Order Statistics: Applications

Preface

The area of Order Statistics received a tremendous attention from numerous researchers during the past century. During this period, several major theoretical advances were made in this area of research. As a matter of fact, two of those have been adjudged as breakthroughs in the field of Statistics [see: Kotz, S. and Johnson, N. L. (1991). Breakthroughs in Statistics, Vols. 1 and 2, Springer-Verlag, New York]. In the course of these developments, order statistics have also found important applications in m a n y diverse areas including life-testing and reliability, robustness studies, statistical quality control, filtering theory, signal processing, image processing, and radar target detection. Based on this immense activity, we decided to prepare this H a n d b o o k on Order Statistics and Their Applications. We feel that we have successfully brought together theoretical researchers working on theoretical and methodological advancements on order statistics and applied statisticians and engineers developing new and innovative applications of order statistics. Altogether, there are 44 articles covering most of the important theoretical and applied aspects of order statistics. For the convenience of readers, the subject matter has been divided into two volumes, the first one ( H a n d b o o k - 16) focusing on Theory and Methods and the second one ( H a n d b o o k - 17) dealing primarily with Applications. Each volume has also been organized according to different parts, with each part specializing on one aspect of Order Statistics. The articles in this volume have been classified into six parts as follows: Part Part Part Part Part Part

I II IIIIV V VI -

Results for Specific Distributions Linear Estimation Inferential Methods Prediction Goodness-of-fit Tests Applications

We have also presented an elaborate Author Index as well as a Subject Index in order to facilitate an easy access to all the material included in the volume. Part I contains five articles - - the first one by A. P. Basu and B. Singh discussed some important properties of order statistics from exponential distribution, the second one by N. Balakrishnan and S. S. G u p t a studies higher order moments of order statistics fi'om exponential and right-truncated exponential distributions and uses them to develop Edgeworth approximate inference in life-

vi

Preface

testing problems, the third one by N. Balakrishnan and P. S. Chan discusses order statistics from the log-gamma distribution and develops linear estimation of parameters of this distribution, the fourth one by N. Balakrishnan and R. Aggarwala establishes several recurrence relations for single and product moments of order statistics from a generalized logistic distribution and illustrates their application to inference and also extends these results to the case of doubly truncated generalized logistic distribution, and the last article by N. Balakrishnan and S. K. Lee similarly discusses order statistics from the Type III generalized logistic distribution and their applications to the estimation of parameters of this distribution. Part II contains four articles - - the first one by S. K. Sarkar and W. Wang discusses some properties of estimators for the scale parameter of the distribution based on a fixed set of order statistics, the second one by M. M. Ali and D. Umbach provides a review of various results on the optimal linear estimation of parameters in location-scale families, the third one by J. R. M. Hosking presents an overview of the L-estimation method, and the last article by S. Alimoradi and A. K. Md. E. Saleh discussed the L-estimation method in the context of linear regression models. Part III contains five articles - - the first one by A. C. Cohen illustrates the role of order statistics in estimating threshold parameters for a variety of life-span models, the second one by F. Kong discusses the estimation of location and scale parameters in the case when the available sample is multiply Type-II censored, the third one by N. Ni Chuiv and B. K. Sinha provides an elaborate review of results available on ranked set sampling and estimation methods based on such a sampling scheme, the fourth one by S. Geisser highlights some uses of order statistics in Bayesian analysis with particular stress of prediction and discordancy problems, and the last article by S. Panchapakesan, A. Childs, B. H. Humphrey and N. Balakrishnan develops some inverse sampling procedures based on order statistics for the purpose of testing for homogeneity in a multinomial population. Part IV contains an expository review article by K. S. Kaminsky and P. I. Nelson on the subject of prediction of order statistics, both one-sample and two-sample situations. Part V contains two articles the first one by R. A. Lockhart and M. A. Stephens provides a review of the probability plot and the related goodness-of-fit test based on the correlation coefficient, and the second article by S. Shapiro illustrates the role of order statistics in distributional assessment problems. Part VI contains six articles - - the first one by H. Schneider and F. Barbera presents applications of order statistics in the development of sampling plans for inspection by variables, the second one by M. Viana uses linear combinations of ordered symmetric observations to analyze visual acuity, the third (by G. R. Arce, Y.-T. Kim and K. E. Barner) and fourth articles (by K. E. Barner and G. R. Arce) overviews filtering methods based on order statistics and their role in smoothing of time-series data, the fifth one by S. T. Acton and A. C. Bovik elaborates the role of order statistics in image processing, and the last article by R. Viswanathan displays the application of order statistics to CFAR radar target detection.

Preface

vii

It needs to be mentioned here that the companion volume (Handbook - 16), focusing on theory and methods of order statistics, has been divided similarly into nine parts. While preparing this volume as well as the companion volume (Handbook 16), we have made a very clear distinction between order statistics and rank order statistics, the latter being an integral part of the area of Nonparametric Statistics. Even though there is an overlap between the two and also that order statistics play a role in Nonparametric Statistics, one of the most important uses of order statistics is in the development of parametric inferential methods, as is clearly evident from this volume. Unfortunately, some researchers still view Order Statistics as part of Nonparametric Statistics. Strangely enough, this view is also present in Mathematical Reviews. We express our sincere thanks to Mr. Gerard Wanrooy (North-Holland, Amsterdam) for his interest in this project and for providing constant support and encouragement during the course of this project. We also thank Mrs. Debbie Iscoe for helping us with the typesetting of some parts of this volume. Thanks are also due to the Natural Sciences and Engineering Research Council of Canada and the U.S. Army Research Office for providing individual research grants to the editors which facilitated the editorial work of this volume. Our special thanks go to all the authors for showing interest in this project and for preparing fine expository articles in their respective topics of expertise. Our final thanks go to Miss Ann Balasubramaniam for helping us with the preparation of the Author Index for this volume. We sincerely hope that theoretical researchers, applied scientists and engineers, and graduate students involved in the area of Order Statistics will all find this Handbook to be a useful and valuable reference in their work. N. Balakrishnan C. R. Rao

Contributors

S. T. Acton, School of Electrical & Computer Eng., 202 Engineering South, Oklahoma State University, Stillwater, OK 74078-0321, USA (Ch. 22) R. A. Aggarwala, Department of Mathematics and Statistics, University of Calgary, 2500 University Drive, N W Calgary, Alberta, Canada T2N 1N4 (Ch. 4) M. M. Ali, Department of Mathematical Sciences, Bell State University, Muncie, IN 47306-0490, USA (Ch. 7) S. Alimoradi, Isfahan University of Technology, POB 69-34, Isfahan, Iran (Ch. 9) G. R. Arce, Department of Electrical Engineering, University of Delaware, Newark, DE 19716, USA (Chs. 20, 21) N. Balakrishnan, Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4KI (Chs. 2, 3,4,5, 14) F. Barbera, Management Department, University of Southeast Louisiana, Hammond, LA 70402, USA (Ch. 18) K. E. Barner, Department of Electrical Engineering, University of Delaware, Newark, DE 19716, USA (Chs. 20, 21) A. P. Basu, University of Missouri, Dept. of Statistics, 222 Math. Sciences Bldg., Columbia, MO 65211, USA (Ch. 1) A. C. Bovik, Center for Vision and Image Sciences, Department of Electrical & Computer Eng., The University of Texas at Austin, Austin, TX 78712-1084, USA (Ch. 22) P. S. Chan, Department of Statistics, The Chinese University of Hong Kong, Shatin, Hong Kong (Ch. 3) A. Childs, Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1 (Ch. 14) A. C. Cohen, 421g Westfield Cornt, Columbus, SA 31907-1837, USA (Ch. 10) S. Geisser, School of Statistics, University of Minnesota-Twin Cities, 206 Church Street SE, 270 Vincent Hall, Minneapolis, MN 55455, USA (Ch. 13) S. S. Gupta, Department of Statistics, Purdue University, West Lafayette, IN 47907, USA (Ch. 2) J. R. M. Hosking, IBM Research Division, Thomas J. Watson Research Ctr., P.O. Box 218, Yorktown Heights, New York, N Y 10598, USA (Ch. 8) B. H. Humphrey, Department of Mathematics, Southern Illinois University at Carbondale, Carbondale, IL 62901-4408, USA (Ch. 14) xvii

xviii

Contributors

K. S. Kaminsky, Department of Math. Stat., School of Mathematics & Computer Sci., Chalmers University of Technology, S-41290 Gothenburg, Sweden (Ch. 15) Y.-T. Kim, Signal Processing R&D Center, Samsung Electronics Co., Suwon, Korea (Ch. 20) F. Kong, Department of Mathematics and Statistics, University of Maryland at Baltimore County, Baltimore, MD 21228, USA (Ch. 11) S. K. Lee, Department of Math. & Stat., McMaster University, Hamilton, Ontario, Canada L8S 4K1 (Ch. 5) R. A. Lockhart, Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6 (Ch. 16) P. I. Nelson, Department of Statistics, Kansas State University, Manhattan, KS 66506, USA (Ch. 15) N. Ni Chuiv, Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada, E3B 5A3 (Ch. 12) S. Panchapakesan, Department of Mathematics, Southern Illinois University at Carbondale, Carbondale, IL, 62901-4408, USA (Ch. 14) A. K. Md. E. Saleh, Department of Mathematics and Statistics, Carleton University, 4302 Herzberg Building, 1125 Colonel By Drive, Ottawa, Ontario, Canada KIS 5B6 (Ch. 9) S. K. Sarkar, Department of Statistics, Temple University, Philadelphia, PA 19122, USA (Ch. 6) H. Schneider, Department of Quantitative Business Analysis, Louisiana State University, Baton Rouge, LA 70803, USA (Ch. 18) S. Shapiro, Department of Statistics, Florida International University, University Park Campus, Miami, FL 33199, USA (Ch. 17) B. Singh, University of Missouri, Dept. of Statistics, 222 Math. Sciences Bldg., Columbia, MO 65211, USA (Ch. 1) B. K. Sinha, Department of Mathematics and Statistics, University of Maryland at Baltimore County, Baltimore, MD 21228-5398, USA (Ch. 12) M. A. Stephens, Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6 (Ch. 16) D. Umbach, Department of Mathematical Sciences, Bell State University, Muncie, IN 47306-0490, USA (Ch. 7) M. Viana, Department of Ophthalmology and Visual Sciences (M/C 648), Eye and Ear Infirmary, Lions of Illinois Eye Research Inst., 1855 West Taylor Street, The University of Illinois at Chicago, Chicago, IL 60612-7243, USA (Ch. 19) R. Viswanathan, Department of Electrical Engineering, College of Engineering, Mailcode 6603, Southern Illinois University at Carbondale, Carbondale, IL 62901-6603, USA (Ch. 23) W. Wang, Clinical Biostatistics Dept., Wyeth-Ayerst Research, 145-B2 King of Prussia Road, Radnor, PA 19087, USA (Ch. 6)

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 17 © 1998 ElsevierScience B.V. All rights reserved.

1 [

Order Statistics in Exponential Distribution

Asit P. Basu and Bahadur Singh

1. Introduction

The exponential distribution is a basic physical model in reliability theory and survival analysis. The properties of the exponential model have been studied widely in the statistical literature. For a survey of exponential and other models in reliability see Balakrishnan and Basu (1995), Lawless (1982) and Sinha and Kale (1980). Order statistics occur naturally in lifetesting and related areas as the weakest unit, among a set of n units, fails first. The second weakest unit fails next and so on giving rise to order statistics. The properties of order statistics have been studied extensively in a number of monographs such as, Balakrishnan and Cohen (1991), David (1981), and Sarhan and Greenberg (1962). In this chapter we consider the properties of order statistics and use these results for estimating the parameters of the one and two parameter exponential distributions. In Section 2 we give a brief summary of some important properties of order statistics from the exponential distribution. In Section 3 various types of censoring are described. The estimates of scale parameter 0 of the exponential distribution for Type I, Type II and randomly censored data are derived. The inferences concerning the two-parameter exponential distribution are also considered. In Section 4 these results are extended to two or more independent Type II censored samples. Order restricted inference for the scale parameters 01, 02,..., Ok (k > 2) of k exponential distributions are considered in Section 5. Bayesian inference is considered in Section 6. Bayesian estimates of 0, for Type I and Type H censored samples, are obtained. Also, Bayesian estimators of # and 0 for the two-parameter exponential family for Type II censored sample are obtained.

2. Order statistics and its properties

In this section we state some well known results of order statistics for the exponential distribution. Let X be a random variable from the one parameter exponential distribution with density function f(x]O) = 0-1 exp(-x/O), distribution function F(x]O)= 1-exp(-x/O), and survival function S(x]O)= 1-F(x]O)=

4

A.P. Basu and B. Singh

e x p ( - x / O ) . We shall denote this one parameter exponential distribution by e(0). Let )((1) _< )((2) _< ..- _< X(,) denote the order statistics in a r a n d o m sample of size n from a population with the standard exponential distribution e(1) with probability density function (pdf) and distribution function, respectively, f(x)=e

-x, F ( x ) = l - e

Then, the marginal pdf of the

f r ( x ) = ( r _ l ) ! ( n _n!r ) !

-x, r th

0_<x
.

(2.1)

order statistics, X(r), is given by

(1 - e x)r-1 e -(n-~+l)x,

O_<x
(2.2)

.

The joint pdf of (X(~),X(s)) for 1 _< r < s <_ n, is given by n! f,-,s(x,y) - (r - 1)!(s - r - 1)!(n - s)t x (1 - e-~)"-I (e -~ - e-Y)*-~-te-Xe -("-s+l)y, 0 _ < x < y < e o

.

(2.2) The pdf of the range, R = (X(n) - X(I)), is given by

fR(x) = (n - 1) e-X(1 - e-X) "-2,

0 _< x < cc .

(2.3)

Note that R and X(1) are independently distributed. Moreover, of X(1) and R characterizes the exponential distribution. Basu provide a survey o f similar characterization results. See also Basu (1995) for some other related results. The joint pdf of all n order statistics, (X(1),X(2)~ ... ,X(n)) is

fl,z,...,n(xl,xa,...,x~) = n!exp

-

the independence and Block (1975) Balakrishnan and given by

O<_xl<x2<...<_x~

xj ,

. (2.4)

Following Sukhatme (1937), with x(0) = 0, the joint pdf (2.4) may be expressed as

f1,2 ...... (Xl,X2,...,x~) = n ! e x p

-~(n-

l+ 1)(x(,/-x(l_l))

.

(2.5)

j=l

If we let y, = ( n - I + 1)(x(0 - x ( , - l ) ) ,

l= 1,2,...,n

,

(2.6)

then 0 _< yl < ec, and y t , y 2 , . . . ,yn, are i.i.d, with pdf (2.1). F r o m (2.6), one may express x(r) as x =

x<,-ll) l=l

=

t+ l=l

II .

(2.7/

Order statistics in exponential distribution

5

Thus, x(r) is a linear combination of i.i.d, exponential random variables with pdf (2.1). As a by-product, we observe an interesting property that the standard exponential order statistics form an additive Markov chain [Renyi (1953), Karlin and Taylor (1975)]. Using an alternative notation for the ith order statistics, X(i), i.e., X(i) = X/:,, in view of (2.6), we have

Xi+~:,-Xi:, D=Xr:, i,

l
,

(2.8)

where D denotes the fact that random variables on the left hand side are equivalent to those on the right hand side in distribution. From (2.7), it follows that the means, variances and covariances of the order statistics are given by F

E(X(r)) = ~--~(n - l + 1) -1

(2.%)

l=l r

Var(Xff)) = Z ( n -

l + 1) 2

(2.9b)

/=l

and

Cov(X(r),X(s)) = Var(X(r)),

for r < s .

(2.9b)

E(R) = E(X(,) - X(x)) = ~--~(n - l + 1) -l

(2.10a)

Also, for the range R, n

/=2

and n

Var(R) = Var(X(,) -X(1)) = ~-~(n - 1 + 1) 2

(2.10b)

l=2

3. Censored data

The failure time data arise naturally in many applications such as, engineering, biological sciences or clinical trials in cancer research. For instance, suppose that some mechanical or electronic components are tested to obtain information about their performance measured in hours, days, weeks, months or years (depending on the type of application) until they fail. Such measurements are called lifetimes or failure times. In clinical trials, these represent a subject's lifetime or time to some non-recurrent event beginning from the time the subject entered the study. In the examples above and many others, the observations naturally occur ordered in time yielding order statistics. Censoring is very common in studies and/or experiments involving lifetime data. The data are said to be censored when only a lower or upper bound on

6

A . P . Basu and B. Singh

lifetime is available. In reliability and survival analysis censoring is very common in lifetime or failure time data because of time and/or cost considerations on data collection. For instance, if n items such as radio tubes, wire fuses, or light bulbs are put through a life test, one may have to terminate the experiment when only r (0 < r < n) items have failed. In cancer research clinical trials involving humans, some patients may leave the study before its completion resulting in censored data. A patient may leave the study because of various reasons such as, moving away from the area, lack of interest in the treatment or some other unknown reasons. There are three major types of censoring: (i) Type I censoring, (ii) Type II censoring, and (iii) Random censoring. We describe them now.

3.1. Type I censoring Suppose that in a lifetime experiment n items are placed on test, and the experiment is terminated after a time L has elapsed. It should be noted here that a decision to terminate the experiment is made well before the beginning of the experiment. Then, the lifetimes will be known exactly only for those items which have failed on or before time L. The data resulting from the above experiment are called singly Type I censored. There are two other versions of Type I censoring experiments involving clinical studies in cancer research, and these are described as follows: in one a clinical study begins at time t = to, terminates at time T > to and all n patients enroll in the study at the beginning of the study at time t = to. During the course of the study a patient may die, be alive or leave the study (i.e., be censored). Another more complicated experiment involving Type I censoring, also, arises in clinical trials in cancer research. Suppose that a clinical study begins at time t = to, and the enrollment period terminates at time t = T > to, and then the patients are followed-up for an additional time z beyond T. Patients enter the study randomly according to time of their diagnosis during the enrollment period I0, T]. Some patients may leave the study (i.e., be censored) randomly due to known or unknown reasons during the enrollment or follow-up periods. Conceptually, there are two measurements (leaving aside the covariates) on each patient, say a failure time ~ or a censoring time Li; i = 1, 2 , . . . , n. However, only one of the ~ or L~ (not both) is observed. If the censoring times LI~L2,..., Ln, are fixed, then the data are said to be Type I censored. The data from such a study can be represented by n pairs of random variables (ti, 6i) where

ti=min(T/;Li)

and

6i=

1 if T~._

L i

That is, 6i = 1 if the failure time is observed, and ~i = 0 if the censoring time is observed. Also, ti = Ti if failure time is observed and ti = Li if the censoring time is observed. The joint pdf of ti and 6i [Lawless (1982, pp. 35-36)] is given by

Order statistics in exponential distribution

7

where S(Li) is the survival function given by P(Ti > Li) = 1 - F(Li). Since n pairs of random variables (ti, 6i) are independent, hence the likelihood function is given by n

L = H f ( t i ) 6 ' S ( L i ) '-~"

(3.1)

i= 1

EXAMPLE 1. If lifetimes are independently distributed as exponential with mean 0, i.e. with distribution e(0) and survival function S(t) = e x p ( - t / O ) , then likelihood function (3.1) becomes

)" ( e - ~ ) -1o,

L=

e 7 i=l

(3.2)

where r = ~ 6i is the observed number of deaths or failures.

3.2. Type H censoring Suppose that in an engineering life testing experiment n items are placed on test, and a decision is made to terminate the experiment when the first r items (1 < r < n) have failed. The number r is pre-selected before the beginning of the experiment. Then, the data are said to be Type II censored. The data consist of the r smallest lifetimes (i.e., r order statistics) T(I) ___T(2) <_ .-. _< T(~) out of a random sample of size n lifetimes TI, T2,..., Tn from the lifetime distribution under investigation. If TI, T2,..., T, are i.i.d, with continuous pdf f ( t ) and survival function S(t), then the joint pdf of T(1), T(2),..., T(r) is given by n!

(n - r)! f(t(1))f(t(2))

" " . f(t(~))

[S(t(~))]"

(3.3)

r

EXAMPLE 2. Let T1, T2,..., T, be a random sample from an exponential distribution e(0). Then the joint pdf of T(1), T(2),.. •, T(r) is given by n! 1 f Tr] (n - r ) ! ~ e x p t - o - J > '

r with Tr = Z t(i) + (n - r)t(,.)

(3.4)

i =1

Then, it follows easily that the maximum likelihood estimator of 0, say 0 is Tr/r. Making the transformation W1 =nt(l),

~=(n-i+ F

1)(t(i)-t(i_l)),

i=2,...,r r

,

it is easy to verify that T~ = ~i=1 t(i) + (n - r)t(r) = ~ i = l W/, and the Jacobian is

8

A . P . Basu and B. Singh

0(w~,..., m~)

n!

O ( t ( , ) , . . . ,t(~))

(n - r)!

'

and therefore the joint pdf of (W1,..., W~) is given by

Moreover,

2/'/0

~

-0=

= ~ e1 x p

f(wl,...,w~)

Wi

,

Wi > 0 .

z22 r '

EXAMPLE 3. Consider now a random sample of size n with Type II censoring from the two-parameter exponential family with the pdf f(t[#,O)=-oe

1

/t ~0 o ,

(3.5)

t>_#

where # _> 0 is a threshold or guarantee time parameter and 0 is a scale parameter. The joint pdf of the r smallest observations, say t(l) < t(2) < ... < t(r), based upon the above random sample of size n from (3.5), is given by (n -n! r)! 01~exp - 0

=

t(i) - #) + (n - r) (t(r) - #

,)}

(3.6)

where # < t(1) < t(2) < ... < t(~). Obviously, the M L E o f # is given by/~ = t(~), and with # = t(1) the M L E of 0 is given by 0 =

t(i) + (n - r)t(r) - nt(1)

r = Tzr/r, say .

(3.7)

Then, ~ and 0 are independently distributed, and 2n(/~- #)/0 and 2rO/O are independently distributed as Z~ and Z22r-2, respectively. It should be noted here that with Type II censoring the number, r, of exact failure times is fixed (non-stochastic) whereas it is a random variable with singly Type I censored data.

