Extensions to some Nonstationary Processes by Yüichirô Kakihara Stochastic Processes in Reliability by Masaaki Kijima, Haijun Li and Moshe Shaked On the supports of Stochastic Processes of Multiplicity Orte by A. Klopotowski and M. G. Nadkarni

860

Contents of previous volumes

17. Gaussian Processes: Inequalities, Small Ball Probabilities and Applications by W. V. Li and Q.-M. Shao 18. Point Processes and Some Related Processes by Robin K. Milne 19. Characterization and Identifiability for Stochastic Processes by B. L. S. Prakasa Rao 20. Associated Sequences and Related Inference Problems by B. L. S. Prakasa Rao and Isha Dewan 21. Exchangeability, Functional Equations, and Characterizations by C. R. Rao and D. N. Shanbhag 22. Martingales and Some Applications by M. M. Rao 23. Markov Chains: Structure and Applications by R. L. Tweedie 24. Diffusion Processes by S. R. S. Varadhan 25. Itô's Stochastic Calculus and Its Applications by S. Watanabe

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