3.3. R a n d o m

censoring

Intuitively, it is reasonable to think of censoring times as random variables. To illustrate the ideas, we generalize the example described above in the first paragraph at the beginning of Section 3.1 for a clinical study in cancer research involving Type I censoring: Assume that a clinical study begins at time t = to, and the enrollment period terminates at time t = T > to, and then the patients are followed-up for an additional time z beyond T. Patients may enter the study at various point in time during the enrollment period, [to, T], more or less in a random fashion, according to their time of diagnosis. A patient may die, be alive or leave the study (i.e., be censored) during the course of the study (i.e., enrollment

Order statistics in exponential distribution

9

or follow-up periods). Then, it is reasonable to assume that the censoring times are independent random variables. Assume now that there are n patients enrolled in the study, and each patient has a lifetime or failure time T and a censoring time L, where T and L are independent continuous random variables with survival functions S(t) and G(t), respectively. Let (T~,Li);i = 1,2,..., n be independent and let ti = min(T~, Li) and 6i = 1 if T/_< Li and 6i = 0 if T / > Li. The data on n patients consist of pairs (ti, 6i), i = 1 , 2 , . . . ,n. Let f(t) and 9(t) be the pdf's of Ti and Li, respectively. Then [Lawless (1982, pp. 37-38)], the sampling distribution of {(ti, 6i); i = 1 , 2 , . . . , n} is given by n

U[f(ti)G(ti)]a~[g(ti)S(ti)]l-6i i=1

(3.8)

If 9(t) and G(t) do not involve any parameters of interest, then the first factor in the parentheses in (3.8) can be dropped, and then, the likelihood function is given by n

L = I-[f(ti)a'S(ti) 1 ~'

(3.9)

i=1

EXAMPLE 4. If T1, T2,..., Tn are a random sample from an exponential distribution with the unknown means 0, i.e., p d f f ( t ) = (1/O)exp(-t/O) and survival function S(t) = exp(-t/O), then likelihood function (3.9) simplifies to [Lawless (1982, p. 105)],

l{l }

L(O) = ~ e x p

- ~ .= ti

,

(3.10)

n

where r = ~ i = l 6i is the observed number of failures, and r is a random variable. Note that n

T~r=Zti=~_Ti+ZLi i=l

iED

(3.11) iEC

is the total observed lifetime for the n patients, where D and C denote the set of individuals for whom lifetimes are observed are censored, respectively. Moreover, (r, T,.') is minimal sufficient for 0, and if r > 0, then 0 = T[/r is the M L E of 0. In case r = 0, likelihood does not have a finite maximum. However, if n is large the probability of r being zero is negligible and, therefore, in the following it is assumed that r > 0. The first and second derivatives of the log-likelihood function of (3.10) are given by

10

A. P. Basu and B. Singh dlnL r 1 " d ~ - 0 t - ~ i~_1 n

d 2 In L

r

2 Z ti

d02

02

03 i=1

Now, we obtain the expected Fisher's information in two steps. First, we assume that the censoring values are L1, L 2 , . . . , L , are fixed and known. Later on in the second step, we shall relax this assumption. So, assume that L1,L2,... ,L, are fixed and known. The probability distribution of ~i is given by Pr(6i = 0) = exp(-Li/O)

Pr(6i = 1) = 1 - exp(-Li/O)

and

.

Moreover, E(til6i = O) = Li, and using integration by parts Lt

E(t,16i= l ) = E ( T i t T i <-Li) =

f0

0-1e 1/0 t (l_e_L,/o) dt=O

Lie Li/O (l_e-L,/O)

Therefore, E(t,) = E(t~lcSi = 0)-Pr(6, = 0) + E(til6i = 1). Pr(6~ = 1) =0(1-e

L,/O) .

and E(r)=ZE(6i)= i=1

1 - e -Ld° = Q ( L 1 , . . . , L , ) = Q

.

i=1

Now, conditional on LI, L 2 , . . . , Ln, the expected Fisher's information is given by

I(O,L1,L2,...,L~) = E ( 1

d21~-~2LL , , L 2 , . . . , L , ) "

Assume, now, that the censoring values, L1, L 2 , . . . , L,, are i.i.d, random variables with censoring pdf 9(t), and censoring and failure times are independent. If ~ is the proportion of observations censored, then it follows that [cf Eq. (2.9) of Lininger et al., (1979) under H0 ], n = P(failure time > censoring time) =

=

P(T > 09(0 dt

/o

S(t)O(t) dt .

Order statistics in exponential distribution

11

Therefore, 1 - zc = f~[1 - S(t)]g(t) dt and obviously,

=E{I(OIL1,L2,...,L,I):LV, E(I_e_LJo)O~~_., g \ n

i=1

g

-0~

where E: denotes the expectation with respect to censoring pdf g(t). g

4. Inference concerning several exponential populations In this section, inferences concerning two or more exponential distributions based on Type II censored random samples from a two-parameter exponential family (3.5) are considered. Both cases of # known and unknown are dealt with.

4.1. Inferences for type H censored data for two populations The two cases of known and unknown threshold parameters are considered separately in the sequel.

Case(i) Two-sample case with known threshold parameters Without any loss in generality it may be assumed that the mean/~ = 0 in the twoparameter family given by (3.5). Assume now that there are two independent random samples of sizes nl, n2 from the two-parameter exponential family (3.5) with parameters (0, 01) and (0, 02), respectively. Let the ri smallest (order statistics) failure time be ti(1) < ti(2) <

""

<

ti(ri),

ri <_ ni,

(i = 1,2)

.

Letting T/= ~ - 1 ti~i)+ ( h i - ri)ti(,~) be the total time on the test in the i th (i = 1,2) population, the M L E of Oi is given by Oi = Ti/ri. Moreover, T1 and T2 are independent, and 2Ti/Oi is distributed as chi-square with 2ri degrees of freedom, Z~r," Therefore, the ratio R1,

R1

2"1"1/2rl 01 2T2/2r202

(4.1)

has a F-distribution with (2rl, 2r2) degrees of freedom and this fact can be used for making inferences about 01 and 02.

Case(ii) Two-sample case with unknown threshold parameters Assume now that there are two independent random samples of sizes nl, n2 from the two-parameter exponential family (3.5) with parameters (/~j, 01) and (Ft2,02), respectively. Let the ri smallest (order statistics) failure time be

12

A. P. Basu and B. Singh

ti(1) < ti(2) < " " < ti(~,),

ri

<_ ni,

(i = 1,2) .

Letting t~ = ~ - 1 tiu) + (n~ - r~) t~(~,) - niti(l ) be the total time on the test in the i th (i = 1,2) population, the M L E ' s of ~i and Oi are given by ]hi = t(1), 0i = Tffri, respectively. Moreover, ~1,01, /~2 and 02 are independent, and 2 n ( f q - #i)/Oi is distributed as X~ and 2riOJOi is distributed as Z~, 2. Therefore, the ratio R2,

2Tl/Zr101 R2 -

(4.2)

2T2/2r202

has a F-distribution with (2rl - 2, 2r2 - 2) degrees of freedom and this fact can be used for making inferences about 0a and 0a.

4.2. Inferences for more than two-populations In this section, the problem of testing hypotheses concerning k(k > 2) exponential means is considered using the likelihood ratio approach when the data are Type II censored. The two cases of known and u n k n o w n threshold parameters are dealt with separately.

Case (i) Type H censored samples with known threshold parameters Assume that there are k independent Type II censored r a n d o m samples of sizes nl, n2~..., nk with pre-specified number of failures rl, r 2 , . . . , rk, respectively, and that t~(l) < ti(2) < ..- < ti(r,)

(i = 1 , 2 , . . . , k )

are the ri smallest ordered observations from the population i with mean 0i (#i is assumed to be zero) with p d f (3.5). The main interest is in testing the null hypothesis

Ho : 01 = 02

Ok versus

.....

HI : Oi ~ Oj

for some i ¢ j .

The likelihood function given the above Type II censored data is given by

L(O,,...,

0k) = I I (., _ r,)! 07' exp i=1

where T / = G~'_-J ti(;) + (hi - ri)ti(rj). Under H0, the M L E of the c o m m o n value of 0, say 0, is given by 0 = p,, and under the full parameter space the M L E of 0i, say Oi, is Oi = Z-/ri. Therefore, the likelihood ratio test of H0 versus H1 rejects H0 for large values of

EL, /EL1

A = - 2 ln[L(0,..., O)/L(01,...., 0k)] =

2

ri

rilnOi .

lnO-2 i=1

(4.3)

Order statistics in exponential distribution

13

Asymptotically for large values of ri (i = 1 , 2 , . . . , k), under H0, A is distributed as 2 Zk-l. For small to moderate size values of r~ s, as Bartlett type correction may be applied. In Bartlett type correction, one approximates the random variable CA by a chi-square distribution with k - 1 degrees or freedom, Z2 1. The constant C is given by C = 1 + 6 ( k - 1~--)

\ i=1 -t

-

"

Case (ii) Type H censored samples with unknown threshold parameters Assume that there are k independent Type II censored random samples of sizes ni, n2,..., nk with pre-specified number of failures rl, r 2 , . . . , rk, respectively, and that

t~(i) < ti(2) < " " < t~(r,)

(i = 1 , 2 , . . . , k )

are the ri smallest ordered observations from the population i from a two-parameter family with parameters (#i, Oi) with pdf (3.5). Then, the likelihood function given the above Type II censored data is given by L(#l, 01,..., #k, Ok) = ]--[ (n~---hi 7=T

exp - - ~

(tio) -- Pj) + (ni - ri)(ti(~i) - #i)

where #i < t~(1) < ti(2) < -.. < ti(r~), (i = 1 , 2 , . . . ,k). Since #i < ti(1), the M L E of#i under full parameter space, fi# is /~i = ti(l)" With ~i = ti(1), the M L E of Oi under the full parameter space, say Oi, is given by ri

Oi = Ti/ri where j=l

Under H0, the MLE of #i is exactly the same a s / 2 i = ti(1) as above for the full parameter space. However, the MLE of the common value of 0~ = 0 under H0, say 0, is given by

0 = Zri i=l

ri /

i=l

The likelihood ratio test of H0 vs. H1 rejects H0 for large values of A = - 2 In [L(/~I, 0,...,/)k, 0)/L(/~I, 0,,...,/)k, 0k)] =

ri

lnO-2

rilnOi . i=l

(4.4)

A. P, Basu and B. Singh

14

and likelihood ratio test statistic (4.4) is in exactly the same form as (4.3) except that the MLE's are slightly different that those in case (i) above. Asymptotically for large values of ri (i = 1,2, ' " "~ k), under H0, A is distributed in )~2 For small k-l' to moderate size values of ri's, a Bartlett type correction may be applied. In Bartlett type correction, one approximates the random variable CtA by a chisquare distribution with k - 1 degrees of freedom, Xk2 1" The constant C I is given by

C'=l+6(k-l__

i=1

Fi

I

r

k

I

Similar results for randomly censored data can be derived, and these will be given elsewhere.

5. Order restricted inference

In this section a very brief summary of inference concerning exponential means is presented when the means are subject to an order restriction. Two cases of complete and censored samples are considered. It turns out that the solutions to a wide class of restricted optimization problems can be found by restricted least squares methods. To put it in perspective, some notation, terminology and background information in introduced in the next section. For more information the reader is referred to Chapter 1 of Robertson, Wright and Dykstra (1988).

5.1. Some notation, terminology and background Letting X = {xl,x2,... ,xk}, suppose that w is a positive weight function defined on X and Y is a restricted family of functions on X. For an arbitrary function 9 defined on X, let g* be the least squares projection of g onto Y . The g* is the solution to the restricted least squares problem if 9* minimizes Z [ 9 ( x ) -f(x)]2w(x)

subject to

f E@ .

xEX

If the family ~ satisfies certain conditions, then 9* provides the solution for a variety of other optimization problems for which the objective functions are quite different from that of least squares. First, the concept of isotonic regression for the simply ordered case is introduced. Assume that the k elements of X are simply ordered i.e., xl _~ x2 _ -. - _~ xk. A function f defined on X is isotonic (with respect to the simple order) if and only if f(xl) _
Order statistics in exponential distribution

-

15

f(x)]2w(x)

x~

in the class of all isotonic functions f defined on X. Next, the concept of isotonic regression over a quasi-ordered set is introduced. To this end, one needs to study the definition of simple order in terms of a binary relation on X which is described next. DEFINITION 5.2. A binary relation _ on X is said to be a simple order on X if and only if 1. 2. 3. 4.

it is reflexive, i.e.; x _~ x for x ~ X; it is transitive, i.e., x , y , z E X , x -< y and y -< z imply x = z; it is antisymmetric, i.e., x , y E X , x ~_ y and y -4 x imply x = y; and every two elements of X are comparable, i.e., x, y E X implies that either x -< y

or y-< x. DEFINITION 5.3. A binary relation _~ on X is said to be a partial order if it is reflexive, transitive and antisymmetric. (But there may be noncomparable elements in X.) DEFINITION 5.4. A binary relation -< on X is a quasi-order if it is reflexive and transitive. (It need not be antisymmetric and it may admit noncomparable elements.) DEFINITION 5.5. A real valued function, f , on X is said to be isotonic with respect to the quasi-order __ on X if x , y E X and x ~ y imply f ( x ) <_f ( y ) . DEFINITION 5.6. Let g be a given function w be a positive weight function both defined on X. An isotonic function g* on X is an isotonic regression of g with weights w if and only if Z [ g ( x ) - g* (x)]2w(x) <_ Z [ g ( x ) - f(x)J2w(x) xcX

xGY

for all functions f on X which are isotonic with respect to the given quasi-order _ on X. One may think of the collection, J , of isotonic functions as a subset of a Euclidean space which has as its dimension the number of points in X. Using this setting the isotonic regression, g*, is the closest point of J of g with distance as the weighted Euclidean distance dw(., .) given by

[dw(9, f)]2 = Z [9(x) - f(x)]2w(x)

(5.1)

xEX

Note that J is a closed convex cone. J is closed in the topology induced by the distance metric given by (5.1). J is convex if f l , f 2 E J and 0 < c~ < 1, then afl+(1-a) EJ;'Jis a cone i f f E J a n d a _ > 0 , then a f E J . It should be

16

A. P. Basu and B. Singh

noted here that the closure property guarantees the existence of the projections whereas the convexity guarantees the uniqueness of the projections. Now, t,;vo more examples of quasi-orders encountered in the applications are introduced. These quasi-orders are known as simple tree and simple loop orders: The k elements of X assume a simple tree order if and only if X 1 ~ [X2,X3,...,Xk]. That is, X 1 precedes xj for j = 1 , . . . , k and the last k - 1 elements may be noncomparable. A function f defined on X is isotonic with respect to simple tree order if and only if f (xl) __f(xj) for j = 2 , 3 , . . . , k - 1 ; and nothing is assumed about the ordering o f f ( x 2 ) , . . . ,f(xk-x). There is also a quasi-order known as matrix order which is not discussed here, and the reader is referred to Robertson, Wright and Dykstra (1988, p. 26) for details. There are some F O R T R A N Subroutines available which compute the restricted least squares estimates of the parameters when ordering among the parameters is specified by simple order, simple tree order, simple loop order or matrix order cones.

5.2. Isotonic regression in samples from exponential family Assume that the failure time t has the probability density function in the exponential form

f(t; O, ~) = exp[pl (O)p2(r)K(t; z) + S*(t;z) + q(O, ~)]

(5.2)

for 0 C ( 0 , 0 ) , ~ E T , and - e c < 0 < 0 _ < o c . The parameter 0 is of interest whereas the parameter v is a nuisance parameter. Assume that the following three regularity conditions hold: (i) pl (') and q(., ~) have continuous second order derivative with respect to 0 on (5.3) (_0,0) for all ~ E T (5.4) (ii) p~(O) > 0 for all 0 E (0,_0), p2(v) > 0 for all E T, (iii) q'(O,z) = -Op~i(O)p2(z) for all 0 E (_0,0) and ~ E T. (5.5) Obviously, f o f(t; 0, ~) d t = 1. Now, differentiating twice with respect to 0 under the integral sign, one may show that E[K(t;~)] = 0 and Var[K(t;v)] = [P~( O)p2('C)] -1. Assume now that we have k (k _< 2) populations characterized by the densities in the exponential form f(.; O(xj), zj) for j = 1 , . . . , k, and that one has k independent random samples {tji; i = 1,... ,nj;j = 1,... ,k} from the k populations. Then, using the regularity conditions (5.3), (5.4) and (5.5), and differentiating the log-likelihood function, the maximum likelihood function of O(xj) is

Order statistics in exponential distribution

17

nj

0(xj) =

ZK(tji; zj)/nj

(5.6)

i=1

Now, assume that -< is a quasi-order on X = {Xl,X2,... ,xk}; assume that one has prior information that 0 is isotonic with respect to this quasi-order; and assume that an estimate of 0 is required with this property. Then, the following theorem can be used to obtain restricted maximum likelihood estimate of 0 which is also isotonic with respect to the quasi-order _~ on X = {Xl , X 2 , . . . ,Xk}. THEOREM 5.1. Assume that one has k independent random samples as described above from the k populations characterized by the exponential densities of the form (5.2) and satisfying the regularity conditions (5.3), (5.4) and (5.5). Then, the restricted maximum likelihood estimate of O(x) = (O(xl), 0(x2),..., O(xk)) subject to the condition that the estimate be isotonic with respect to the quasi-order _~ on X is given by the isotonic regression of 0(x) = (0(Xl), 0(x2),..., 0(xk))~with weight vector w(x) = (W(Xl), w(x2),... ,w(xk)) where w(xj) = njp2(zj) and O(xj) is given by (5.6). For the proof of Theorem 5.1 the reader is referred to Robertson, Wright and Dykstra (1988, p. 34). Theorem 5.1 is, also, true for the Type II censored data and the corresponding proof is straightforward. Assume, now that zj's are known and let H0:01

Ok; HI:

= 02 . . . . .

andH2:(_0<0j<0) The unrestricted MLE of of 0j under H0 is

(01,02,..., Ok) is isotonic with respect to -4; forjEX .

Oj is given by

(5.6) and the M L E of the common value

00 = 2jk=l w(xs)O(xj) 2~=, w(xj) '

(5.7)

and the M L E of 0 = (01,02,...,0k) subject to 0 C H1 is 0* = Pw(0(xj)lY

(5.8)

where J is the closed convex cone of functions defined on X which are isotonic with respect to ~, and Pw(O(xj)lJ ) is the least squares projection of the unrestricted MLE's {0(xj)} onto the cone J . Let A01 denote the likelihood ratio test (LRT) statistics for testing H0 versus H1 - H0. Then L R T rejects H0 for large values of T01 = - 2 In A01 where l

_-2 Z j=l

(0>- (00/]

k

j=l

(5.9) Similarly, if A~2 denotes the L R T statistics for testing H L R T rejects H1 for large values of T12 = - 2 In Ai2 where

l versus//2

- H~, then,

18

A. P. Basu and B. Singh k k T12 = 2 Z wjOj [/91( 8 j ) - pl (0;) 1 + 2 E j=l j=l

nj[q(Oj,zj)-

q(8),

zj)]. (5.10)

The LRT of Ho versus/-/2 - Ho rejects Ho for T02 m - 2 In Ao2 where

k k To2 = 2 Z wjOj[Pl (Oj) -- Pl (80)] -}- 2Z nj[q(Oj, V) - q(Oo, V)] • j=l j=l

(5.11)

Now, assume that nj-~ cc and N ~ ~ (with ~ nj = N) in such a way that nj/N ~ aj E (0, 1) for j E X. Let P(I, k; w) denote the probability that, under H0, there are I distinct coordinates in the vector 0* of the restricted M L E of 0. Then, we have the following theorem which can be used to compute the tail probabilities of the appropriate test statistics. THEOREM 5.2. If H0 is true, for any real number C1,

k lim P[Tol >_ C1] = ZP(l,k;w)P[z~_l >_ C1] • N---+c~ l-1

(5.12)

Moreover, H0 is asymptotically least favorable within Ha, and under H0, for any real number C2,

k lim P[T12 >_ (72] = E P(l,k;w)P[)~_l >_ C21 • l=1

(5.13)

Furthermore, under H0 and for any real number C3, lim P[T02 >_ C3] =P[z~-I -> C3] .

(5.14)

The coefficients P(l,k; w) are the level probabilities associated with the usual normal means problem which are discussed extensively in Chapters 2 and 3 in Robertson, Wright and Dykstra (1988). For a proof of Theorem 5.2 the reader is referred to Robertson et al. (1988, pp. 164-166).

5.3. Application to one parameter exponential distributions In this section we give a very brief account of the results for the case of complete and Type II censored samples from one parameter exponential distributions. The case of one parameter exponential distribution is included in Section 5.2 as a special member of the exponential family (5.2) with the following substitutions: pl(0)-

1

0'

p2(z)=l,

K(t;z)=t,

The case of compete samples is discussed next.

S*(t;~)=Oandq(O,z)=-lnO

Order stat•tics in exponential distribution

19

Case I. Complete samples Assume k independent random samples {tji; i = 1,..., nj;j = 1 , . . . , k} from k one parameter exponential distribution e(0j). In this case the unrestricted M L E of Oj, Oj (5.6) simplifies and is given by

Oj = Z t#nj ,

(5.15)

i=1

and the weight vector is w = (w~,..., wk) with wj = nj. The M L E of the common value of 0j in (5.7) is given by

00 = Z nj j=l

(5.16)

.j /j=l

and the M L E of 0 = (01,02,..., Ok) subject to 0 E H1 is given by 0* = Pw(OjlJ) where Oj is now given by (5.15). Similarly, the LRT statistics (5.9), (5.10) and (5.11) simplify. The level probabilities P(l, k; w) now depend on the sample sizes ( n 2 , . . • , nk) and the null distributions of T01, T12 and T02 are given by Theorem 5.2 with these level probabilities.

Case II. Type censored samples Assume now that there are k independent random samples of size nj from one parameter exponential distributions e(Oj),j = 1 , . . . , k. Let the rj smallest (order statistics) failure times for the jth sample be tj(1) < tj(2) < - . .

<

tj(rj),

rj ~ nj, (j = 1 , . . . , k )

.

Now that rl,r2,...,rk are prespecified numbers. Letting Tj=~i~J=ltj(i)+ (nj -rj)tj(,j) be the total time on the test in the jth (/. = 1,... ,k) population, the unrestricted M L E of Oj is given by 0j = Tj/rj. The MLE of the common value of 0j under H0, say 00, is 00 = ~ = l rjOj/~=l rj. The M L E of 0 = (Ol,O2,...,Ok) subject to 0 E Ha is given by 0* = Pw(0jlJ)

with

w = (F1,F2,..., rk)

where J is the closed convex cone of functions defined on X which are isotonic with respect to quasi order _~, and Pw(Oj[J) is the least squares projection of the unrestricted MLE's {0j} onto cone J . The likelihood ratio test statistics T01, T12 and T02, respectively, are given by (5.9), (5.10) and (5.11) with appropriate substitutions as specified above. Moreover, the null distributions of Tol, T12 and T02 are given by Theorem 5.2 where the level probabilities P(I, k; w) now depend on w = (rl, r 2 , . . . ,

rk).

A. P. Basu and B. Singh

20 6. Bayesian inference

In this section we consider Bayesian inference for the scale parameter 0 of the exponential distribution. In the Bayesian approach the parameter 0 is considered a random variable. The density function of the exponential distribution, conditional on 0, is given by

f(tlO) = O a exp(_t/O),

t > O, 0 > 0

.

We shall denote this exponential density by tlO ~ e(O). Using (3.2) and (3.4) the likelihood functions, L1 (0) = f(T~lO ) and L2(0) = f(T~lO ) for Type I and Type II censoring are given by

L1 (0) ~ 0 -~ exp(-T~r/O)

(6.1)

L2(0) oc 0 -r exp(-Tr/O)

(6.2)

and

respectively, where Tr and T[ are given by (3.4) and (3.11) respectively. Note that likelihoods (6.1) and (6.2) are of the same functional form. Let 9(0) denote the prior density of 0 which reflects the prior knowledge about 0 before the data are collected. We obtain the posterior density of 0 given the data using Bayes' theorem. Since the likelihoods (6.1) and (6.2) are of the same functional form, the posterior densities are of the same functional form except that the data consist of sufficient statistics (r, T/) in case of Type I censoring and Tr in case of Type II censoring. We consider the following two prior distributions of 0. First, we consider the noninformative prior, 91(0) given by 1 91(0) oc~,

0<0
(6.3)

when we do not have any prior information about 0. To reflect a prior opinion, the inverted gamma distribution, which is the conjugate prior, is used. The corresponding conjugate prior density, 92(0) is given by

g2(O) = ~ v ) 0-(~+l)e-~'

0> 0 .

(6.4)

We denote this inverted gamma density by 0 ~ IG(c~, v). Here the hyper parameters c~ (c~ > 0) and v (v < 0) are chosen to reflect the prior belief on 0. Note that the noninformative prior distribution given by gl(0) is a special case of the conjugate prior distribution given by g2(0) when c~= v = 0 with the convention that 0° = 1 and F(0) = 1. Combining the prior density g2(0) in (6.4) with likelihood L2(O) in (6.2), the posterior density of 0 given Tr is given by

21

Order statistics in exponential distribution

t(rrlO) O2(0) 92( OITr) = f o f ( Tr[O) 92(0) dO

(6.5)

- (~ + T')~+~ 0 (~+~+l/e ~r~,

0 > 0 .

r(v + r)

The posterior density (6.5) is again an inverted gamma density with parameters + Tr and v + r. That is, O[Tr ~ IG(c~ + Tr, v + r). In this case the Bayes estimator of 0, assuming the squared loss function, is given by the posterior mean, 0B2 = E(0IT~) --

c~+ T~ v+r- 1 '

(6.6)

provided v + r > 1. The posterior variance of 0 is given by

E 0-0B2

(v+r_ 1)2(v+r_2 )

Similarly, the Bayes estimator of 0, say 0e t , assuming squared error loss function, for the Type I censored sample is given by (6.6) with Tr replaced by T~. The corresponding posterior variance of 0 for the Type 1 censored sample is similarly given by (6.7) with T~ replaced by T~. The Bayes estimator and posterior variance of 0 using noninformative prior 91 (0) can be obtained in an analogous manner from (6.6) and (6.7) with the substitution ~ = v = 0. We next consider the two parameter exponential distribution with density (3.5). Here the conditional density of t given # and 0 is 1

f ( t l # ,O)=-~e

(t-~/

o ,

t>_# .

As in Example 3 in Section 3 consider a Type II censored sample t = (t0) < t(2) < ... < t(r)) size r based upon a random sample on size n. The iikelihood function L(#, O) = f ( t [#, 0)

(6.8)

is given by (3.6). Now, consider the class of noninformative prior distribution 1 gl(#,0) o(0,

--o(~<#
0>0

.

The posterior distribution of (#, 0) is then given by f 1 1)0 r+l e x p ' - 0 (Tzr + n(t(l) - #)) }

n T,~;-l

g(#,O It) = F ( r -

The marginal posterior distribution of 0 is then given by

(6.9)

22

A. P. Basu and B. Singh

t(l)

hi (0] t ) =

g(#, 01 t ) d # • ,' - - 0 0

T~ 1 { Y2r~ = OrF(r - 1) exp - - ~ - j ,

0>0

(6.10)

The Bayes estimator of 0 is then

rzr

OB--r_2,

r>2

.

(6.11)

Similarly, the marginal posterior density of # is given by ( r - 1)nr~r 1 h 2 ( ~ l t = fo0 ° g(#,O)lt)dO={Tzr+n(t(1)_#)}r

, -cx~<#
Bayes estimator of # is then given by

fiB=t(1)

T2r

n(r--2)'

r>2

.

(6.12)

As in the case of one parameter exponential distribution. Bayes estimators of 0 and # will be similar to (6.11) and (6.12) with the appropriate modifications for Type I censored data. For any other arbitrary prior distribution g(O) the posterior distribution may not be tractable in a closed form. In such situations, one may consider appropriate approximate methods put forth by Lindley (1980), Tierney and Kadane (1986) and others. See O'Hagan (1994) and Press (1989) for some of these and other related developments. We can similarly carry out Bayesian tests of hypotheses. For details see Chapter 10 in Balakrishnan and Basu (1995). Moreover, inference procedures for the two sample and k-sample problems from a Bayesian view point can be developed in a similar manner.

Acknowledgement This research has been supported partially by the U.S. Air Office of Scientific Research under the grant No. F49620-95-1-0094.

References Balakrishnan N. and A. C. Cohen (1991). Order Statistics and Inference: Estimation Methods. Academic Press, San Diego. Balakrishnan, N. and A. P. Basu (eds.) (1995). The Exponential Distribution: Theory, Methods and Applications. Gordon and Breach, Newark, New Jersey. Bartholomew, D. J. (1963). The sampling distribution of an estimate arising in life testing. Teehnometrics 5, 361-374.

Order statistics in exponential distribution

23

Basu, A. P. and H. Block (1975). On characterizing univariate and multivariate exponential distributions with applications. Statistical Distribution in Scientific Work 3, 379M21. ed. Patil, G. P. et al. Elsevier. David, H. A. (1981). Order Statistics. 2nd edn. John Wiley & Sons, New York. Karlin, S. and H. M. Taylor (1975). A First Course in Stochastic Processes. 2nd edn. Academic Press, New York. Lawless, J. F. (1982). Statistical Models and Methods for Lifetime Data. John Wiley & Sons, New York. Lindley, D. V. (1980). Approximate Bayesian methods. Bayesian Statistics. pp. 223-245. eds. J. M. Bernardo, M. H. De Groot, D. V. Lindley, and A. F. M. Smith. Valencia Press, Spain. Lininger, L., M. H. Gail, S. B., Green and D. P. Byar (1979). Comparison of four tests for equality of survival curves in presence of stratification and censoring. Biometrika 66, 419428. O'Hagan, A. (1994). Bayesian Inference, Kendall's Advanced Theory o f Statistics. Vol. 2B, Edward Arnold, London. Press, S. J. (1989). Bayesian Statistics: Principles Models and Applications. John Wiley & Sons, New York. Renyi, A. (1953). On the theory of order statistics. Acta. Math. Acad. Sci, Hung. 4, 191-231. Robertson, T., T. W. Wright and R. L. Dykstra (1988). Order Restricted Statistical Inference. John Wiley & Sons, New York. Sarhan, A. E. and B. G. Greenberg (eds.) (1962). Contributions to Order Statistics. John Wiley & Sons, New York. Sinha, S. K. and B. K. Kale (1980). Lifetesting and Reliability Estimation. Wiley, New Delhi. Sukhatme, P. V. (1937). Tests of significance for samples of Z2-population with two degrees of freedom. Ann. Eng. 8, 52-56. Tierney, L. and J. Kadane (1986). Accurate approximations for posterior moments and marginals. J. Amer. Statist. Assoc. 81, 82-86.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics Vol. 17 © 1998 Elsevier Science B.V. All rights reserved.

.~

Higher Order Moments of Order Statistics from Exponential and Right-truncated Exponential Distributions and Applications to Life-testing Problems

N. Balakrishnan and Shanti S. Gupta

1. Introduction Let X 1 , X 2 , . . . ,Xn be a r a n d o m sample of size n from a standard exponential population with probability density function f ( x ) = e -x,

0_<x
,

(1.1)

and cumulative distribution function f(x)=l-e

-x,

0_<x<ec

.

(1.2)

It is important to observe here that 0<x
f(x)=l-F(x),

.

(1.3)

Let Xl:n _< X2:n ___ ,.- < Xn:~ be the order statistics obtained by arranging the above sample in increasing order of magnitude. Let us denote the single moments E[X~n] a b (a b) by #~) for 1 < r < n and a _> 1, the double moments E[X¢:~J:,] by ~r,s:n for a b c (a.b,c) 1 _< r < s <_ n and a, b >_ 1, the triple moments E[X¢:,XJ:nX;:,] by IJr,;,t: n for 1 <_ r < s < t _< n and a, b, c _> 1, and the quadruple m o m e n t s E[X¢:nXJ:nX~:nX~u:n]a b c d (a,b,c.d)

by#r,s,tl,;~ for 1 < r < s < t < u _ < n a n d a , b,c,d_> 1. It is well known that the normalized spacings $1 = nXl:,, 32

:

(n - 1)(X2:, - X I : , ) , . . . ,

Sn = Xn:n - Xn-I:n

are all statistically independent and have standard exponential distributions; see Sukhatme (1937), This result, as pointed out by David (1981) and Balakrishnan and Cohen (1991), will enable one to express the r th order statistic in a sample of size n as a linear function of r independent standard exponential r a n d o m variables and thus to prove that the exponential order statistics form an additive M a r k o v chain. One may use this representation to derive exact and explicit expressions for moments of exponential order statistics. 25

26

N. Balakrishnan and S. S. Gupta

Alternatively, by using the relation in (1.3), Joshi (1978, 1982) derived some simple recurrence relations satisfied by the single and product moments of order statistics from a standard exponential distribution. Joshi pointed out that these recurrence relations are very easy to use and that one could write a simple computer program to evaluate the first and second single moments and the product moments of all order statistics without introducing serious rounding errors, at least up to moderately large sample sizesl These results have been generalized by Joshi (1979, 1982) and Balakrishnan and Joshi (1984) for the case when the order statistics arise from right-truncated and doubly truncated exponential distributions. All these results for the exponential and truncated exponential distributions and also similar results for some other distributions including the power function, Pareto, logistic, half logistic, log-logistic and truncated logistic are presented in the monograph by Arnold and Balakrishnan (1989) on this topic of research. In this paper, we derive several recurrence relations satisfied by the single, double, triple and quadruple moments of order statistics from the standard exponential distribution in Sections 2-5, respectively. In Section 6, we consider the scale-parameter exponential distribution and use the results derived in Sections 2-5 to determine the mean, variance and coefficients of skewness and kurtosis of the best linear unbiased estimator of the scale parameter based on doubly Type-II censored samples. While it is well known that for the case when the available sample is Type-II right censored the best linear unbiased estimator of the scale parameter has exactly a chi-square distribution, we show in Section 6 that a chi-square distribution provides a very close approximation to the distribution of the best linear unbiased estimator even when the available sample is doubly Type-II censored. In Section 7, we generalize all the results presented in Sections 2-5 to the case when the order statistics arise from a righttruncated exponential distribution. Finally, in Section 8 we consider three examples involving life-testing data and illustrate the usefulness of the chi-square approximation for the distribution of the best linear unbiased estimator discussed in Section 6 by constructing approximate confidence intervals for the mean life-time.

2. Relations for single moments

The probability density function of Xr:n is (see David, 1981, p. 9) given by n! IF(w)]r-1 [1 - F(w)]n-~f(w), f~:,(w) = ( r - 1)!(n - r ) ! 0_<w<ec, l
(2.1) ,

using which the single moments of Xr:n may be computed by /~(a) = r : n

/o

waf~:,(w)dw,

a = 1,2,

"

"

"

~

l
"

(2.2)

27

Higher order moments of order statistics

, (a) satisfy the recurrence relations presented in the Then, the single m o m e n t s ~r:n following two theorems. THEOREM 1. F o r n _> 1 and a = 1 , 2 , . . . , #(a) (a 1),/ n l:n = a#l:n

(2.3)

with #(o) l:n ~ 1. PROOF. F r o m Eqs. (2.1) and (2.2), for n > 1 and a _> 1 let us consider ttl~n-1) = n

/7

wa-l[1 - F ( w ) ] n - l f ( w ) d w

(2.4) [1

F(w)] n d(w a)

u p o n using the relation in (1.3). The recurrence relation in Eq. (2.3) follows u p o n integrating the right h a n d side of (2.4) by parts and simplifying the resulting expression. THEOREM 2. F o r n _> 2, 2 < r < n and a = 1 , 2 , . . . , (a-l) #(a) r:n = #(a) r-l:n _I_ a#r:n /(?./ _ r -}- 1)

(2.5)

with t*r:n , (0) =- 1. PROOF. F r o m Eqs. (2.1) and (2.2), for n _> 2, 2 < r < n and a _> 1 let us consider n!

/0

W a-1 [F(w)] ~-1 [1 - F ( w ) ] "

rf(w) dw

n~ (r - l)!(n - r)!a

(2.6)

where I is the integral

I =

/0

[F(w)l r-1 [1 - F(w)] n r+l d(w a)

u p o n using the relation in (1.3). Integration by parts now yields I = (n - r + 1)

/7

waiF(w)] r 1[1 - F ( w ) ] n - r f ( w ) d w

- (r - 1) f0 ~ w a [ f ( w ) ] r - 2 [ 1 -- F(w)]n-r+If(w)dw

=

It- l l!(ea-r + 11!{# r : , (

r-l:,j

•

28

N. Balakrishnan and S. S. Gupta

Upon substituting this expression of I in (2.6) and simplifying the resulting equation, we derive the recurrence relation in Eq. (2.5). It should be mentioned here that Theorems 1 and 2 have been proved by Joshi (1978) and have been presented here for the sake of completeness and a better understanding of the results derived in subsequent sections. The recurrence relations presented in Theorems 1 and 2 will enable one to compute all the single moments of order statistics for all sample sizes in a simple recursive way.

3. Relations for double moments

The joint density function of Xr:n and Xs:n is (see David, 1981, p. 10) given by n!

1)!(s- r - 1)!(n- s)! [F(w)]~-i

fr,s:,(w,x) = ( r -

x IF(x) - F(w)] *-r-1 [1 - F ( x ) ] " - S f ( w ) f ( x ) , 0_<w<x<ec,

l
(3.1) ,

using which we may compute the double moments as #!a,s.b.2 :

/[ ti

waxbfr,s:n(W,x) d w d x ,

1
< s < n, a , b 2> 1 .

~]

0<w<x
(3.2) (a.b) Then, the double moments p~,s:, satisfy the recurrence relations presented in the following two theorems. THEOREM 3. For 1 < r < n - 1

anda, b= 1,2,...,

= [2(a+b) . . ( a b - 1 ) H + O#r,;+l:,/kn - r) ;

x(a,b)

r,~+1. . . . .

and for 1 <_ r < s <_ n, s - r

> 2 and a , b = 1 , 2 , . . . ,

~,asb2 = ]/~a,b_)l:n + bl~,i':. (ab 1) / ( n - s + 1) , (a,0)

(3.3)

(3.4)

, (a)

where /2r,s:n ~ i~r:n . PROOF. From Eqs. (3.1) and (3.2), for 1 _< r < s _< n and a , b _> 1 let us consider ]2(a,b_ ..... 1) = ( F - 1 ) [ ( s - r n! - 1)[(n-s)[b i ~ wa[f(w)]r l f ( w ) I ( w ) d w

'

(3.5) where I(w) =

fOG

IF(x) - F(w)] . . . . 111 - F(x)]" s+l d(x b)

Higher order moments of order statistics

29

u p o n using the relation in (1.3). Integrating by parts, we obtain for s = r + 1 that

I(w) = (n - r)

/?

xb[1 -- F(x)ln-r-l f ( x ) dx - xb[1 -- F(x)] "-r ,

and for s - r > 2 that

I(w) = ( n - s + 1) --(s--r--

/? /?

xb[F(x) -F(w)]S-r-l[1 - F(x)]n-S f ( x ) d x xb[F(x)_F(w)lS

1)

.

r 2[l_f(x)]n-s+lf(x)dx

U p o n substituting the above expressions of I(w) in Eq. (3.5) and simplifying the resulting equations, we derive the recurrence relations in Eqs. (3.3) and (3.4). THEOREM 4. F o r n >_ 2 and a, b = 1 , 2 , . . . , (a,b) = #(a+b) (a-l,b) (a+b) 1,2:n 2:n +a#1,2:n --]'/#1:n 1 ; forn_>3,2
(3.6)

anda, b=l,2,...,

(a+b)+! r

. = #~+l:,

r [a#~x+]':n-

,,S.(o+b) .(a,b) "Ll

,tier:n-1--.t~r

l f]

l ....

;

(3.7)

for n _> 3,3 < s < n and a,b = 1 , 2 , . . . ,

a" (a-l,b) #I;~2 = #~s~2 -[- ,Ul,s:n

-

-

O,b) n#1,s l:n-I ;

(3.8)

and for n > 4,2 < r < s < n , s - r > 2 and a, b = l , 2 , . . . ,

1 a#!~;, ~,b)

r-r,s:n

r

r (a b)

--'~.#r,s-l:n-I

--

,-

}]

,

(3.9)

(0 b) where #~,s':, - #~,) . PROOV. F r o m Eqs. (3.1) and (3.2), for 1 < r < s _< n and a,b _> 1 let us consider #(a-l,b) r,s:n = ( r -

1 ) ! ( s - rn! - 1 ) ! ( n - s ) ! a f0 ~ xb[1 - F ( x ) ] " " f ( x ) J ( x ) d x

,

(3.10) where

J(x) =

fox[F(w)] r

'IF(x) - F(w)] s-r-] f ( w ) d(w a)

= fX[F(w)]~-] [F(x) - F(w)] s-~-I d(w a)

- foX[F(w)l [F(x)

- F ( w ) ] s-~-I d(w a)

30

N. Balakrishnan and S. S. Gupta

u p o n using the relation in (1.3). Integrating by parts, we obtain for r = 1 and s = 2 that

foxw a f ( w )

J ( x ) = x ~ - x~F(x) +

forr_>2ands=r+l

that

J ( x ) = x~[F(x)] r-1 - (r - 1)

-xa[F(x)]r +r

forr=l

dw ,

fo Xwa[F(w)]r-2 f ( w )

~0xwa[F(w)]r-l f ( w ) d w

dw

,

and s _> 3 that J ( x ) = (s - 2)

+

foxwa[F(x)

fo Xw a IF(x) -

- (s - 2)

- F(w)]S-3 f ( w ) d w

F(w)] ~-2 f ( w ) d w

fo Xw a F ( w ) [ F ( x )

- F(w)] s J f ( w ) d w ,

and for r_> 2 and s - r >_ 2 that J ( x ) = (s - r - 1)

fo Xw~[F(w)] r-1 IF(x) -

- (r- l) foXwaIF(w);-2[F(x) + r

foXwa[F(w)] r

- (s - r - 1)

F(w)]S-r-Z f ( w ) d w

- F(w)]~-r-lf(w) dw

l[F(x) - F(w)] s-r-1 f ( w ) dw

~0Xwa[F(w)]r[F(x)

_ F(w)]S

r

2 f ( w ) dw .

U p o n substituting the above expressions of J ( x ) in Eq. (3.10) and simplifying the resulting equations, we derive the recurrence relations given in Eqs. (3.6)(3.9). Either the recurrence relations presented in T h e o r e m 3 or those presented in T h e o r e m 4 m a y be used in a simple systematic recursive way to c o m p u t e the double m o m e n t s o f order statistics (of all order) for all sample sizes. The relations in T h e o r e m 3 have been proved by Joshi (1982) for the special case when a=b=l.

31

Higher order moments o f order statistics

4. Relations for triple moments T h e j o i n t d e n s i t y f u n c t i o n o f Xr:n,Xs:, a n d Xt:~ is g i v e n b y

n! f~,s,t:,(w,x,y) = (r - 1)!(s - r - l ) ! ( t - s - 1)!(n - t)! [F(w)]r-1 × IF(x) - F(w)] s-r-1 IF(Y) - F ( x ) ] t-~-I × [1 - F ( y ) ] n - t f ( w ) f ( x ) f ( y )

,

O<w<x
~r,s,tln :

(4.1)

, l <_r<s
the t r i p l e m o m e n t s

as

waxbyCfr,s,t:n (W, x, y) d w d x dy, O<_w<x
(4.2)

1 <_r<s
Then, the triple moments following three theorems. THEOREM 5. F o r n _ > 3 , lz(a,b,c)

r,s,s+l:n :

a,b,c>

1 .

s a t i s f y the r e c u r r e n c e r e l a t i o n s p r e s e n t e d in the

l_
anda,

b,c=l,2,...,

1),/ #r,~,s~l:n / ( n -- s) ;

p(a,bWc) ' e (abe

.....

l
andforn>4,

±

t-s>_2anda,

(4.3)

b,c=l,2,

.,

(a,b,c) = #(a,b,c) . (a,b,c-1),, #,-,~,t:n r,~,t-l:,, + c #~,~,t:. / ~n - t + 1) , • (a,b,0)

,

(4.4)

(a b)

where 12r,s,t:n ~ ]Ar,~:n • PROOF. F r o m consider

Eqs. (4.1) a n d (4.2), f o r 1 _< r < s < t _< n a n d a , b , c >_ 1 let us

(a,b c-1) ~r,s,tln =

(r - 1)!(s - r - 1)!(t - s - 1)!(n - t)!c X IF(x) - F(w)] s - r - 1 1 ( x ) f ( w ) f ( x )

dxdw

IF(Y) - F(x)] t-s-1 {1 - F ( y ) ] n - t f ( y )

d ( y c)

where

I(x) = =

fx °~

[ F ( y ) - F(x)] t-s-I [1 - F ( y ) ] n-t+~ d ( y c)

waxb[F(w)]r-~ ,

(4.5)

N. Balakr&hnan and S. S. Gupta

32

u p o n using the relation in (1.3). I n t e g r a t i n g by parts, we o b t a i n for t = s + 1 that

I(w) = ( n - s )

/x

j{1

-F(y)}n-s-lf(y)dy-yC{1

-F(y)} n ' ,

a n d for t - s _> 2 that

yC{f(y) - f(x)}t-s-l{1 - f(y)}n-tf(y) dy

I(x) = (n - t ÷ 1)

- (t-s-

yC{F(y) -F(x)}t-s-2{1 - F ( y ) } n - t + l f ( y ) d y

1)

.

U p o n substituting the a b o v e expressions o f I(x) in Eq. (4.5) a n d simplifying the resulting equations, we derive the recurrence relations in Eqs. (4.3) a n d (4.4). THEOREM 6. F o r n _> 3, 3 < t < n a n d a,b,c = 1 , 2 , . . . , Id(a,b,c) 1,2,t:n

forn_>4,

=

(a+b,c) (a-l,b,c) n (a+b,c) ]'12,t:n + al~l,2,t:n -~l,t-l:n-1

2 <_r < t <_n, t - r

(4.6)

;

>_2 and a, b , c = l , 2 , . . . ,

#(
((a+b.c)

iA(a,b,c)

1] (4.7)

for n _> 4, 3 < _ s < t < _ n a n d a , [A( a ' b ' e ) l,s,t:n --

and forn_>5,

]A(a'b'c)

a (a-l,b,c) t'l (a,b,c) ]Al,s,t:n -- [Al,s-l,t-l:n

2,s,t:n ÷

2
(a,b,c) = p(a,b,c)

#r,s,t:n

b,c=l,2,...,

s-r_>2anda, 1 [

(a-i b,c)

r+l,s,t:n ÷ - [ a l t r , s . t : n F

--

(4.8)

i ;

b,c=l,2,...,

n f (a,b,¢)

~r,s-l,t-l:n-1

_ #(a,b,c)

r-l,s-l,t

}] l:n

1

'

(4.9) (0.b#) (b,c) where #r,s,t:n = Ps,t:n PROOF. F r o m Eqs. (4.1) a n d (4.2), for 1 _< r < s < t _< n a n d a, b, c _> 1 let us write (a-l,b#) = n! #r,s,t:n (r -- 1)!(s -- r -- 1)[(t -- s -- 1)!(n -- t)!a

× f°°foYxbyC[F(y)- F(x)]t-x-~[1- F(y)]n-t x J(x) f(x) fO') dx dy , where

(4.1 O)

Higher order moments of order statistics

33

J(x) = foX[F(w)l r-I [F(x) - F ( w ) ] s-r-1 f ( w ) d ( w a) : foX[F(w)] r-' IF(x) - F(w)] s-r-1 d ( w a)

_

foX[F(w)] [F(x)

_ F(w)]S r 1 d ( w ~)

u p o n u s i n g the r e l a t i o n in (1.3). I n t e g r a t i n g by p a r t s , we o b t a i n for r = 1 a n d s = 2 that

J(x) = x a - xaF(x) + fo x w ~ f ( w ) dw , for r >_ 2 a n d s = r ÷ 1 t h a t

J(x) = xa[F(x)] r-1 - (r - 1)

foXW°[x(w)] -2f(w)dw

- x~[F(x)] ~ + r fo x wa[F(w)] r-1 f ( w ) dw , forr=l

ands>3that

J(x) = (s - 2) ~o x waiF(x) - X(w)]S-3 f ( w ) d w

+ foXwa[F(x) - F(w)l s 2y(w) dw - (s - 2) fo x waF(w)[F(x) - F(w)] s 3 f ( w ) dw , a n d for r > 2 a n d s -

r > 2 that

J(x) = (s - r - 1) fo x w~IF(w)l r-1 [F(x) - F(w)]S-r-2 f ( w ) dw - (r - 1) foXwa[F(w)]~-ZIF(x) - F ( w ) l S - ~ - l f ( w )

dw

+ rfoXwa[F(w)] ~ 1IF(x) - F(w)] s r - l f ( w ) dw - (s - r - 1) fo x w~[F(w)lr[F(x) - F ( w ) ] ~ - r - Z f ( w ) dw . U p o n s u b s t i t u t i n g the a b o v e e x p r e s s i o n s o f J(x) in Eq. (4.10) a n d s i m p l i f y i n g the r e s u l t i n g e q u a t i o n s , we d e r i v e the r e c u r r e n c e r e l a t i o n s given in Eqs. ( 4 . 6 ) (4.9).

34

N. Balakrishnan a n d S. S. Gupta

THEOREM 7. F o r n _> 3, 1 < r < n -- 2 a n d a,b,c = 1 , 2 , . . . , ]A(a,b,c) #(a+b,c) , . (a.b-l,c) r,r+l,r+2:n = r,r+2:n -t- OlAr,r+l,r+2: n --

(Iv/

.x ((a,b+c)

-- r -- l ) ~ # r , r + l : n

--

[2 (a+b'c) 1

r,r+l:nJ

'

(4.11) forn_>4,

l <_r
t-r>

3anda, b,c= l,2,...,

(a+b,c) 1 F. ( a b - l , e ) . #r,t:n ' -[- t -- r -- 1 L°#r'~+l't:" - - t n -- t + 1)

x f (~ b c)

I #r,r~+'l,t- l :n --

forn>4,

l <_r < s < < _ n - 1 , s - r > _ 2 a n d a , ]l(a,b,c) = u(a,b,c) r,s,s+l:n r,s 1,s+l:n

(4.12)

p(~+b,~) l~] . r,t- l:n J ]

'

b,c= l,2,...,

(a,b-l,e) _ ( B - - S ) { /2!,°s:bn +c) _/X(a'b,cl r,s l,s:nJI , ÷ .O#r,s,s+l:n

(4.13) andforn>

5, l < r < s < t < _ n ,

s-r>2,

t-s>>_2anda, b,c=l,2,...,

.(a,b,c) /A(a,b,c) ÷ ~ ~ . (a.b-I c) 1,f (abc) ' -- (n -- t ÷ )t#r.s'.tSl:n-#r,s,t:n ~ r,s 1,t:n t -- S LO~tr.s,t:n ' ''

ix(a,b,c)

~]

r,s_l,t_l:nj j

(4.14)

(a,O,c)

(a,c)

w h e r e ]Ar,s,t: n ~ #r,t:n

•

PROOF. F r o m Eqs. (4.1) a n d (4.2), f o r 1 _< r < s < t _< n a n d a, b, c _> 1 let us w r i t e (a,b-l,c) =

n! (f' -- 1)!(S -- r -- 1)!(t -- s -- 1)!(n -- t)!b

#r,s,t:n

x

/07;

way~[F(w)] ~-1 [1 - F ( y ) ] n - t K ( w , y ) f ( w ) f ( y )

dw dy , (4.15)

where K(w,y)

= fwY[f(x) - f ( w ) ] s r l [ f ( y ) _ f ( x ) ] t - s - 1

f(x)

d ( x b)

= LY[F(x) - F(w)] s-r-1 [F(y) - F(x)] t-s d ( x b) + LY[F(x) - F(w)] s-r-1 [F(y) - F(x)] t-s-1 [1 - F ( y ) ] d ( x b) upon

u s i n g t h e r e l a t i o n in (1,3) a n d t h e n w r i t i n g 1 - F ( x ) as I F ( Y ) - F ( x ) ] ÷ by parts now, we obtain for s = r + 1 and t = r + 2 that

[1 - F ( y ) ] . I n t e g r a t i n g

K(w,y) = -wb[F(y) -- F ( w ) ] ÷ LYxb f(x) dx ÷ y b [ 1 -- F ( y ) ] -- wb[1 -- F ( y ) ]

,

Higher order moments of order statistics fors=r+l

35

andt-r_>3that K ( w , y) = - w b [F(y) - F(w)]t-r-1 + (t - r - 1)

xb[F(y) - F(x)] t-r 2 f ( x ) dx

- wb[F(y) - F(w)]t-r-z[1 - F(y)] ÷ (t -- r -- 2)

fors-r>2andt=s+l K(w,y) =

xb[F(y) -- F(x)lt-r-311 - F ( y ) ] f ( x ) dx ,

that

2

xb{F(x) -- F(w)]" r l f ( x ) d x - ( s - r -

1)

× J Y x b [ F ( x ) -- F(w)] s-~ 2[g(y) - F ( x ) ] f ( x ) dx + yb[F(y) - F(w)] ~-~-l [1 - F(y)] - (s - r - 1) x

a n d for s -

xb[F(x) - F(w)]S-~-2[1 - F ( y ) ] f ( x ) dx ,

r_> 2 a n d t - s K ( w , y ) = (t - s)

- (s-r-

+ (t-s

_> 2 that xb[F(x) - F(w)] s-~-I [F(y) - F(x)] t-s-1 f ( x ) dx

1)

- 1)

xb[F(x) - F ( w ) ] ~ r 2{F(y)

2

× {1 - F ( y ) ] f ( x )

x~{F(x) _ F ( w ) ] ~

dx - (~ - r - l)

× IF(y) - F(~)] t-~-I [1 - F ( y ) ] f ( x )

-F(x)]t-'f(x)d~

~-IEF(y ) _ F ( x ) ] t - ,

£

2

x~{F(x) - F ( w ) ] ' - ~ - 2

d~.

U p o n substituting the a b o v e expressions o f K ( w , y) in Eq. (4.15) a n d simplifying the resulting equations, we derive the recurrence relations given in Eqs. (4.11)(4.14). The recurrence relations presented in a n y one o f T h e o r e m s 5, 6 a n d 7 m a y be e m p l o y e d in a simple systematic recursive m a n n e r to c o m p u t e the triple m o m e n t s o f o r d e r statistics (of all order) for all sample sizes.

N. Balakrishnan and S. S. Gupta

36

5. Relations for quadruple moments The joint density f u n c t i o n ofXr:., Xs:., Xt:. a n d X~:. is given by

n! 1 ) ! ( t - s - 1 ) ! ( u - t - - 1 ) [ ( n - u)[

1)[(s- r-

fr,s,t .... ( W , X , y , z ) = ( r -

x IF(w)] r - l [ F ( x ) - F(w)]" x IF(z) - F(y)]U

t-I

l[F(y) - F(x)] t - , - I

"

[1 - F ( z ) ] n - U f ( w ) f ( x ) f ( y ) f ( z )

O<_w<x
,

l<_r<s
(5.1) using w h i c h we m a y c o m p u t e the q u a d r u p l e m o m e n t s as

dw,,.,d.,,dz

,.,,..,,,,= r,s,t,u;n 0_<~<x
(5.2)

1 <_r<s
a,b,c,d>

1 .

(abcd)

T h e n , the q u a d r u p l e m o m e n t s #r,;,ti~':n satisfy the recurrence relations p r e s e n t e d in the following four theorems. THEOREM 8. F o r n _> 4, 1 _< r < s < t _< n - 1 and a, b, c, d = 1 , 2 , . . . , #(a,b,c,a) (a,b.c+d) . (a,b,c,d 1 ) H r,s,t,t+l:. = #r,s,tln ~- a#r,s,t,t._l: n /~./'/

andforn_>5,

--

t) ;

(5.3)

1 <_ r < s < t < u <_ n, u - t >_ 2 and a, b , c , d = 1 , 2 , . . . ,

(abcd)

#.,s',G':. =

#(a,b,c,d)

.

(ab.cd

,,s,,,.-~:. + auG',,;.':.

1)..

/~n - u + 1) ,

(5.4)

(a,b,c,0) (a,b,c) w h e r e #r,s,t,u:n ~ #r,s,t:n • PROOF. F r o m Eqs. (5.1) and (5.2), for 1 _< r < s < t < u _< n a n d a , b , c , d > 1 let us write (a,b,c,d 1 ) =

n!

(r- 1)!(s-,'- 1)!(t-s- 1)!(u- t - 1)!(n- u)Id

.... ' ....

X fo°Cfw~fx°°WaxbyC[F(w)]r-~[F(x) - F(w)]S-r-1

× [F(y) - F(x)] t s-I I ( y ) f ( w ) f ( x ) f ( y ) where , OO

I(y) = =

IF(z) - F(y)] u t-, [1 - F(z)]" Uf(z) d(z d) [f(z)

-

f ( y ) ] u-t

1[1 -

F(z)]n-U+ld(zd)

dydxdw

,

(5.5)

37

Higher order moments of order statistics

u p o n u s i n g the r e l a t i o n in (1 •3). I n t e g r a t i n g b y p a r t s , we o b t a i n for u = t + 1 t h a t

I(y) = - j [ 1 -

F(y)] n t + ( n - t)

/7

z a [ 1 - F(z)]n t l f ( z ) dz ,

a n d for u - t > 2 t h a t I(y) = (n-u+

1)

/7 /7

zd[F(z)-F(y)]U-t-l[1

- - ( u - - t - - 1)

zd[f(z)-F(y)]

u

-F(z)]"-Uf(z)dz

t-2[1-F(z)]

u+lf(z) dz •

n

U p o n s u b s t i t u t i n g the a b o v e e x p r e s s i o n s o f I ( y ) in Eq. (5.5) a n d s i m p l i f y i n g the r e s u l t i n g e q u a t i o n s , we d e r i v e the r e c u r r e n c e r e l a t i o n s given in Eqs. (5.3) a n d (5.4). ThEOrEM 9. F o r n _> 4, 3 _< t < u _< n a n d a , b , c , d

#(a,b,c,a)

1,2,t,u:n =

Ft(a+b,c,d) 2,t,u:n

(~-l,b,c,a)

q- a#l,2,t,u:n

for n_> 5, 2 <_ r < t < u <_ n, t -

#(a,b,c,d)

=

r,r+l,t,u:n

#(a+b,c,d)

r+l,t .... + -

= 1,2,...,

(a+b,c,a)

-- nPl,t-l,u-l:n-1

.

(5.6)

'

r > 2 and a, b , e , d = l,2,..., 1 F (a-l,b,c,a) r

[aHr,r+l,t,u:n

f (~+bX,a)

-- n ~ # r . t - l , u

l:n-1

(5.7)

_ t.t(~,b#, r 1 , ra) ,t-l,u-l:n-l}]; forn>5,

3_<s
b, c, d = l, 2, ...,

(5•8)

~(a,b,c,d) ]~(a,b,c,d) (a l,b,c,d) (a,b,c,d) . 1,s,t,u:n ~--- 2,s,t,u:n -~- alll,s,t,u:n --l'tlll,s l,t-l,u--l:n--l ,

and forn>6,

2_
s-r>2anda,

b,c,d=l,2,...,

(a,b.c,d) ],.l(a'b'¢'d) Jr- ! F (a-1 b c d) ((a.b,c.d) #r,s,tlu:n = r+l,s,t,u:n r ~allr's't'u:n ' -- ne(ktr,s-llt

_kt(~,b#, d)

r 1,s-l,t-l,u-l:n

• (o,b,c,~)

1}]

1,u-l:n-I

(5.9)

'

(b,c,a)

w h e r e I~r,s,t,u:n = #s,t,u:n

PROOF. F r o m Eqs. (5.1) a n d (5.2), for 1 _< r < s < t < u <_ n a n d a , b , c , d us write (a,b,c,a) = #r,s,t ....

___ 1 let

n!

(r -- 1)!(s -- r -- 1)!(t -- s -- 1)!(u -- t - 1)!(n - u)!a f~fzfy

× Jo JoJo xbYCza[F(Y) -- F(x)]t-s l IF(z) - F(Y)]~-t-1 × [1 - F ( z ) ] n - ~ J ( x ) f ( x ) f ( y ) f ( z )

dxdydz

, (5.10)

38

N. Balakrishnan and S. S. Gupta

where

~0xIF(w)] r-I [F(x)

J(x) =

- F ( w ) ] ~-r ' d(w ~)

/o x[F(w)] r-1 IF(x) -

=

-

F ( w ) ] s-r-1 d(w a)

foX[F(w)]~[g(x)

- F ( w ) ] s-~-I d(w a)

u p o n using the relation in (1.3). I n t e g r a t i n g by parts now, we o b t a i n for r = 1 a n d s = 2 that J(x) = x a - x"F(x) +

forr>2ands=r+l

w ~f ( w ) d w ,

j0.~

that

J ( x ) = xa[F(x)] r - ' - (r - 1)

- xaIF(x)] r + r

j0 ~

wa[F(w)]r-2 f ( w ) d w

wa[F(w)] r-I f ( w ) d w ,

for r = 1 a n d s _> 3 that J ( x ) = (s - 2)

/o ~

w a[F(x) - F ( w ) ] s 3 f ( w ) d w

+ f o x wa[F(x) - F ( w ) ] S - Z f ( w )

- (s - 2) a n d for r > 2 a n d s -

/o xw ~ F ( w ) [ F ( x )

dw

- F(w)]S-3f(w)

dw ,

r_> 2 that

J ( x ) = (s - r - 1)

waiF(w)] ~-1 IF(x) - F ( w ) ] s ~ - 2 f ( w ) dw

- (r -

/0 ~ 1) j0 x wa[F(w)]r-2[F(x)

4- r

w a [F(w)] r-1 IF(x) - F(w)] * - r - ' f ( w ) d w

/o ~

- (s - r - 1)

- F(w)] s ~ - l f ( w )

wa[g(w)]~[F(x)

/o ~

_ F(w)]s

r

dw

2 f ( w ) dw ,

U p o n substituting the a b o v e expressions o f J(x) in Eq. (5.10) a n d simplifying the resulting equations, we derive the recurrence relations given in Eqs. (5.6)-(5.9).

39

Higher order m o m e n t s o f order statistics

THEOREM 10. F o r n _ > 4 ,

1
u-r_>3anda,

b,c,d=

1,2,3,...,

(~ b+c.d)n li(a,b,c,d) = li(a+b,c,d) . . (a,b-l,c,d) -- (U -- F -- ~, Z) fl lir,r+l.u: r,r+l,r+2,u:n ,.,r+2,u:n t Dlir,r+l,r+2,u: n . (5.11) -- [2 (a+b'c'd)l --

(n--

r,,.+l,u:nJ

forn>

5, 1 < _ r < t < u < _ n ,

U

](li(a,b+c,d)

+ l J l ,.,,.+1,, ~:n

t-r>_3anda, 1 t-r-1

li (a:b,c,d) (a+b,c,d) r,r+ l,t,u:n = lir,t,u:n

(u

r,r+l,u-l:,} ;

b,c,d=l,2,3,...,

[. (a,b-l.c,d) [eti,.,.+l/.u:, ' ' t, U

I

_ li(a+b,c,d)

~li(a,b,c,d) r,r+l,t

1....

_ li(a+b,c,d)\

(5.12)

r,t-l,u:n J

[ li(a,b,c,d) __ + 1~1.) r,~+l,t-l,,-1 . . . .

li(a+b,c,d)

~]

t-l,~-l:,JJ

.

,

and a,b,c,d = 1,2,3,...,

for 1 < r < s < u < _ n , s - r > 2

li(a,b,c,d) = li(a,b,cfl) .. (a,b 1,c,d) r,s,s+l,u:n r,s-l,s+l,u:n ~- Olir,s,s+l,u:n

(u - s __ 1){lit,;,.:.' (a b+c d) --/o,b,c,d/ r,s-1,s,u:n 1 J

-

• (a.b+c,d)

--

(n -- u -t- 1) { tXr,.s.u_l:n

a n d for 1 <_ r < s < t < u <_ n, s - r > _ 2 , (a.b,c,d) lir,s,t,u:n =

li(a,b,c,d)

li(a.b,c,d) r.s-l,s,u

l:n} ;

t - s >_ 2 and a, b , c , d = l , 2 , 3 , . . . ,

__1

[ Dlirs't'u:n["(a b l.c,d)

(u

"f (~ b,~ a)

r,s-l,t,u:n + t -- S --

--

-- l ) l l i r , s , t : l , u : n

--

p(a,b,c,d)

r , s - l , t - I ....

,, f (a b.c,d) -- ( n - - U + (a,O,c,d)

(5.13)

}

p(a,b,c,a)

l)~lir,s',t2i,u-l:n--

r,s 14 l,u

(5.14)

l:n}] ,

(a c d)

w h e r e li,.,s,t.... - lir,t',~,':n PROOF. F r o m Eqs. (5.1) and (5.2), for 1 _< r < s < t < u _< n a n d a , b , c , d _> 1 we can write (a,b l,c,d) lir,s,t,u:n

n! (r-

1)!(s- r-

x [~[7

1)!(t- s-

1)!(u- t-

1 ) ! ( n - u)!b

y wayCS[F(w)] r-1 IF(z) - F(y)] u-` 111 - F(z)] '~-u

JO dOdO

x K(w,y)f(w)f~/)f(z) where

dwdydz ,

(5.15)

40

N. Balakrishnan and S. S. Gupta

K(w,y) =

[F(x) - F(w)] s-r-1 IF(y) - F ( x ) ] t - s - l f ( x ) d(x b)

= fwY[F(x) - F(w)] s-r-1 [F(y) - F(x)] t-* d(x b) +

yw[F(x)

+

- F(w)] s-r-1 IF(y) - F(x)] t-*-I [F(z) - F ( y ) ] d(x a)

IF(x) - F(w)] s-r-1 [F(y) - F(x)] t-s-1 [1 - F(z)] d(x b)

u p o n u s i n g the r e l a t i o n in (1.3) a n d t h e n writing 1 - F ( x ) as [ F ( y ) F(x)] + IF(z) - F ( y ) ] + [1 - F ( z ) ] . I n t e g r a t i n g b y p a r t s n o w , we o b t a i n for s = r+l andt=r+2that

K ( w , y ) = - w~[FO,) - F(w)] +

//

- w~[F(z) - F C v ) ] + y ~ [ 1 fors=r+l

xb f ( x ) d~ + y ~ [ F ( z ) - F 0 , ) I - F(z)] - w~[1 - F ( z ) ]

,

andt-r>3that K ( w , y ) = - wb[F(y) -- F(w)] t r-1 + (t - r - 1)

/w

xb[F(y)-- F(x)] t-r-2

x f ( x ) d x - w a [ F ( y ) - F(w)] t-~ 2IF(z) - F ( y ) ] + (t - r - 2)

x b [ f ( y ) -- f ( x ) ] t-r 3If(z) - f ( y ) ] f ( x )

dx

- w ~[F(y) - F(w)] t-r-2[1 - F(z)] + (t - r - 2)

fors-r_>2andt=s+l

/w"

~b[F(y)

- F(~)]'

~-3 [1 - F ( z ) ] f ( ~ ) d ~ ,

~[F(x)

- F(w)] .... 2IF(y) - F(x)]f(~)

that

X ( w , y ) = - ( s - r - 1)

£

d~

÷ fwYxb[F(x) -- F(w)] s-r-1 f ( x ) dx + yb[F(y) -- F(w)] s-r-1 IF(z) - F ( y ) ] - (s - r - 1)

/J

xb[F(x) -- F ( w ) ] . . . . 2[F(z) - F(y)] f ( x ) dx

+ y b [ F ( y ) - F(w)] s-r l[1 - F(z)] - (s - r - 1)

xb[F(x) -- F ( w ) ] . . . . 211 - F ( z ) ] f ( x ) dx ,

41

Higher order m o m e n t s o f order statistics

and for s-

r >_ 2 a n d t - s

>_ 2 t h a t

X ( w , y ) = - ( s - r - 1)

2

z~[F(x) - F(w/] '-~ 2IF(y) - F(x)]' ' f ( x ) cU

+ (t - ~) . / ~ x ~IF(x) - F(w)I'-r-~[FO,)

- (s-r-

- F(x)l'-"

i f ( x ) d~

x b [ F ( x ) - F ( w ) ] ' ~ 2[F(y) - F ( x ) ] t ,-I

1)

x IF(z) - F(y)]f(x) dx + (t - s - 1)

xb[F(x) -- F ( w ) ] . . . . 1

x IF(Y) - F(x)lt-s-zIF(z ) - F(y)]f(x) dx

(s - r - 1)

-

x b f F ( x ) - F(w)] s ~ - 2 [ F ( y ) - F(x)] '-'-~ [1 - F ( z ) ] f ( x )

+(t-s-1

xb[F(x)-F(w)l s-r l[F(y)-F(x)]'-"

dx

2[1-F(z)lf(x)dx.

U p o n s u b s t i t u t i n g t h e a b o v e e x p r e s s i o n s o f K(w, y) in Eq. (5.15) a n d s i m p l i f y i n g the r e s u l t i n g e q u a t i o n s , w e d e r i v e the r e c u r r e n c e r e l a t i o n s g i v e n in Eqs. ( 5 . 1 1 ) (5.14). THEOREM 1 1. F o r n >_ 4, 1 _< r < s < n -- 2 a n d a, b, c, d = 1 , 2 , . . . ,

#(~,b,c,4

#(~,b+~,d)

r,s,s+l,s+2:n =

r,s,s+2n

(a,b.~-1,a)

@ C~r,s,s+l,s+2:n

-- ( n -- S -- 1 ) f # (a'b'c+d) -- # (a'b+c'd) l J ( r,s,s+l:n r,s,s+l:n J

forn

> 5, l < _ r < s < u < n , ]A(a'b'c'd)

u-s>_3anda, 1

= ]l(a'b+c'd) Jr-

..... +1,,:,

r~s....

5, l < _ r < s < t < _ n - 1 ,

b,c,d=l~2,...,

V (a 6,c-l,d)

LC#r,s,s+l .....

u - s - 1 ., f

' (a b c d)

(5.17)

#(a,b+c,d) ~ - l ) l ] 2 r , s ' s ' + ' l , u _ l : n -- r .... l:n }]

--(gl--b!

forn>_

(5.16)

" ,

t-s>_2anda,

;

b,c,d=l,2,...,

]A(a,b,c,d) #(a,b,c,d) (a,b,c 1 .d) ..... +l,u:n = r,s,t-l,t+l:n ~- C]Ar,s~tlt+l:'n (g/ " [" (a bc+d) #(a,b,c,d) I --- t ) l llr,s,tln -- .... t - l ,t:n J i

andforn

> 6, 1 < _ r < s < t < u < _ n , (a,b,c,a) _ #(,,b,c,a) ,tlr.s,t.u:n

--

t-s>_2, 1

F

(5.18) • '

u-t>_2anda,b,c,d=l,2,...,

(, b c-i,a)

r,s,t-l,u:n -w b/ -- t

. , f (a.b.~d) . (~,b,c,d)}] - (n - u + 1) J ,u~,s,tf_h, , -- #~,s,t-l,,-l:n

(~,b,O,a)

w h e r e ~lr,s, t ....

(~ b a)

~ #r4,u:n •

(5.19) ,

N. Balakrishnan and S. S. Gupta

42

PROOF. F r o m Eqs. (5.1) a n d (5.2), for 1 _< r < s < t < u _< n a n d a , b , c , d _> 1 we can write

(a,b,c- l,d) = #r's'tlu:n (r -×

n! 1)!(S

-- r --

1)!(t

-- S --

1)!(H

-- t -- 1)!(n

I7oYo xwaxbzd[F(w)] r-1 [F(x) -

-- b/)!C

F(w)] s-r-1

x [1 - F(z)]n-"L (x, z ) f ( w ) f

( x ) f ( z ) d w dx dz , (5.20)

where

L ( x , z ) = ~x z IF(y) =

F(x)] t-"

/xzIF(y) +

I[F(z)-F(y)jUtlf(y)d(y c)

F(x)] t-s-1 IF(z) - F(y)] u-t d ( y c)

/x z[F(y) -

F(x)] t ,-1 [F(z) - F ( y ) ] "-t-1 [1 - F(z)] d ( f f )

u p o n u s i n g the r e l a t i o n in (1.3) a n d t h e n w r i t i n g 1 - F ( y ) as [ f ( z ) - F ( y ) ] + [1-F(z)]. I n t e g r a t i n g b y p a r t s now, we o b t a i n for t = s + 1 a n d u = s + 2 that

L ( x , z ) = -xC[F(z) - F ( x ) ] fort=s+l

+

/x yOf(y)

dy+S[1

-F(z)]-x~[1

- F(z)] ,

andu-s_>3that

L(x, z) = - x~[F(z) - F ( x ) ] " - s - I + ( u - s -

1

y°[F(z)-F(y)]"-'-2f(y)

dy

- x°[F(z) - F(x)] u-s 211 - F(z)] + (u - s - 2) fort-s_>2andu=t+l

that

L ( x , z ) = - ( t - s - 1) -+-

_> 2 a n d u -

/x zyC[F(y) -

F(x)]t-s-2[F(z) - F ( y ) ] f ( y ) dy

yC[F(y)-F(x)]t-s-lf(y)dy+zC[F(z)-F(x)]

- (t - s - 1) a n d for t - s

yC[F(z) - F(y)] u-s-3 [1 - F ( z ) ] f ( y ) dy ,

/z yC[F(y) -

t_> 2 t h a t

t-s l [ 1 - F ( z ) ]

F ( x ) ] t - ' - 2 [ 1 - F ( z ) ] f ( y ) dy ,

Higher order moments of order statistics

L(x, z) = - ( t - s - 1)

~XZ

yCIF(y ) - F ( x ) ] t - s - 2 [ F ( z )

43

- F(y)]"-tf(y)

yC[F(y ) _ F(x)]t ,-1 IF(z) - F ( y ) ] " - t - l f ( y )

+ (u - t)

-- (t -- S --

1)

//

yC[f(y ) _ f(x)]t-s-2[f(z)

x [1 - F ( z ) ] f ( y ) d y + (u - t - 1)

dy dy

_ f(y)]U t 1

yC[F(y ) _ F(x)]t , 1

x IF(z) - F(y)] u t 2[1 _ F ( z ) l f ( y ) d y . Upon substituting the above expressions of L(x, z) in Eq. (5.20) and simplifying the resulting equations, we derive the recurrence relations given in Eqs. (5.16)(5.19). The recurrence relations presented in any one of Theorems 8, 9, 10 and 11 may be employed in a simple systematic recursive way to compute the quadruple moments of order statistics (of all order) for all sample sizes.

6. Applications to inference for the one-parameter exponential distribution By using the results presented in Sections 2-5, we computed the single, double, triple and quadruple moments of order statistics (of order up to 4) for sample sizes up to 12. As will be displayed in this section, these quantities may be successfully used in order to develop a chi-square approximation for the distribution of the best linear unbiased estimator of the scale parameter of an exponential distribution based on doubly Type-II censored samples. In this section we assume that Yr+l:n <- Yr+2:n < _ ' "

<~ Y, s:,

(6.1)

is a doubly Type-II censored sample from the one-parameter exponential population with probability density function g(y;a)=~e

-y/~,

y_>0, G > 0 .

(6.2)

The estimation of the parameter a based on the doubly Type-II censored sample in (6.1) has been considered by several authors including Epstein (1956, 1962), Epstein and Sobel (1953, 1954), Sarhan and Greenberg (1957, 1958, 1962), Saleh (1966), Cohen and Helm (1973), Mann et al. (1974), Kambo (1978), Lawless (1982), Nelson (1982), and Balakrishnan and Cohen (1991). By using the facts that E[~:,]=a

~ g=n-i+ l

1/e

N. Balakrishnan and S. S. Gupta

44

and Var[Y/:~1 = Cov[Y,.:~, Yj:n] = 0-2 ~

g=n

1/g 2 , i+1

it can be shown that the best linear unbiased estimator of 0- based on the doubly Type-II censored sample in (6.1) is given by n-s

aiYi:, ,

0-*= ~

(6.3)

i=r+l

where

ai = a*/K with K = (n-r-s-

1)-}-

{-~-1}2/[--~-r g

r

1/g2 1

g

and

1/gl/I ~

l/g:] - ( n - r - l )

ai

for i = r + l for r + 2 < i < n - s - 1 for i = n - s ,

and the variance of this estimator is Var(o*) = 0-2/K ;

(6.4)

for example, see Balakrishnan and Cohen (1991, pp. 76-80). In the special case of the right-censored sample (that is, r = 0), the estimator 0-* in (6.3) simplifies to

and the variance of the estimator in (6.5) is simply Var(0-*) = 0-2/(n - s) .

(6.6)

In this special case of right-censoring, by writing Y,:nas 0-X,.:nwe may immediately write from (6.5) that 0-* 0-

1 n-s-1 I = n - - S { i~_l Xi:nq-(S-- l)Xn s:n _

~& n

_1

n

s

s

i=1

(6.7)

Higher order moments of order statistics

45

where Si = (n - i + 1)[Xi:n - X / ~:n] are the normalized spacings as defined earlier in Section 1. By using the property that these S~'s are independent standard exponential random variables, we immediately observe from Eq. (6.7) that 2(n - s ) a * / a has a chi-square distribution with 2(n - s) degrees of freedom. This result can be used to obtain confidence intervals for o- and carry out tests of hypotheses concerning a. However, such an elegant distributional result does not extend to the best linear unbiased estimator of a in (6.3) based on a doubly TypeII censored sample. For the doubly Type-II censored sample case, by considering the best linear unbiased estimator of a in (6.3) and writing 0,~

n --S

--=

Z

o"

ai~:. ,

(6.8)

i=r+l

we may determine the mean, variance and the coefficients of skewness and kurtosis of the linear function in (6.8) by making use of the tabulated values of the single, double, triple and quadruple moments of order statistics (of order up to 4). For example, we computed the values of the mean, variance, coefficient of skewness (V~l) and coefficient of kurtosis (f12) of a*/o- for sample size n = 8(1)12 and various levels of censoring both on left and right sides of the sample. These values are presented in Table 1. Also given in the last column of this table are the values of 7 =/32 - 1.5/31 - 3. It is observed that 7 = 0 whenever the sample is just right-censored and this is expected because of the fact that 2 ( n - s ) a * / a has exactly a chi-square distribution with 2(n - s) degrees of freedom and that 7 = 0 represents the chi-square line in the Pearson (/31,/32)-plane. But interestingly enough, we also observe that 7 is very nearly 0 even in the case of doubly censored samples even for a sample of size as small as 8. Thus, Table 1 suggests strongly that the distribution of a*/a in the case of doubly Type-II censored samples may be very closely approximated by a chi-square distribution with degrees of freedom determined approximately (for example, by looking at the variance). The chisquare approximation for the distribution of o-*/o- may be utilized to construct approximate confidence intervals for a and carry out tests of hypotheses concerning a. This application is well illustrated in Section 8 through three examples involving life-time data.

7. Generalized results for the right-truncated exponential distribution In this section, we generalize the results established in Sections 2-5 to the case when the population distribution is exponential truncated on the right. As pointed out by Cohen (1959), in life-testing, dosage-response studies, target analyses, biological assays, and in other related investigations, sample selection or observation is often restricted on the right side of the range of possible population values. With this in mind, Joshi (1978, 1979, 1982), Saleh et al. (1975), and Balakrishnan and Joshi (1984) have all studied order statistics from truncated

N. Balakrishnan and S. S. Gupta

46

Table 1 Mean, variance, i f 1 , n

f12 and

7 = fi2 - 1.5 fll - 3 values of ~*/o for doubly Type-II censored samples

r

s

Mean

Variance

ffl

f12

7

8 8 8 8 8 8 8 8 8 8 8 8

0 0 0 1 1 1 2 2 2 3 3 3

0 1 2 0 1 2 0 1 2 0 1 2

1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

0.125000 0.142857 0.166667 0.125138 0.143038 0.166913 0.125645 0.143701 0.167816 0.126929 0.145383 0.170114

0.707107 0.755929 0.816497 0.708271 0.757352 0.818290 0.712513 0.762540 0.824838 0.723164 0.775601 0.841378

3.750000 3.857143 4.000000 3.753285 3.861434 4.005843 3.765300 3.877152 4.027282 3.795792 3.917184 4.082145

0.000000 0.000000 0.000000 0.000813 0.001061 0.001445 0.003788 0.004951 0.006745 0.011343 0.014849 0.020270

9 9 9 9 9 9 9 9 9 9 9 9

0 0 0 1 1 1 2 2 2 3 3 3

0 1 2 0 1 2 0 l 2 0 1 2

1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

0.111111 0.125000 0.142857 0.111196 0.125108 0.142998 0.111500 0.125493 0.143501 0.112243 0.126434 0.144733

0.666667 0.707107 0.755929 0..667428 0.708016 0.757040 0.670135 0.711247 0.760991 0.676703 0.719101 0.770614

3.666667 3.750000 3.857143 3.668693 3.752566 3.860495 3.675917 3.761717 3.872462 3.693582 3.784140 3.901863

0.000000 0.000000 0.000000 0.000503 0.000636 0.000831 0.002296 0.002909 0.003801 0.006692 0.008481 0.011094

10 10 10 10 10 10 10 10 10 10 10 10

0 0 0 1 1 1 2 2 2 3 3 3

0 1 2 0 1 2 0 1 2 0 1 2

1~000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

0A00000 0.111111 0.125000 0.100055 0.111179 0.125086 0.100249 0.111418 0.125389 0.100708 0.111986 0.126108

0.632456 0.666667 0.707107 0.632977 0.667278 0.707836 0.634795 0.669408 0.710379 0.639098 0.674454 0.716410

3.600000 3.666667 3.750000 3.601317 3.668293 3.752059 3.605919 3.673978 3.759259 3.616872 3.687525 3.776445

0.000000 0.000000 0.000000 0.000327 0.000403 0.000511 0.001472 O.001817 0.002302 0.004203 0.005193 0.006580

11 11 11 11 11 11 11 1l 11 1l II 11

0 0 0 0 1 1 1 1 2 2 2 2

0 1 2 3 0 1 2 3 0 1 2 3

1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

0.090909 0.100000 0.111111 0.125000 0.090947 0.100045 0.111167 0.125071 0.091075 0.100201 0.111359 0.125314

0.603023 0.632456 0.666667 0.707107 0.603393 0.632883 0.667167 0.707704 0.604665 0.634351 0.668887 0.709758

3.545455 3.600000 3.666667 3.750000 3.546347 3.601080 3.668000 3.751688 3.549417 3.604796 3.672590 3.757500

0.000000 0.000000 0.000000 0.000000 0.000222 0.000269 O.000332 0.000421 0.000987 0.001194 0.001475 0.001865

Higher order moments of order statistics

47

Table 1 (Contd.) n

r

s

Mean

Variance

~

/~2

?

11 ll 11 11 11 11 11 11

3 3 3 3 4 4 4 4

0 1 2 3 0 1 2 3

1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

0.091375 0.100564 0.111808 0.125883 0.091970 0.101285 0.112700 0.127015

0.607618 0.637761 0.672885 0.714534 0.613443 0.644494 0.680792 0.723999

3.556574 3.613466 3.683310 3.771093 3.570803 3.630734 3.704705 3.798294

0.002775 0.003357 0.004149 0.005255 .0006335 0.007675 0.009488 0.012032

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

0 1 2 3 0 1 2 3 0 1 2 3 0 l 2 3 0 1 2 3

1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

0.083333 0.090909 0.100000 0.111111 0.083360 0.090940 0.100038 0.111158 0.083449 0.091046 0.100166 0.111316 0.083653 0.091289 0.100460 0.111680 0.084049 0.091762 0.101033 0.112388

0.577350 0.603023 0.632456 0.666667 0.577622 0.603332 0.632812 0.667085 0.578542 0.604381 0.634022 0.668502 0.580645 0.606779 0.636791 0.671748 0.584714 0.611422 0.642157 0.678045

3.500000 3.545455 3.600000 3.666667 3.500626 3.546200 3.600902 3.667780 3.502752 3.548730 3.603964 3.671562 3.507628 3.554538 3.610999 3.680258 3.517118 3.565854 3.624724 3.697252

0.000000 0.000000 0.000000 0.000000 0.000155 0.000186 0.000226 0.000276 0.000686 0.000815 0.000988 0.001220 0.001905 0.002267 0.002745 0.003390 0.004282 0.005099 0.006176 0.007635

e x p o n e n t i a l d i s t r i b u t i o n s . K j e l s b e r g (1962), T a r t e r (1966), B a l a k r i s h n a n a n d K o c h e r l a k o t a (1986), a n d G u p t a a n d B a l a k r i s h n a n (1992) h a v e all s i m i l a r l y exa m i n e d o r d e r statistics f r o m t r u n c a t e d logistic d i s t r i b u t i o n s . L e t Xl:n < X2:, _ < . . . < X,:, be the o r d e r statistics o b t a i n e d f r o m a r a n d o m s a m p l e o f size n f r o m a r i g h t - t r u n c a t e d e x p o n e n t i a l p o p u l a t i o n w i t h p r o b a b i l i t y density function

l

x

f(x)={~e

'

0<x<& elsewhere

(7.1)

and cumulative distribution function

F(x)= {(ll,-e

x)/p,

x > plO<X<,P1

(7.2)

w h e r e 1 - P is t h e p r o p o r t i o n o f t r u n c a t i o n o n the r i g h t o f the s t a n d a r d e x p o n e n t i a l d i s t r i b u t i o n in (1.2), a n d the t r u n c a t i o n p o i n t PI = -gn(1 - P). It is easily o b s e r v e d f r o m E q s . (7.1) a n d (7.2) t h a t

N. Balakrishnan and S. S. Gupta

48

f(x) = ( l / P ) - F ( x ) ,

0 < x < PI ,

(7.3)

and

f(x) = [1- F(x)] + I 1 p P l ,

0<x
.

(7.4)

These two relations will be used successfully in this section in order to generalize all the results derived in Sections 2-5 to the right-truncated exponential distribution in (7.2).

(i) Relations for single moments The probability density function of Xr:n is given by

n!

fr:,(w) = (r - 1)!(n - r)! [F(w)]r-1 [1 - F(w)]"-rf(w),

O<_w<_P1, 1 < r < n

(7.5)

,

using which the single m o m e n t s m a y be c o m p u t e d as

#(a)r:n=

wafr:~(W)dw,

1 < r < n, a = 1,2, . . . .

(7.6)

Then by proceeding on lines very similar to those used in proving T h e o r e m s 1 and 2, we m a y prove the following two theorems. THEOREM 12. F o r n _> 2 and a = 1 , 2 , . . . , (a) a l..an 1) l:n = # --

[l~_J

. (a)

(7.7)

/~l:n- 1

with /Xl: • (0)n = 1 THEOREM 13. F o r n _ > 3 , 2 < r < n r:n

=

tZr-l:n

-~

1 anda=

n- - r- +"l

|a#r:n

--

n

1,2,..., r:n-1

--

t~r

l:n-I

(7.8) and f o r n _ > 2 a n d a =

1,2,...,

n-l:n #(a) = #(~) + au(a-1)

--

n

P~ -

n-l:n l J

with #!0) _= 1. It should be mentioned here that T h e o r e m s 12 and 13 have been proved by Joshi (1978) and are presented here for the sake o f completeness. The recurrence relations presented in T h e o r e m s 12 and 13 will enable one to c o m p u t e the single m o m e n t s (of all order) o f order statistics for all sample sizes in a simple recursive way.

Higher order moments of order statistics

49

(ii) Relations for double moments The joint density function of Xr:, and Xs:n is given by n[ f~s:,(w,x) = ( r - 1 ) ! ( s - r -

1 ) ! ( n - s)( [F(w)]r 1

× IF(x) - F(w)] . . . . 1[1 _ F(x)j,_Sf(w)f(x), O<w<x<_P~,

(7.10)

1 <_r<s<_n ,

using which we may compute the double moments as #!~,;~2 =

/f waxbf~,s:n(w, x) dw dx, O<w<x<_Pi

1 < r < s <_ n, a, b >_ 1

a

(7.11) Then by proceeding on lines very similar to those used in proving Theorems 3 and 4, we may prove the following two theorems. TnEOREM 14. For n > 3, 1 < r < n - 2 and a,b = 1 , 2 , . . . , 1

#rr+l(:n, • , =b u(~+b) ) r-r:n

Fb(ab_l) [l~_]f.(a,b) [ #rr+l:n, -- f/ I t~rr+l'n, ,

-}-fl_r

1 --

#(a+b)/] r:n 1JJ

. '

(7.12) forn>2anda,

b= 1,2,...,

(a,b) #(a+b)bl~,]i-,~ n-l,n:n : n l:n + '

"

- n

[l~_nP](pb(a) [.

F

J

(a+b) "l #n l:n-lJ~

el l ~ n - l : n - 1

(7.13) forn> 4,1
s-r

>_2anda, b = l , 2 , . . . ,

1 n--s-l-

[

(a 6_1)

[ 1 p P] {

1

}] . "

•

,

(7.14) and for n > 3,1 < r < n - 2 a n d a ,

b= 1,2,...,

#~o~,), . : #(a,b)r,n_l:~+ b#~,a,;:b~-1) - n

~(ri #r:n_ 1 - /~.~,~b)l.~ } ,. 1

,

(7.15)

where ,I,lr,s:n THEOREM 15. For n _> 2 and a,b = 1 , 2 , . . . , (a,b)

. (a+b)

#l,2:n = ,U2:n

(a l,b)

+ a#l,2:n

n . (a+b)

-- ePtl:n-I

;

(7.16)

N. Balakrishnan and S. S. Gupta

50

for n > 3,2 < r < n ,

1 and a , b = 1 , 2 , . . . , r kapr'r+hn

r+l:.

~ ~.Ur:n 1 --

,.

'

for n _> 3,3 < s < n and a , b = 1 , 2 , . . . , (a-l,b) #I;~2 = #~s~2 -~ a"#l,s:n and for n >_ 4,2 <_ r < s <_ n , s -

e ]21;b-)l'n-1, , ;

(7.18)

r_> 2 and a , b = 1 , 2 , . . . ,

#(a' rall(a-"b)--p{#(ra, r,s:nb)'~--#:a+bl ' " !s'rn-t-l k-r-r's:n 'sb-)l • 'n1--#:a'-t!s-l' ' • n 1}]

(7.19)

'

(o b) where #r,;:n =- #!b). The recurrence relations presented in either T h e o r e m 14 or T h e o r e m 15 m a y be used in a simple systematic recursive m a n n e r in order to c o m p u t e the double m o m e n t s (of all order) o f order statistics for all sample sizes. The relations presented in T h e o r e m 14 have been proved by Joshi (1982) for the special case when a=b-1. (iii) Relations f o r triple m o m e n t s The joint density function of Xr:n, Xs:n and Xt:n is given by n! f~,s,t:,(w,x,y) = (r - 1)!(s - r - 1)!(t - s - 1)!(n - t)! x [F(w)] r-1 [F(x) - F(w)] s-r-1

(7.20)

x IF(y) - F(x)] t-s l[1 - F ( y ) ] n - t f ( w ) f ( x ) f ( y ) , O<_w<x
1 <_r<s
,

using which we m a y c o m p u t e the triple m o m e n t s as #r,s,l:n =

J/f

waxbyCfr,,,t:, (w, x , y ) d w dx dy,

O<w<x
(7.21 ) l<_r<s
a,b,c>

1 .

Then by proceeding on lines very similar to those used in proving Theorems 5-7, we m a y prove the following three theorems. THEOREM 16. F o r n > 4, 1 < r < s < n - 2 and a , b , c = 1 , 2 , . . . , #(a,b,c) ..... u(a'b+c) 4-1[n -- S (a b,c l ) [~_] +l:, = r~,s:, _ LC#r,s',+l:, - n

f

r,s,s+hn-1

_ #(a,b+c) \ j r,s:n-1

(7.22)

;

Higher order moments

for n > 3, 1 < r < n -

2

and a,b,c

qf order

51

statistics

= 1,2,...,

• (a,b,c) . (a,b+c) -~- C ]"lr,n-l,n:n (a,b,c-l) _ n ]~r,n--l,n:n = #r,n-l:n

[~Tp_.ff_l.fI.*r,~1 #r,n .(a,b)l:n-1

--

p(~.b+~) r,n-l:n-I }

"'

(7.23) forn>5,1

<_r < s

< t
l,t-s

> 2 anda, b,c=

(a,b,c) ].l(a,b,c) 1 F (abc-1) #r,s,t:n = r.s,t, l:n -~ El--7-~- 1 [C]2r,sS n

--El

1,2...,

[~@-~-] f ].l (a'b'c) [2 (a'b'e) 1. r,s,t:n 1-r,s,t-l:n

l]

•

i jj

(7.24) andforn_>4,1

<_ r < s <_ n - 2 a n d a, b , c = l , 2 . . . ,

la(~.b#) p(a,b,c) c ( a b ~ l ) [ 1 - P ] f p c , O,b) r,s.n:n = r.s,n--l:n -}- ~r,s,'n':n --El L t" j I. l t*r,s:n-l -- #r,s,n'(a'b'C)

(7.25) where

(
THEOREM 17.

Forn _> 3,3

and

< t
a,b,c=

~(a,b,c) = ]l(a+b,c) (a l,b,c) 1,2,t:n 2,t:n ~- al'tl,2,t:n

for n >_ 4 , 2 <_ r < t < n , t -

1,2,...,

El [,l(a+b,c) 1,t-l:n-1

p

(7.26)

;

r >_ 2 a n d a, b , c = 1 , 2 , . . . ,

El f ].t(a+b,c) (a b c) "l 7 Id(a'b,c) 11(a+b'c) _{_ ! F (a-l,b,c) r,r+l,t:n = r+l,t:n r La#r,r+l,t:n -- p I. r,t l:n-1 -- ]2r"-'l',r,t-l:n-1 ~J ;

(7.27) forn>4,3_<s
l,s,t:n =

and forn>

5,2
(a b.c)

t2r,s,tn

:

p(a,b,c)

b,c=

1,2,...,

(~ l,b,c)

nl~(a,b,~)

2,s,t:n -I-al~l,s,t: n

<s
e

>_2anda,

i~(~,b,~) + 1 r (~-i b ~) r+l,s,t:n

r [ al2r's't:n '

(7.28)

1,s 1,t-l:n 1 ;

b,c=

1,2,...,

n f p(.,b,c)

_ i~(a,b,~)

e ~ r,s-l,t-l:n-1

].]

r-l,s-l,t-l:n-l

JJ

,

(7.29) (O,b,c) . (b,c) where [Ar,s,t: n ~ lds,t:n .

and a,b,c = t~l(a+b'c) ' b (a,b l,c) (n -- r r.r+2:n ~- [Ar,r+l,r+2: n --

THEOREM 18. For n > 3, 1 < r < n - 2 ]A(a'b'c) r'r+l'r+2:n =

- n

[

r,r+hn-1

1,2,..., " \ ((a.b+c) ]2(a+b,c)'~ -- l)el.12r,r+l:n -- r,r+l:n J

r,r+l:n-I j

,

(7.30)

N. Balakrishnan and S. S. Gupta

52

forn>4,1<_r_3anda,

b,e=

[2(a,b,c) (a+b,c) 1 r,r+l.t n = ~r,t:n + t- __- r -

r.

(ab

1

1.c)

,

--[Ar,t-lln ~ - -

-- (n--,+

1)

1. r,r+l,t-l:n-1 --

forn>4,1<_r<s<_n-l,s-r>_2anda, #(a,b,c)

1,2,...,

= #(a,b,c)

f ]..l(a,b,c) [ r,r+l,t-l:n

r,t-l:n-1

'

b,c= 1,2,...,

~ (a.b-l,c) F~s- ] ~s:n )

- n

/#r,;':,-~ -

and for n >_ 5, 1 <_ r < s < t <_ n , s - r (a,b,c) ,tl(a,b,c) 1 ~r,s,t:n = r,s 1.t n + ~ - - S --n • (a,O,~)

[

r.

~,,-1,,:, 1} ;

>_ 2, t - s _ > 2 and a , b , c = 1 , 2 , . . . , (ab

it)

[o~,r,s,,:° '

r,s,t-l:n

(7.32)

1-

- (n -t+ r,s-l,t

l:n-1

.\((abc) .(a,b,c) ,)t~,~;,:,:, . . . - . ~,_~,

}]

"1

~:'l'

,7.33,

'

(~,c)

where t.tr,s,t:n ~ ].lr,t:n . The recurrence relations presented in any one of Theorems 16, 17 and 18 may be used in a sample systematic recursive fashion in order to compute the triple moments (of all order) of order statistics for all sample sizes. (iv) Relations for quadruple moments The joint density function of Xr:,,Xs:n,Xt:, and X~:, is given by L , , , , .... ( w , x , y , z )

=

(r- 1)!(s- r-

n! 1 ) ! ( t - s - 1 ) ! ( u - t - 1 ) ! ( n - u)!

× IF(w)] r-I [F(x) - F(w)] s ~ 1

x [F(y) - F(x)] t ~-l[F(z) - F(y)] ~ t

1

x [1 - X ( z ) ] ' - U f ( w ) f ( x ) f ( y ) f ( z ) , O<w<x
l<_r<s
, (7.34)

using which we may compute the quadruple moments as Pr,,,t.,, =

WaXbyCJfr,,,, .... (W,X,y, Z) dw dx dy dz, (7.35)

O<w<x
<_r<s
a,b,c,d>_ 1 .

Then by proceeding on lines very similar to those used in proving Theorems 8-11, we may prove the following four theorems.

53

Higher order moments of order statistics

THEOREM 19. F o r n >_ 5, 1 _< r < s < t _< n #(a,b,c,d)

2 and a,b,c,d = 1,2,...,

(~,b,c+d)

r,s,t,t+l:n = ['tr,s,t:n

1 [ . (a,b,e,d-1) q- l ' l - - t [ altr's't

(7.36) --l'l

~ r,s,t,t+l:n-1-

for n > 4,1 < r < s < n - 2 a n d a ,

r,s,n-l,n:n =

b,c,d=

r,s,n-l:n ~-

l

r,s,t:n-I

;

1,2,...,

#r,s,n-l,n:n -- rt

(7.37) fnd

(~ b c)

l~(~,b,c+d)

× { r i ~r,s,n'-l:n-1for n > 6, 1 ~ r < s < t < u <_ n - 1 , u (a.b,c.d)

#r,s,t,d:n =

1} ;

r,s,n-l:n

t _> 2 a n d a , b , c , d = 1 , 2 , . . . ,

p(a,b,c,d) 1 F . (a.b c d-l) r,s,t,u-l:n -~ n -- u + l La/~r'&iu':n

(7.38) --n

a n d for n _> 5, 1 _< r < s < t_< n -

~ r,s,t,u:n-1 -- t~r,s,t,u-l:n-I

;

2 and a , b , c , d = 1 , 2 , . . . ,

(a,b,c,d) (a,b,c,d) ,. (a,b,c,d 1) [1-P]f,,4 fir.s,t,n:n, = #r,s,t,n_l:n ~- a#r,s,t,n:n -- l"t

(a,b,c)

k-7-JUi r,s,,:.-1

#(a,b,c,d) "~ r,s,t,n l:n 1J '

(7.39) (a,b,c,O) (a,b,c) w h e r e #r,s,t,u:n = #r,s,t:n "

THEOREM 20. F o r n _> 4, 3 _< t < u _< n a n d a, b, c, d = 1 , 2 , . . . , (a,b,c,d) = #(a+b,c,d) (a-l,b,c,d) 24.... q- aPl,2,t,u:n

/~l,2,t ....

n (a+b,e,d) ~Pl,t-l,u-l:n-I

;

(7.40)

for n > 5,2 <_ r < t < u <_ n , t - - r_> 2 a n d a , b , c , d = 1 , 2 , . . . , #(a,b,c,a) r,r+l,t,u:n

= #(a+b#,d)

1 I a (,~ l,b,~,d)

n f#(a+b,c,d)

r+l,t,u:n q---IF ~r,r+l,t,u:n r-l,r,t-l,u-l:n-l

?( I1

J]

r,t-l,u-l:n-1 •

(7.41)

,

for n _> 5,3 _< s < t < u _< n a n d a , b , c , d = 1 , 2 , . . . , #(
(~-l,b#,d)

2,s,t,u:n ~- a~l,s,t,u:n

and for n >_ 6 , 2 <_ r < s < t < u <_ n , s -

n/~(~,b#,d)

p

1,s-l,t-l,u-l:n-1

;

r >_ 2 and a, b , c , d = l , 2 , . . . ,

(7.42)

54

N. Balakrishnan and S. S. Gupta (ab.c d) #(a,b,c,d) - t - ! r (a-l b.c,d) n f f(a,b,c,d) #r,s',tlu':n = r+l,s,t .... r [ aflr's't'u:n ' -- p I r,s--l,t-l,u

l:n 1 (7.43)

_ # (~,b,c,d)

':n 1}] '

r-l,s-l,t-i,u

• (o,b,c,d)

. (
where tZr,s,t,u:n ~ tZs,t,u:n . ThEOReM 21. F o r n _> 4, 1 <_ r < u <_ n , u - r _ >

2) ft #(a,b+c,d) r,r+l,u:n

#(a,b~c,d) fl(a+b,c,a) _ _ . (a,b-l,c,d) -- (u - r r,r+l,r+2,u:n = r,r+2,u:n ~- O#r,r+l,r+2,u:n (~+b#,d)'~ _ (n -- + 1) f (a b+c d) - # r ,~+l .... u , . j t#rr+l.u-l:n --n

1 - P] f (~ b+~,d) T i #r'r+l'u l:n-I-

(~+b,c,d)

r,r+l,t,u:n = #r,t,u:n

1 r-

Avt

(a+b.c,a) "l --#rt-i'u:n, , j~ - ( n t forn>_5,1

<_r < s < u <_ n , s - r

;

(7.44)

= 1,2,,..,

[ . (ab-l,cd) t, f (~b.cd) -- ( I t - ) t # r , r + l l t _ l , u : n 1 [O#rr+l.tu':n ' ,'

( (a.b c d) (a+b,c,d) u + 1) t#r,r+'llt-l,u-l:n--#r,t 1,u 1:.}

r,r+l,t-l,u-l:n-1 >_ 2, u -

--

s_> 2 a n d a, b , c , d

= 1,2,...,

1){#r,s,u:n'(ab+cd)

f fi(a.b+c,d)

-- (n -- u + 1 ) t

(7.45)

i}1 ;

r,t-l,u-l:n

#(a,b,c,d) = #(a,b,c,d) _ _ . (a,b-l,c,d), -- (hi -- S r,s,s+l,u:n r,s i,s+i,u:n "~ Oflr,s,s+l.u:n _ #(a,b,c,d) r,s- 1,s,u:n

#(a+b,c,d)

-- r,r+l,u-l:n}

#(a+b,~,d) r,r+l,u-l:n-1}

for n > 5, 1 < r < t < u < n, t - r _> 3 a n d a , b , c , d l~(a,b,
= 1,2,...,

3 and a,b,c,d

r,;,u-I:. -

#(a,b,c,d) ] r,s-l,s#-l:nJ

{ #(a,b+~,a) _ #(a,b#,a) \ . r,s,u-l:n-1 r,s 1,s,u-l:n-l J , a n d f o r n _> 6, 1 < _ r < s < t < u

<_ n, s - r _ >

2, t - s

_> 2 a n d a , b , c , d

(a.b,c,d) (~.b#,d) 1 [ . (a,b-1 c d) , = #r,s l,t,u:n ~- t - s [O#r,s,t,u:n' -- ( U #r,s,t.u:n -

#(a,b,c,a)

-

1" f

., f

(a,o#,d) (a c d) w h e r e #r,s,t .... = #r,t',~,':,, •

r,s,t 1,u-l:n-1-

= 1,2,...

(~ b c.d)

' ~j
(a b c.d)

--r,s-l,t-1 .... }--("--U-I-)t#r,s,t'~i,u_ t

(7.46)

r,s-'l,t-l,u

_ #(a,
1....

l:n 1

s-l,t-l,u

)}

l:n}

<7.47,

Higher order moments of order statistics

55

THEOREM 22. For n >_ 4, 1 _< r < s _< n - 2 and a, b, c, d = 1 , 2 , . . . , (~b c+d) ~(a,b~c,d) (a,b,c-l,d) -- (n - s - 1,f )t#r,s,s'+l:n r,s,s+l,s+2:n = ~(a,b+c,d) r,s,s+2:n __ + C]~r,s,s+l,s+2:n _ ]l(a,b+e,d) • [~_l r,s,s+l:, j -- n

f o r n > 5, l < _ r < s < u < _ n ,

u-s>_3anda,

~(a,b,c,d) ,u(a,b..c,d) + , , , = r,s,u:n rss+l.u:n IA

f ]A(a,b,c+d ) _ #(a,b+c,d) 1. . I. ..... , , l : n - I ..... +l:n-1 j ' b,c,d=l,2,...,

1 [ (~ b,c l.d) s - - 1 [C]gr,s,s+l,mn - - (H

_#(a,b--c,d) I -- [~-} r,s,u-l:n j n

(7.48)

--

U

-~-

S. (~,b,
1) I.t'~rss+l.u-l:n " ,

I]l(a'b'c'd) _~(a,b+c,d) }]. t r~s,s+l,u-l:n-1 r,s,u l : n - 1

(7.49) f o r n >- 5, l <_ r < s < t <_ n - 1 ,

t-

#(a,b,c,d) ~(a.b,c,d) r,s,t,t+l:n : r,s,t-l,t+l:n'

s >_ 2 a n d a, b , c , d

= l , 2 , . .. ,

. (a,b.c-l,d) .,((a,b,c+d) t C]gr,s,tlt+l: n -- (gl -- l)l gr,s,t:n

(7.50) ,,

,

[ r,s,t:n-I

andforn>_6,1<_r<s_2, (a,b.cm)

#x(a,b,c,d)

[2r,s,tlu:n :

-

-

1

r,s~t-l,u:n @ U -- t

r

IXr,s,t-l,t:n-1 ~.j ;

u-t>_2anda, (a b c 1, d) '

__ /1(a,b'c'd) r,s,t l,u_l:n} _#1[~_]

t

-

(.

-

u

f iA(a,Kc,d ) r,s.tu-l:n-1-

b,c,d=l,2,..., , x(

(a.b c d)

]~(a,b,cM) r,s,t-l,u-l:n

"(] 1jj

,

(7.51) (a,b,O,d)

(
where Igr,s,t,u:n ~ Igr,s,u:n . The recurrence relations presented in any one of Theorems 19, 20, 21 and 22 may be used in a simple systematic recursive manner in order to compute the quadruple moments (of all order) of order statistics for all sample sizes. In particular, as shown earlier in Section 6 for the exponential case, the results presented in Theorems 12-21 may be used to compute the mean, variance and the coefficients of skewness and kurtosis for any linear function of order statistics from the righttruncated exponential distribution in (7.2) which may then be used to approximate the distribution of that linear function of order statistics. It is of interest to mention here that Theorems 1-11 may be deduced from Theorems 12-12 by letting P ---+ 1.

8. Illustrative examples In this section, we consider three examples involving life-test data and illustrate how we may use the results presented in Section 6 in order to develop approxi-

N. Balakrishnan and S. S. Gupta

56

mate confidence intervals and carry out tests of hypotheses a p p r o x i m a t e l y for the scale p a r a m e t e r 0. o f the exponential distribution based on doubly T y p e - I I censored samples. EXAMPLE 1. Twelve c o m p o n e n t s were placed on a life-test and the time-to-fail of the first eight c o m p o n e n t s that failed were recorded and the experiment was terminated as soon as the eighth failure occurred. The T y p e - I I right-censored sample that resulted f r o m this experiment, as reported by Lawless (1982), is given below: 31,58,157,185,300,470,497,673 . . . . In this case, we have n = 12, r = 0 and s = 4. The best linear unbiased estimate of 0. is obtained from (6.3) to be 0.* = g

~ - ' Yi:12 + 4Y8:12

= 632.875 .

Furthermore, since the censoring in this data is only on the right, it is k n o w n that 2 ( n - s)0.*/0. = 16o-*/o- has exactly a chi-square distribution with 16 degrees of freedom. By using this distributional result, we obtain the 95% lower confidence limit for the true expected life-time 0. to be 160.*/26.296 = 385.078. Similarly, we obtain the 95% confidence interval for 0. to be _ 160.* 160.*] 28.845' 6.9-~J = I351.049, 1465.8371

EXAMPLE 2. Let us consider the following data which represent failure times, in minutes, for a specific type of electrical insulation in an experiment in which the insulation was subjected to a continuously increasing voltage stress (see Lawless, 1982, pp. 138): - , - , 24.4, 28.6, 43.2, 46.9, 70.7, 75.3, 95.5, 98.1,138.6, Here, the smallest two observations are censored as their failure times were not recorded due to experimental difficulty and the largest observation is censored because the experiment was stopped as soon as the eleventh failure occurred. In this case, we have n = 12, r = 2, s = 1, and =

IV0 ~- 1 -I- 1] 2 = 10.9834

1

I

The best linear unbiased estimate of a is obtained from (6.3) to be 0., _ _ _ 1 10.9834

1.8788I~:12 +

i=4

Y/.12. + 2Y1H2

= 71.1385 minutes . F r o m Table 1, we observe that the value of 7 in this case is 0.000815 indicating that the distribution of a * / a can be a p p r o x i m a t e d very closely by a chi-square

Higher order moments of order statistics

57

distribution. Specifically, we m a y approximate the distribution of 22o-*/0- by a chi-square distribution with 22 degrees of freedom. By making use of this approximate distributional result, we obtain the 95% lower confidence limit for the true expected life-time a to be 22o.*/33.924 = 46.1339 mins. Further, we obtain an approximate 95% confidence interval for o. to be 22o.* _22o.* = [42.5504,142.5102] . 36.781'10.982J Proceeding similarly, we obtain the approximate 90% lower confidence limit for the true expected life-time o. to be 22o-*/30.813 = 50.7918 mins. We also obtain an approximate 90% confidence interval for o. to be 22o.* 22o.*1 = [46.1339,126.8477]. 3~.~4'12.338J EXAMPLE 3. Nelson (1982, Table 2.1) has presented a data arising from an experiment on insulting fluid breakdowns. A m o n g the n = 12 specimens tested at 45 kV, 3 failed before 1 second and the times to breakdown (in seconds) of the remaining 9 specimens were 2, 2, 3, 9, 13, 47, 50, 55 and 71. Though the type of censoring is not strictly of Type-II, let us consider the observed data to be doubly Type-I1 censored with n = 12, r = 3 and s = 0. In this case we have K=

11.9542 ,

and the best linear unbiased estimate of o. is obtained from (6.3) to be

o* _ _ _ 1 11.9542

{

2.2612Y4:12 -~

1± } i=5

Y/'12

= 21.2915 seconds . F r o m Table 1, we observe that the value of 7 in this case is 0.001905 indicating that we m a y approximate closely the distribution of 24o.*/o. by a chi-square distribution with 24 degrees of freedom. Consequently, we obtain an approximate 90% confidence interval for o. to be 24o-*

24o-*

36.4151' 13.8484j

= [14.0325,36.8993] .

It is of interest to mention here that, by using the conditional method of inference and exact numerical integration, Viveros (1995) has recently obtained a 90% conditional confidence interval for o. in this case to be [14.00, 36.79]. The accuracy of the approximate confidence interval proposed in this paper is quite evident here.

58

N. Balakrishnan and S. S. Gupta

Acknowledgements T h e first a u t h o r t h a n k s t h e N a t u r a l S c i e n c e s a n d E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada while the second author thanks the National Science Foundation for f u n d i n g t h i s p r o j e c t . T h e a u t h o r s w o u l d a l s o like t o t h a n k D o m e n i c a M a z e p a f o r the excellent typing of the manuscript.

References Arnold, B. C. and N. Balakrishnan (1989). Relations, Bounds and Approximations For Order Statistics. Lecture Notes in Statistics No.53, Springer-Verlag, New York. Balakrishnan, N. and A. C. Cohen (1991). Order Statistics and Inference: Estimation Methods. Academic Press, San Diego. Balakrishnan, N. and P. C. Joshi (1984). Product moments of order statistics from the doubly truncated exponential distribution. Naval Res. Logist. Quart. 31, 27-31. Balakrishnan, N. and S. Kocherlakota (1986). On the moments of order statistics from doubly truncated logistic distribution. J. Statist. Plann. Inf. 13, 117 129. Cohen, A. C. (1959). Simplified estimators for the normal distribution when samples are singly censored or truncated. Technometrics 1, 217-237. Cohen, A. C. and R. Helm (1973). Estimation in the exponential distribution. Technometrics 14, 841-846. David, H. A. (1981). Order Statistics, Second edition. John Wiley & Sons, New York. Epstein, B. (1956). Simple estimators of the parameters of exponential distributions when samples are censored. Ann. Inst. Statist. Math. 8, 15-26. Epstein, B. (1962). Simple estimates of the parameters of exponential distributions. In Contributions to Order Statistics (Eds., A. E. Sarhan and B. G. Greenberg), pp. 361-371. John Wiley & Sons, New York. Epstein, B. and M. Sobel (1953). Life testing. J. Amer. Statist. Assoc. 48, 48(~502. Epstein, B. and M. Sobel (1954). Some theorems relevant to life testing from an exponential distribution. Ann. Math. Statist. 25, 373-381. Gupta, S. S. and N. Balakrishnan (1992). Logistic order statistics and properties. In Handbook of the Logistic Distribution (Ed., N. Balakrishnan). Marcel Dekker, New York. Joshi, P. C. (1978). Recurrence relations between moments of order statistics from exponential and truncated exponential distributions. Sankhy~, Set. B 39, 362-37l. Joshi, P. C. (1979). A note on the moments of order statistics from doubly truncated exponential distribution. Ann. Inst. Statist. Math. 31, 321 324. Joshi, P. C. (1982). A note on the mixed moments of order statistics from exponential and truncated exponential distributions. J. Statist. Plann. Inf. 6, 13-16. Kambo, N. S. (1978). Maximum likelihood estimators of the location and scale parameters of the exponential distribution from a censored sample. Commun. Statist. - Theory Meth. 7(12), 1129 1132. Kjelsberg, M. O. (1962) Estimation of the parameters o f the logistic distribution under truncation and censoring. Ph.D. Thesis, University of Minnesota. Lawless, J. F. (1982). Statistical Models & Methods For Lifetime Data. John Wiley & Sons, New York. Mann, N. R., R. E. Schafer and N. D. Singpurwalla (1974). Methods .for Statistical Analysis o f Reliability and Life Data. John Wiley & Sons, New York. Nelson, W. (1982). Applied Life Data Analysis. John Wiley & Sons, New York. Saleh, A. K. Md. E. (1966). Estimation of the parameters of the exponential distribution based on order statistics in censored samples. Ann. Math. Statist. 37, 1717-1735. Saleh, A. K. Md. E., C. Scott and D. B. Junkins (1975). Exact first and second order moments of order statistics from the truncated exponential distribution. Naval Res. Logist. Quart. 22, 65-77.

Higher order moments of order statistics

59

Sarhan, A. E. and B. G. Greenberg (1957). Tables for best linear estimates by order statistics of the parameters of single exponential distributions from singly and doubly censored samples. J. Amer. Statist. Assoc. 52, 58-87. Sarhan, A. E. and B. G. Greenberg (1958). Estimation problems in exponential distribution using order statistics. Proc. of the Statistical Techniques in Missile Evaluation Symposium, Blacksburg, Virginia, pp. 123-173. Sarhan, A. E. and B. G. Greenberg (Eds.) (1962). Contributions to Order Statistics. John Wiley & Sons, New York. Sukhatme, P. V. (1937). Tests of significance for samples of the chi-square population with 2 degrees of freedom. Ann. Eugen. 8, 5~56. Tarter, M. E. (1966). Exact moments and product moments of order statistics from the truncated logistic distribution. J. Amer. Statist. Assoc. 61, 514-525, Viveros, R. (1995). Conditional inference and applications. In The Exponential Distribution: Theory, Methods and Applications (Eds., N. Balakrishnan and A. P. Basu). Gordon and Breach, Langhorne, PA.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 17 © 1998 ElsevierScienceB.V. All rights reserved.

.~

Log-gamma Order Statistics and Linear Estimation of Parameters

N. Balakrishnan and P. S. Chan

1. Introduction

Let X be a log-gamma random variable with probability density function 1

~x_e~

f*(x)=F-~e

,

-ec<x<ec,

~c>0

(1.1)

and cumulative distribution function

F*(x)=Iex(~c),

-oc<x
to>0 ,

(1.2)

where It(K) is the incomplete gamma function defined by t 1 = f--e-Zz K-ldz, 0 < t < o % to>0 . I,(~c) J0 r(t¢) Figure 1 shows log-gamma distributions for several values of ~c. The special case of ~c= 1 is the extreme value distribution which has been discussed in great detail by many authors including Mann, Schafer and Singpurwalla (1974), Bain (1978), Lawless (1982) and Cohen and Whitten (1988). Note that, for positive integral values of ~:, the cumulative distribution function F*(x) in (1.2) can be written as oo

F*(x)= Z e-e~eix/i!, -oc < x < oc, ~c= 1,2,... , i--K

or, equivalently, t¢

1

1-F*(x)=~e-eeiX/i!,

-ec<x
~:=1,2,....

(1.3)

i--0

From (1.1), we derive the moment generating function of the log-gamma random variable X to be

Mx(t) = E(e tx) -- r(tc + t)

r(~)

61

62

N. Balakrishnan a n d P. S. Chan

1.4

fix)

1.2

~<=8

~:=12

1

0.8

i~=4

0.6 ~=1

0.4

x

i -5

-4

-3

-2

-1

1

2

3

4

5

-0.2 Fig. 1. L o g - g a m m a density functions.

from which we immediately obtain the mean and variance of X to be

E(X) -

rr(x) r ( x ) - O(x)

and Var(X) -

Ur(x)

(Fr(x)) 2

r(x)

\r-~j

= 0r(x) '

where

d Fl(z) ~(z) = 5Z-ln F(z) -tlZ

is the digamma function and O r(z) = d O(z)

is the trigamma function. As a matter of fact, by using the asymptotic formula that 0(x) ~ log x and 0r(x) ~ 1/x (see Abramowitz and Stegun, 1965), Prentice (1974) suggested a reparametrized model of the density function in (1.1) as

f(x)

--~ X ~ 2 _~/~x_~:e~/~

r(x) c

,

-oc<x
~c>0 ,

(1.4)

and showed the asymptotic normality (as x ~ ec ) of this family. It can be easily seen that the cumulative distribution function for the density function in (1.4) is given by

63

Log-gamma order statistics and linear estimation F ( x ) = I~e~/¢~(~c),

- o c < x < oo, ~c > 0 .

(1.5)

Let Y be a log-gamma random variable with probability density function

9(Y) - aF(~c) exp x/~

-oc
- xe('%")/~/~ , -oc<#
a>0, K>0

(1.6) ,

where # is the location parameter, a is the scale parameter, and ~c is the shape parameter. Lawless (1980, 1982) has illustrated the usefulness of the log-gamma model in (1.6) as a life-test model and discussed the maximum likelihood estimation of the parameters; see also Prentice (1974). Young and Bakir (1987) discussed the log-gamma regression model. One may also refer to Lawless (1980) and DiCiccio (1987) for some valuable work on the inference for a related generalized gamma distribution. In this paper, by assuming that the shape parameter ~: in (1.5) is known, we discuss the best linear unbiased estimation of the parameters/~ and a based on complete and Type-II censored samples. We can provide the necessary tables for complete and right-censored samples for sample sizes up to twenty five and K = 0.5(0.5)2.0(1.0)4.0(2.0)12.0 upon request. This is made possible after determining the single and product moments of order statistics in terms of digamma and trigamma functions and then computing the means, variances and covariances of order statistics. Then we consider a life-test data given by Lawless (1982) and illustrate the method of estimation discussed in this paper. 2. Single moments of order statistics

Let XI:, <_ X2:, <_ ... <_ X,:n be the order statistics obtained from a random sample of size n. Then the density function of Xr:n is (David, 1981, p. 9; Balakrishnan and Cohen, 1991, p. 12) given by n! {F(x))r_l { 1 _ F ( x ) } , , _ r f ( x ) , f,:,(x) = (r - 1)[(n - r)! 1
-oc<x
(2.1)

.

Upon using the expressions o f f ( x ) and F ( x ) in Eqs. (1.1) and (1.2), respectively, we may compute the single moments of order statistics from (2.1) as c~F(k) :n =

xkf,:,(x) dx

(2.2)

OO

directly by using numerical integration methods. Although explicit algebraic expressions cannot be derived from (2.1) in general, exact explicit formulae may be derived for the single moments of order statistics for positive integral values of by following steps similar to those of Gupta (1960, 1962).

N. Balakrishnan and P. S. Chart

64

For positive integral values of ~c, upon using the expression of 1 - F ( x ) in Eq. (1.3) we can write the moment generating function of XI:, as

Mi:.(t) = E(e ix':')

n

;~

I n-I

(K-1

- F-(~)~ooetX(j~=oe-e~e/~/J[) rt

oo ( K-1

e e~e'~Xdx

].-1

-- F(K)~ Ij~=oe-)'YJ/J'~

e-Yy~+t-ldy

(~ 1)(,-1)

n

(2.3)

aj( c,n-1)r( c +j +t)/n

where F(.) denotes the complete gamma function and aj(~c,m) denotes the coefficient o f / i n {~i~-01yi/i!}". The single moments of the smallest order statistic may be derived from Eq. (2.3). For example, from Eq. (2.3) we immediately obtain

oq:, = d Ml:.(t) t=0 (K-1)(n-l) _

F(t< + j ) {O(~c + j ) - lnn}

n

rtx+J (~-l)(n-1) __

n

aj(~c,n- 1)F(~c+j)O(~c +j)/n'~+J-lnn

F(tc)

(2.4)

and c~(2) l:n - - dd-~2MI:" (t) t=o /'/

~c-1)(n-1)

-- F(~c) ~

F(tc+j) aj(~c,n- l) n,~+j [O'(~c+j)+{O(tc+j)-lnn} 2]

n ~-1)(.-0 -- F(K) ~ aj(K,,n-H

- 21nnF---~

l)

F(K + j ) n~C~{~/(~c+j)+(~(g+j))

2}

(~-l)(n-1)

~

aj(~c,n -1)F(tc+j)O(~c+j)/n'~+J+(lnn) 2 ; (2.5)

hence, the variance of XI:, is given by

Log-gamma order statistics and linear estimation

65

ill,l:. = Var(Xl:.) .(2)n _ ~2:n mECXl: -- C(/¢)/'/(~-l)(n-1)~[] aJ(~c,n _ t )"F(K + J) { o' (~ + j) + (O(K + j) )2 j'=O (,v-l)(n 1)

-

r-~)

.=

}2

aj(K,n- 1)C(~:+j)O(~c+j)/n ~+; (2.6)

In the above formulae, 0(z) is the psi (or digamma) function defined by O(z) =dlnF(z)/dz=U(z)/F(z) and O'(z) is its derivative given by O'(z)= • • F It (z)/F(z) - (0(z)) 2 commonly termed as the trlgamma function. Hence, the first two single moments of XI:, may be easily computed from Eqs. (2.4) and (2.5) at least for small values of n. Furthermore, upon using the recurrence relation (David, 1981, p. 48; Arnold and Balakrishnan, 1989, p. 8) that

o~(k)

i=n~r+l (--1)i-n+r 1(i--1) -- r

I ~\i/O~l:i,(k)2 < r < n

,

(2.7)

the single moments of all other order statistics may be computed in a simple recursive manner. Also, the coefficients aj(~:, n - 1) involved in Eqs. (2.3)-(2.6) can be generated very easily as follows. First of all, we note that ao(~C, 1 ) = 1, al(tC, l ) = 1, a2(K, 1 ) = ~ , •

• "',

a~ l(K:, 1 ) -

(2.8)

1

(~c-l):

Next, for m >_ 2 we have

a;(~c,m) = Coefficient of y in l i~_oY'/i' ~ =Z[C°efficient°fyt.in{i~=oYi/i!=o fK-1

x

Coefficientofy j-i in

I~yi/i,~

~m-l~

J

K 1

= Zaj:i(K,m - 1)/i! . i=0

(2.9)

N. Balakrishnan and P. S. Chan

66

Thus, by starting with the values o f aj(~c, 1) given in (2.8), we can c o m p u t e the coefficients aj(~c, m) for any value o f m by repeated a p p l i c a t i o n o f the recurrence relation in (2.9). M e a n s and variances o f all order statistics have been c o m p u t e d for sample sizes up to twenty five for choices o f 1<=0.5(0.5)2.0(1.0)4.0(2.0)12.0. Interested readers m a y write to us directly to receive a c o p y o f these tables. It should be m e n t i o n e d here that for K = 1 the values o f m e a n s and variances agreed with those o f o r d e r statistics f r o m the extreme value distribution c o m p u t e d earlier by Lieblein and Salzer (1957), W h i t e (1969) and B a l a k r i s h n a n and C h a n (1992a). F o r the sake o f the illustrative example in Section 5, we present the m e a n s and variances o f order statistics when n = 23 in Tables 1 and 2, respectively.

Table 1 Means of the log-gamma order statistics when n = 23 r

l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

K

3.0

4.0

6.0

8.0

10.0

12.0

-0.47939 -0.07784 0.14000 0.29494 0.41808 0.52215 0.61367 0.69643 0.77289 0.84476 0.91333 0.97960 1.04443 1.10863 1.17297 1.23832 1.30570 1.37643 1.45241 1.53669 1.63486 1.75968 1.95532

0.08980 0.41453 0.59310 0.72115 0.82356 0.91055 0.98738 1.05713 1.12181 1.18282 1.24120 1.29780 1.35335 1.40852 1.46398 1.52048 1.57891 1.64045 1.70679 1.78065 1.86707 1.97755 2.15210

0.79497 1.04021 1.17726 1.27648 1.3564l 1.42471 1.48533 1.54063 1.59212 1.64087 1.68771 1.73329 1.77817 1.82290 1.86803 1.91417 1.96206 2.01270 2.06751 2.12881 2.20091 2.29370 2.44175

1.24554 1.44867 1.56326 1.64669 1.71418 1.77205 1.82357 1.87068 1.91466 1.95641 1.99660 2.03578 2.07446 2.11308 2.15213 2.19214 2.23376 2.27786 2.32571 2.37939 2.44272 2.52454 2.65589

1.57372 1,75011 1,85026 1,92345 1.98282 2.03384 2.07936 2.12106 2.16005 2.19712 2.23285 2.26775 2.30224 2.33674 2.37166 2.40749 2.44482 2.48444' 2.52750 2.57589 2.63311 2.70725 2.82677

1.83058 1.98821 2.07812 2.14401 2.19756 2.24366 2.28485 2.32264 2.35801 2.39168 2.42417 2.45593 2.48736 2.51882 2.55070 2.58345 2.61761 2.65390 2.69340 2.73785 2.79049 2.85884 2.96938

Log-gamma order statistics and linear estimation

67

Table 2 Variances and covariances of the log-gamma order statistics when n = 23 f

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 t 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3

S

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 3 4

/£

3.0

4.0

6.0

8.0

0.23798 0.09970 0.06282 0.04578 0.03590 0.02942 0.02482 0.02136 0.01866 0.01648 0.01468 0.01315 0.01184 0.01069 0.00967 0.00875 0.00791 0.00713 0.00640 0.00568 0.00497 0.00422 0.00332 0.10702 0.06765 0.04938 0.03877 0.03179 0.02683 0.02310 0.02019 0.01783 0.01588 0,01424 0,01282 0.01157 0,01047 0.00948 0.00857 0.00772 0.00693 0.00616 0.00539 0.00458 0.00359 0.07170 0.05241

0.15310 0.06553 0.04173 0.03064 0.02417 0.01990 0.01686 0.01457 0.01277 0.01131 0.01011 0.00909 0.00820 0.00743 0.00674 0.00612 0.00555 0.00502 0.00452 0.00403 0.00354 0.00302 0.00239 0.07126 0.04554 0.03349 0,02645 0.02179 0.01847 0.01597 0.01400 0.01241 0,01109 0.00997 0.00900 0.00815 0.00740 0.00672 0.00609 0.00551 0.00496 0.00442 0.00389 0.00332 0.00263 0.04865 0.03583

0.08562 0.03760 0.02425 0,01796 0.01426 0.01181 0,01005 0.00872 0.00768 0.00683 0.00613 0.00553 0.00501 0.00455 0.00414 0.00377 0.00343 0.00312 0.00282 0.00253 0.00223 0.00192 0.00153 0.04153 0.02689 0.01994 0,01586 0.01314 0.01119 0.00972 0.00856 0.00761 0.00683 0.00616 0.00558 0.00507 0.00462 0.00421 0.00383 0.00348 0,00314 0.00282 0.00249 0.00214 0.00171 0.02900 0.02154

0.05807 0.02588 0.01682 0.01252 0.00998 0.00829 0.00708 0.00616 0.00543 0.00485 0.00435 0.00394 0.00357 0.00325 0.00297 0.00271 0.00247 0.00225 0.00204 0.00183 0.00162 0.00140 0.00113 0.02886 0.01882 0.01403 0.01120 0.00931 0.00796 0.00693 0.00611 0.00545 0.00490 0.00443 0,00402 0.00366 0.00334 0.00305 0.00278 0.00253 0,00229 0.00206 0,00183 0.00158 0,00127 0,02041 0.01524

10.0 0.04346 0.01957 0.01278 0.00954 0.00763 0,00635 0.00543 0.00474 0.00419 0.00374 0.00336 0.00304 0.00277 0,00252 0.00231 0.00211 0.00193 0.00175 0.00159 0.00143 0.00127 0.00110 0.00089 0.02195 0.01439 0,01077 0.00862 0.00718 0.00614 0.00536 0.00474 0,00423 0.00381 0.00345 0.00314 0.00286 0,00261 0.00239 0.00218 0.00199 0.00180 0.00163 0.00144 0.00125 0.00101 0.01566 0.01174

12.0 0.03451 0.01565 0.01026 0.00768 0.00615 0.00513 0.00439 0.00383 0,00339 0.00303 0.00273 0.00248 0.00225 0.00206 0.00188 0.00172 0.00157 0.00144 0.00131 0.00118 0.00105 0.00091 0.00074 0.01764 0.01161 0.00870 0.00698 0.00583 0.00499 0,00436 0.00386 0.00345 0,00311 0.00282 0.00256 0.00234 0.00214 0.00196 0,00179 0.00164 0.00149 0.00134 0.00119 0.00103 0.00084 0.01267 0,00951

68

N. Ba~kr~hnan and P. S. Chan

Table 2 (Contd.) r

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5

S

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 5 6 7 8 9 10 11 t2 13

/£

3.0

4.0

6.0

8.0

10.0

12.0

0.04118 0.03379 0.02853 0.02458 0.02148 0.01898 0.01691 0.01516 0.01365 0.01233 0.01115 0.01009 0.00913 0.00823 0.00738 0.00656 0.00574 0.00488 0.00383 0.05526 0.04346 0.03568 0.03014 0.02597 0.02270 0.02007 0.01788 0.01603 0.01443 0.01304 0.01180 0.01068 0.00966 0.00871 0.00781 0.00694 0.00608 0.00516 0.00405 0.04571 0.03755 0.03173 0.02735 0.02392 0.02115 0.01884 0.01690 0.01522

0.02832 0.02336 0.01980 0.01712 0.01502 0.01332 0.01190 0.01070 0.00966 0.00875 0.00794 0.00721 0.00654 0.00592 0.00533 0.00475 0.00418 0.00357 0.00282 0.03801 0.03007 0.02481 0.02105 0,01821 0.01597 0.01416 0.01266 0.01139 0.01028 0.00932 0.00845 0.00768 0.00696 0.00630 0.00567 0.00506 0.00445 0.00380 0.00301 0.03179 0.02624 0.02227 0.01927 0.01691 0.01500 0.01341 0.01206 0.01090

0.01714 0.01422 0.01211 0.01052 0.00927 0.00825 0.00740 0.00668 0.00605 0.00550 0.00501 0.00456 0.00415 0.00377 0.00341 0.00306 0.00270 0.00232 0.00185 0.02301 0.01833 0.01521 0.01297 0.01127 0.00993 0.00884 0.00793 0.00715 0.00649 0.00590 0.00537 0.00489 0.00445 0.00404 0.00366 0.00328 0.00290 0.00249 0.00199 0.01949 0.01618 0.01380 0.01200 0.01057 0.00941 0.00845 0.00762 0.00691

0.01218 0.01013 0.00866 0.00754 0.00666 0.00594 0.00534 0.00483 0.00438 0.00399 0.00364 0.00333 0.00304 0.00276 0.00250 0.00225 0.00199 0.00172 0.00138 0.01635 0,01308 0.01089 0.00931 0.00811 0.00716 0.00639 0.00574 0.00520 0.00472 0.00430 0.00392 0.00358 0.00327 0.00297 0.00269 0.00242 0.00215 0.00185 0.00149 0.01395 0.01162 0.00994 0.00866 0.00765 0.00683 0.00614 0.00556 0.00505

0.00940 0.00784 0.00671 0.00586 0.00518 0.00463 0.00417 0.00377 0.00343 0.00313 0.00286 0.00261 0.00239 0.00218 0.00198 0.00178 0.00158 0.00137 0.00110 0.01262 0.01012 0.00845 0.00723 0.00631 0.00559 0.00499 0.00449 0.00407 0.00370 0.00338 0.00309 0.00282 0.00258 0.00235 0.00213 0.00192 0.00171 0.00148 0.00119 0.01082 0.00904 0.00774 0.00676 0.00598 0.00535 0.00482 0.00436 0.00397

0.00764 0.00638 0.00547 0.00478 0.00423 0.00378 0.00341 0.00309 0.00281 0.00257 0.00235 0.00215 0.00197 0.00180 0.00163 0.00147 0.00131 0.00114 0.00092 0.01025 0.00824 0.00689 0.00591 0.00516 0.00457 0.00409 0.00369 0.00334 0.00304 0.00278 0.00254 0.00233 0.00213 0.00194 0.00177 0.00159 0.00142 0.00123 0.00100 0.00882 0.00738 0.00633 0.00554 0.00490 0.00439 0.00396 0.00359 0.00327

Log-gamma order statistics and lhTear estimation

69

Table2(Contd.) r

s

14 15 16 17 18 19 20 21 22 23 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 8 9 10

K

3.0

4.0

6.0

8.0

10.0

12.0

0.01375 0.01244 0.01126 0.01018 0.00918 0.00824 0.00732 0.00641 0.00544 0.00428 0.03946 0.03336 0.02877 0.02517 0.02225 0.01983 0.01779 0.01602 0.01447 0.01310 0.01186 0.01073 0.00967 0.00868 0.00772 0.00675 0.00574 0.00451 0.03506 0.03025 0.02647 0.02341 0.02087 0.01872 0.01686 0.01524 0.01379 0.01249 0.01129 0,01019 0.00914 0.00813 0.00711 0.00604 0.00475 0.03181 0.02784 0.02463

0.00987 0.00896 0.00814 0.00738 0.00668 0.00601 0.00536 0.00472 0.00403 0.00319 0.02770 0.02352 0.02036 0.01787 0.01585 0.01418 0.01275 0.01152 0.01044 0.00948 0.00861 0.00781 0.00706 0.00636 0.00568 0.00499 0.00426 0.00337 0.02481 0.02149 0.01887 0.01674 0.01497 0.01347 0.01217 0.01103 0.01002 0.00910 0.00825 0.00747 0.00672 0.00600 0.00528 0.00451 0.00357 0.02268 0.01992 0.01768

0.00628 0.00572 0.00522 0.00475 0.00431 0.00390 0.00349 0.00309 0.00265 0.00212 0.01717 0.01465 0.01274 0.01123 0.01000 0.00897 0.00810 0.00735 0.00668 0.00608 0.00555 0.00505 0.00459 0.00415 0.00372 0.00328 0.00282 0.00226 0.01552 0.01350 0.01191 0.01061 0.00952 0.00860 0.00780 0.00709 0.00646 0.00589 0.00536 0.00487 0.00440 0.00395 0.00349 0.00300 0.00240 0.01431 0.01263 0.01125

0.00460 0,00420 0.00383 0.00350 0.00318 0.00288 0.00259 0.00230 0.00198 0.00160 0.01236 0.01058 0.00922 0.00815 0.00728 0.00654 0.00592 0.00538 0.00490 0.00447 0.00409 0.00373 0.00339 0.00308 0.00276 0.00245 0.00212 0.00170 0.01124 0.00981 0.00867 0.00774 0.00696 0.00630 0.00572 0.00522 0.00476 0.00435 0.00397 0.00361 0.00327 0.00294 0.00261 0.00225 0.00181 0.01042 0.00921 0.00823

0.00362 0.00331 0.00303 0.00277 0.00252 0.00229 0.00206 0.00183 0.00158 0.00128 0.00963 0.00826 0.00721 0.00638 0.00571 0,00514 0,00466 0,00424 0.00387 0,00353 0,00323 0,00295 0,00269 0,00245 0.00220 0,00196 0.00169 0.00137 0.00879 0.00768 0.00680 0.00608 0.00548 0.00497 0.00452 0.00412 0.00377 0.00345 0.00315 0.00287 0.00261 0.00235 0.00209 0.00181 0.00146 0.00818 0.00724 0.00648

0.00298 0.00273 0.00250 0.00229 0.00209 0.00190 0.00171 0.00152 0.00132 0.00107 0.00788 0.00676 0.00592 0.00524 0.00469 0.00423 0.00384 0.00349 0.00319 0.00292 0.00267 0.00245 0.00223 0.00203 0.00183 0.00163 0.00141 0.00114 0.00721 0.00631 0.00559 0.00501 0.00451 0.00410 0.00373 0.00341 0.00312 0.00286 0.00261 0.00239 0.00217 0.00196 0.00174 0.00151 0.00122 0.00672 0.00596 0.00534

N. Balakr~hnan and P. S. Chan

70 Table 2 (Contd.) r

8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 I0 10 10 10 10 10 10 lO 10 10 10 10 10 11 11 11 ll 11 11

s

11 12 13 14 15 16 17 18 19 20 21 22 23 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 10 11 12 13 14 15 16 17 18 19 20 21 22 23 11 12 13 14 15 16

K

3.0

4.0

6.0

8.0

10.0

12.0

0.02196 0.01970 0.01775 0.01605 0.01453 0.01315 0.01190 0.01073 0.00963 0.00856 0.00750 0.00637 0.00500 0.02933 0.02595 0.02315 0.02077 0.01872 0.01692 0.01532 0.01387 0.01255 0.01132 0.01016 0.00904 0.00791 0.00672 0.00528 0.02739 0.02444 0.02193 0.01977 0.01787 0.01618 0.01466 0.01326 0.01197 0.01074 0.00955 0.00837 0.00711 0.00558 0.02586 0.02322 0.02093 0.01893 0.01714 0.01553

0.01582 0.01424 0.01287 0.01166 0.01059 0.00962 0.00873 0.00790 0.00711 0.00635 0.00558 0.00477 0.00377 0.02105 0.01869 0.01673 0.01506 0.01361 0.01234 0.01120 0.01018 0.00924 0.00836 0.00753 0.00672 0.00591 0.00505 0.00400 0.01979 0.01772 0.01595 0.01442 0.01307 0.01187 0.01079 0.00979 0.00886 0.00798 0.00713 0.00627 0.00535 0.00424 0.01881 0.01693 0.01531 0.01389 0.01262 0.01147

0.01010 0.00912 0.00827 0.00753 0.00686 0.00625 0.00569 0.00517 0.00468 0.00419 0.00371 0.00319 0.00255 0.01339 0.01194 0.01072 0.00969 0.00879 0.00799 0.00728 0.00664 0.00605 0.00550 0.00497 0.00446 0.00394 0.00339 0.00271 0.01269 0.01140 0.01030 0.00934 0.00850 0.00775 0.00707 0.00644 0.00585 0.00529 0.00474 0.00419 0.00361 0.00288 0.01214 0.01098 0.00996 0.00906 0.00826 0.00754

0.00740 0.00670 0.00609 0.00555 0.00507 0.00463 0.00422 0.00385 0.00349 0.00313 0.00278 0.00240 0.00193 0.00980 0.00875 0.00788 0.00713 0.00648 0.00591 0.00540 0.00493 0.00450 0.00410 0.00371 0.00334 0.00296 0.00256 0.00206 0.00932 0.00839 0.00760 0.00691 0.00630 0.00575 0.00526 0.00480 0.00437 0.00396 0.00356 0.00316 0.00273 0.00220 0.00896 0.00811 0.00738 0.00673 0.00615 0.00562

0.00584 0.00529 0.00481 0.00439 0.00402 0.00368 0.00336 0.00306 0.00278 0.00251 0.00223 0.00193 0.00156 0.00771 0.00690 0.00622 0.00564 0.00513 0.00469 0.00428 0.00392 0.00358 0.00327 0.00297 0.00267 0.00238 0.00206 0.00166 0.00736 0.00664 0.00602 0.00548 0.00500 0.00457 0.00419 0.00383 0.00349 0.00317 0.00286 0.00254 0.00220 0.00178 0.00709 0.00643 0.00586 0.00535 0.00490 0.00448

0.00481 0.00437 0.00398 0.00364 0.00333 0.00305 0.00279 0.00255 0.00231 0.00209 0.00186 0.00161 0.00131 0.00635 0.00569 0.00514 0.00466 0.00425 0.00388 0.00355 0.00325 0.00298 0.00272 0.00247 0.00223 0.00199 0.00172 0.00140 0.00608 0.00549 0.00498 0.00454 0.00415 0.00380 0.00348 0.00318 0.00291 0.00264 0.00238 0.00212 0.00184 0.00149 0.00587 0.00533 0.00486 0.00444 0.00407 0.00373

Log-gamma order statistics and linear estimation

71

Table2(Contd.) r

11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 14 14 14 15 15 15 15 15 15 15 15

s

17 18 19 20 21 22 23 12 13 14 15 16 17 18 19 20 21 22 23 13 14 15 16 17 18 19 20 21 22 23 14 15 16 17 18 19 20 21 22 23 15 16 17 18 19 20 21 22

3.0

4.0

6.0

0.01406 0.01268 0.01138 0.01013 0.00887 0.00754 0.00592 0.02465 0.02223 0.02011 0.01822 0.01651 0.01494 0.01348 0.01211 0.01077 0.00943 0.00802 0.00630 0.02371 0.02145 0.01944 0.01762 0.01595 0.01440 0.01293 0.01150 0.01008 0.00857 0.00673 0.02300 0.02085 0.01890 0.01712 0.01546 0.01388 0.01235 0.01082 0.00920 0.00724 0.02251 0.02042 0.01849 0.01670 0.01500 0.01336 0.01171 0.00996

0.01041 0.00942 0.00849 0.00758 0.00667 0.00569 0.00451 0.01804 0.01632 0.01480 0.01345 0.01222 0.01110 0.01005 0.00905 0.00808 0.00711 0.00608 0.00481 0,01745 0,01584 0,01439 0,01308 0.01188 0.01076 0.00969 0.00866 0.00762 0.00651 0.00516 0.01703 0.01548 0.01408 0.01279 0.01158 0.01044 0.00933 0.00821 0.00702 0.00556 0.01677 0.01525 0.01386 0.01256 0.01132 0.01011 0.00890 0.00761

0.00687 0.00624 0.00564 0.00506 0.00448 0.00385 0.00308 0.01173 0.01065 0.00969 0.00884 0.00806 0.00735 0.00668 0.00604 0.00542 0.00479 0.00412 0.00330 0.01143 0.01041 0.00949 0.00866 0.00789 0.00718 0.00649 0.00582 0.00515 0.00443 0.00355 0.01123 0.01025 0.00935 0.00853 0.00775 0.00701 0.00629 0.00557 0.00479 0.00384 0.01114 0.01017 0.00927 0.00843 0.00763 0.00685 0.00606 0.00522

8.0 0.00513 0.00467 0.00424 0.00381 0.00338 0.00292 0.00235 0.00869 0.00791 0.00721 0.00659 0.00602 0.00550 0.00501 0.00454 0.00409 0.00363 0.00313 0,00252 0,00850 0,00776 0.00709 0.00648 0.00592 0.00539 0.00489 0.00440 0.00391 0.00337 0.00272 0.00839 0.00767 0.00701 0.00641 0.00584 0.00530 0.00477 0.00423 0.00365 0.00294 0.00835 0.00764 0.00698 0.00637 0.00578 0.00520 0.00461 0.00399

10.0

12.0

0.00410 0.00374 0.00339 0.00306 0.00272 0.00235 0.00190 0.00690 0.00629 0.00574 0.00525 0.00481 0.00440 0.00401 0.00365 0.00329 0.00292 0.00253 0.00204 0.00677 0.00619 0.00566 0.00518 0.00474 0.00433 0.00393 0.00354 0.00315 0.00273 0.00221 0.00670 0.00613 0.00562 0.00514 0.00469 0.00426 0.00384 0.00342 0.00296 0.00239 0.00669 0.00613 0.00561 0.00512 0.00465 0.00420 0.00373 0.00323

0.00341 0.00312 0.00283 0.00256 0.00228 0.00198 0.00160 0.00572 0.00522 0.00477 0.00437 0.00400 0.00367 0.00335 0,00305 0,00275 0,00245 0,00212 0,00172 0,00563 0.00515 0.00471 0.00432 0.00396 0.00362 0.00329 0.00297 0.00264 0.00229 0.00186 0.00558 0.00511 0.00469 0.00429 0.00392 0.00357 0.00322 0.00287 0.00249 0.00202 0.00558 0.00512 0.00469 0.00429 0.00390 0.00352 0.00314 0.00273

N. Balakr~hnan and P. S. Chan

72 Table 2 (Contd.) r

15 16 16 16 16 16 16 16 16 17 17 17 17 17 17 17 18 18 18 18 18 18 19 19 19 19 19 20 20 20 20 21 21 21 22 22 23

S

23 16 17 18 19 20 21 22 23 17 18 19 20 21 22 23 18 19 20 21 22 23 19 20 21 22 23 20 21 22 23 21 22 23 22 23 23

K

3.0

4.0

6.0

8.0

10.0

12.0

0.00783 0.02224 0.02015 0.01820 0.01636 0.01457 0.01277 0.01086 0.00855 0.02220 0.02006 0.01804 0.01607 0.01409 0.01199 0.00944 0.02245 0.02020 0.01800 0.01580 0.01345 0.01060 0.02310 0.02060 0.01809 0.01542 0.01216 0.02436 0.02142 0.01827 0.01443 0.02675 0.02286 0.01809 0.03179 0.02527 0.04712

0.00604 0.01666 0.01515 0.01373 0.01238 0.01106 0.00974 0.00833 0.00661 0.01674 0.01518 0.01369 0.01224 0.01079 0.00923 0.00732 0.01704 0.01538 0.01376 0.01213 0.01038 0.00825 0.01766 0.01581 0.01394 0.01195 0.00950 0.01877 0.01657 0.01421 0.01131 0.02080 0.01788 0.01426 0.02502 0.02004 0.03780

0.00418 0.01115 0.01017 0.00925 0.00838 0.00752 0.00666 0.00573 0.00459 0.01128 0.01027 0.00930 0.00835 0.00740 0.00637 0.00511 0.01157 0.01049 0.00943 0.00835 0.00720 0.00577 0.01210 0.01088 0.00964 0.00832 0.00668 0.01298 0.01152 0.00994 0.00799 0.01455 0.01258 0.01013 0.01775 0.01436 0.02745

0.00321 0.00840 0.00767 0.00700 0.00635 0.00572 0.00508 0.00439 0.00354 0.00853 0.00779 0.00707 0.00637 0.00566 0.00489 0.00394 0.00880 0.00799 0.00720 0.00640 0.00554 0.00447 0.00924 0.00833 0.00741 0.00642 0.00518 0.00997 0.00888 0.00770 0.00622 0.01125 0.00977 0.00792 0.01385 0.01127 0.02174

0.00262 0.00674 0.00617 0.00564 0.00513 0.00462 0.00411 0.00357 0.00289 0.00687 0.00628 0.00571 0.00516 0.00459 0.00398 0.00322 0.00711 0.00647 0.00584 0.00520 0.00451 0.00366 0.00749 0.00677 0.00603 0.00524 0.00425 0.00812 0.00724 0.00630 0.00511 0.00920 0.00801 0.00652 0.01140 0.00931 0.01808

0.00221 0.00564 0.00517 0.00472 0.00430 0.00388 0.00346 0.00301 0.00244 0.00576 0.00527 0.00480 0.00434 0.00387 0.00336 0.00273 0.00597 0.00544 0.00492 0.00439 0.00382 0.00310 0.00631 0.00571 0.00510 0.00443 0.00361 0.00686 0.00613 0.00534 0.00435 0.00780 0.00681 0.00555 0.00971 0.00796 0.01551

3. Product moments of order statistics T h e j o i n t d e n s i t y f u n c t i o n o f Xr:n a n d X~:n is ( D a v i d , 1981, p. 10; B a l a k r i s h n a n a n d C o h e n , 1990, p. 11 ) g i v e n b y

n! fr,s:.(x,y) = (r - 1)l(s - r - 1)!(n - s)! { F ( x ) } r - 1 x {g(y)-

F(x)}S-~-l{1

l<_r<s<_n,

-F(y)}"-Sf(x)f(y), -cc<x
(3.1) .

Log-gamma orderstatisticsand linearestimation

73

Now upon using the expressions of f ( . ) and F(.) in Eqs. (1.1) and (1.2), respectively, we may compute the product moments of order statistics from (3.1) as

O~r,s:n=

f /

xyfr,s:n(X,y)dydx

(3.2)

-oo<x
directly by making use of numerical integration methods. In general, an explicit algebraic expression cannot be derived for e ..... from (3.2); however, for positive integral values of ~c, an explicit expression may be derived as follows. Upon expanding the term {F(y) - F ( x ) } s-~ 1 in (3.1) binomially in powers of F ( x ) and F ( y ) , we can write (r-1)!(s-r--1)!(n-s)! x

f

~ (-1)i s - r - I i=o i

oo

y{F(y)}S

r 1-i{F(y)}n-Si(y)f(y

) dy ,

(3.3)

oo

where I(y) =

f

x{F(x)}~+i-lf(x)

dx

oo

- r 1+ i Y { F ( y ) } r + i

1 r + i J(y)

(3.4)

where J(y) =

//

{F(x)}r+idx. oo

For positive integral value of ~c, upon using the expression ofF(u) in Eq. (1.3) we have

I J(y) =

e-e'e]x/j] ~oo

=

=

1. j = t c

e (~+')" "

j=(r+i)~

j=(r+i)c

bj(~c,r+i)

"~r+i dx )

u j"/ j "!

J0

~

-1 du

u

e -v

1-~e-(r+i)e('r+i)ke~/k!/~=o

r+ i (3.5)

74

N. Balakrishnan and P. S. Chan

where bj(~c, m) denotes the coefficient of uj in {Y~.I~ ul/l!} ~" Upon substituting the above expression of J ( y ) into (3.4), we get

I(y) = ,.@;y{F(y)} r+i-

bj(~c,r + i)F(l')/(r + i)J+l

~

(3.6)

j=(r +i)tc

j-1

+ ~ bj(~c,r + i) F(li'lj+~ 1 ~ e -(r+i)e;'(r+ i)ke~/k' . j=(r+i)~ (r + k=o Upon substituting the expression of l(y) in (3.6) into (3.3) and simplifying the resulting equation, we obtain

(:)(

7r.... = ~ ( _ 1 ) i=0

i r+

- ~(-1)'

-1

i

~

r + i J s:n

r+/-~

i=0

x

s-l~c~(2)

n r + i

O~s-r-i:n r i

bj(~c, r + i)V(j')/(r + i)j

j=(r+i)~c s r-I

+ Z i=0

x

111!

(-1)' ( r - 1)!(n - s)!i!(s - r - 1 - i)!(r + i)

~

bj(tc, r + i )

j=(r+i)t;

j-I

.k

F(j) (r + i)j

oo

x~-~ (r +~. ,) J-[Y {F(y)} , r- - l -, {1 -F(y)}n-Sf(y)e-(r+i)eYekYdy . k=0

(3.7) Now i n order to determine the integral in the last term on the right-hand side of (3.7), let us consider the function

H(!) =

J?

e'e{F(y)} *-r

1-/{1 -

F(y)}n-*f(y)e-(r+i)e'e ky dy

oo

s r-l-i

(K l)(n-s+p)

= Z p=O x

Z q=O

F

e -(n s+p+r+i+l)eVe(~C+k+t+q)dy

oo

1 --

s-r-l-i

F(~c) ~ p:O

(K-1)(n-s+p)

(-1)P( s rp1-i)

~

aq(tr, n - x + p )

q=O

F(~c + k + q + t) x (n - s + p + r + i + 1) ~:+k+q+t "

(3.8)

Log-gamma order statistics and linear estimation

75

Upon differentiating the expression of H(t) in (3.8) with respect to t and then setting t = 0, we obtain

f~y{

F(y) }s-r-1-i{

F (y) }"-Sf(y)e-(r+i)e'e~dy

1 --

1 s r 1 i (~:-l)(n-s+p) --F(K) E (--1)P(s rP l-i) Z aq(K,n--s+p) p=0 X

q=0

F(t~ + k + q) (n-s+p+r+i+ 1) '~+k+q { O ( K + k + q ) l)} .

-ln(n-s+p+r+i+

(3.9)

We, therefore, obtain from (3.7) an exact explicit expression for the product moments as srl O;r's:n "= E (--l)i(r+[-1) (s-l'~'(2)\r]-s:n +` i=0

s-r-I -- E (--1)i(r+iz1)(rn+i)O~s-r-i:n i=o 1

s-r- I

r i

r(j)

bj(K,r+i)

j=(r+i)~

(r + i) j

oe _(~+/ bj(K, r ÷ i) F(j) 1)!(n - s ) ! i ! j )K ( r + i) j+l n!

+ F - - ~ Ei=0 ( - 1 ) i ( r -

j-1 / , .\ks-r-l-i kr-t-t) V ' ( 1 ) p 1 x Ek=0 ~. z_.., p ! ( s r l - i - p)! p=0 x

(x-1)(n-s+p) F(K + k + p) E aq(K,n - s + p) q=0 (n-s+p+r+i+ 1) ~+k+q

x{O(~c+k+q)-ln(n-s+p+r+i+l)}

, l
.

(3.1o) From the explicit expression of er,s:n in Eq. (3.10), the product moments may be computed at least for small values of n. Note that the coefficients bj(~c, m) involved in Eq. (3.10), like the coefficients aj(~, m), may be easily generated in the following recursive form. First of all, we observe that 1

b,~(x, 1) = ~ ,

bK+l

(tO, 1)

1

--

1

(tO -t- 1)!' b,~+2(K, 1) = (K + 2 ' ! " " )

Next, for m _> 2, we have for j = m~c,m~c + 1,. • -,

(3.11)

N. Balakrishnan and P. S. Chan

76

bj(tc, m) = Coefficient o f u j in

=

ul/l!

~

]Coefficient of u i in

i=tc

L

uZ/l! (

oo

]

m-11

x CoefficientofuJiinlz~-~_uZ/l'l J j-(m-1)c

=

Z

~. b j - i ( t c ' m - 1 )

"

(3.12)

i=/¢

Thus, by starting with the values of bj(K, 1) given in (3.11), we can compute the coefficients bj(~c, m) for any value of m by repeated application of the recurrence relation in (3.12). Product moments, and hence the covariances fl...... of order statistics have been computed for sample sizes up to twenty five for choices of ~: = 0.5(0.5)2.0(1.0) 4.0(2.0) 12.0. These tables are available upon request. It should be mentioned here that for ~c= 1 the values of covariances in these tables agreed with those of order statistics from the extreme value distribution given by Lieblein and Zelen (1956) and Balakrishnan and Chan (1992a). Also for the sake of the illustrative example in Section 5, we present the covariances of order statistics when n = 23 in Table 2. 4. Best linear unbiased estimators

In this section, we shall assume that Yr+l:n --< Y~+2:n _<"" _< Yn-s:n is a doubly Type-II censored sample available from the log-gamma population, with/~ as the location parameter and a as the scale parameter, and its density function as presented in (1.5). It is now easy to note that E(Y/:n) = # + aei:,

and

Var(Yi:n)

=

ff2fli,i:n

and Cov(Yi:~, ~:,) = cr2fl i,j:, ,

r + 1 < i < j <_ n - s .

Let us denote

=(~r÷l:n

~r+2:n

"' "

O~n-s:n)l~

1 = ( 1 1 ... 1)i,_r_sxl, B =((fli,j:n)), and

r + 1 < i < j < n -- s,

n =((~o~j)) = B 1

Then, by minimizing the generalized variance

Log-gamma order statistics and linear estimation

(Y - #1 - a~)'g~(Y - #1 - ~ )

77

,

we may derive the best linear unbiased estimators of # and a (see Lloyd, 1952; David, 1980, p. 129; Balakrishnan and Cohen, 1991, pp. 80-81) to be #*

= / 0t'g2~¢l'~ - ~'F210~'g2 ] %

= - a/Ay n--S

= ~

aiYi:,

(4.1)

i=r+l

and

a* = {

l'g21~c'g2 -- 1'~o¢1'g2 "[y

= fAY n--s

= E

bYi:, ,

(4.2)

i=r+ 1

where A is a skew-symmetric matrix of order n - r - s given by ~ ( l e ' - al')t] A = ( ~ ' f ~ ) ( l ' n l ) - (~'f~l) 2 '

(4.3)

Further, the variances and covariance of the estimators/~* and o-* are (see Lloyd, 1952; David, 1981, p. 130; Balakrishnan and Cohen, 1991, p. 81) given by

Var(#,) : a2~

Var(a*)

~'~_

[ (~'~)(1'~1)

-

(4.4)

(ot,£21)2}

(4.5)

-

and

C o v ( / t * , o-*) =

[ (~t~['~)(lt~r~l)

_ (~,~..~1)2

}

.

(4.6)

By making use of the values of means, variances and covariances of order statistics, we computed the coefficients ai and bi from Eqs. (4.1) and (4.2) and the values of Var(#*)/o-2, Var(o-*)/~ 2 and Cov(#*, o-*)/(fl from Eqs. (4.4)-(4.6), respectively. In Table 3, we have presented the coefficients of the best linear unbiased estimator #*, ai, and the coefficients of the best linear unbiased estimator a* for

N. Balakrishnan and P. S. Chan

78

I

ii

II

I l l l

I

I I ~ l

I

?,

I

I I I

i

~D

H

I

79

Log-gamma order statistics and linear estimation

the complete sample for sample size n = 23, for ~ = 3, 4, 6, 8, 10, 12. The corresponding variances of the estimators are tabulated on the second last row of the table and the covariances between the two estimators are given in the last row of the table. Similarly, we have presented in Table 4 the coefficients of the best linear unbiased estimators,/~* and a*, ai and bi and their variances and covariances for sample size 23 and right-censored sample (r = 0, s = 2), for tc = 3,4, 6, 8, 10, 12. It should be mentioned here that we have prepared more elaborate tables for sample sizes up to twenty five and interested readers may write to us directly to receive a copy of these tables.

5. Illustrative example Let us consider here the example of Lawless (1982) which gives the number of million revolutions to failure for each of a group of 23 ball bearings in a fatigue test. The 23 log-lifetimes are 2.884, 3.950, 4.534,

3.365, 3.991, 4.591,

3.497, 4.017, 4.655,

3.726, 4.217, 4.662,

3.741, 4.229, 4.851,

3.820, 4.229, 4.852,

3.881, 4.232, 5.156.

3.948, 4.432,

Assume that the data came from log-gamma distribution with ~c = 3, 4, 6, 8, 10, 12. Q-Q plots of the data in Figure 2 indicate that these models appear reasonably consonant with the data (see also Lawless (1982) based on relative maximum likelihood) and hence they will be used here to illustrate the method of estimation discussed in this paper. We should mention here that the correlation values from the Q-Q plots were at least 0.99, thus providing a strong evidence towards the appropriateness of the models considered here. It is of interest to point out here that Lieblein and Zelen (1965) assumed a twoparameter Weibull distribution for the original data and hence an extreme value distribution for the log-lifetime data given above. Here, we proceed along the lines of Lawless (1980, 1982) who assumed a generalized gamma distribution for the given log-lifetime data. The required coefficients of the BLUEs for this example are given in Table 3. We have summarized the findings in the following table. K

#*

a*

Expected lifetime

Std. dev. of the lifetime

3.0 4.0 6.0 8.0 10.0 12.0

3.314(0.208) 2.828(0.270) 1.934(0.392) 1.112(0.509) 0.343(0.621) -0.385(0.728)

0.903(0.143) 1.051(0.165) 1.299(0.202) 1.507(0.233) 1.691(0.261) 1.857(0.285)

4.147(0.118) 4.148(0.116) 4.150(0.115) 4.150(0.115) 4.151(0.115) 4.151(0.114)

0.568(0.090) 0.560(0.088) 0.553(0.086) 0.550(0.085) 0.548(0.084) 0.547(0.084)

Note here that we have reported the BLU estimates #* and a* as well as the mean lifetime and the standard deviation of the mean lifetime where the mean lifetime is # + a~k(tc) and the standard deviation of the mean lifetime is a ~ .

N. Balakrishnan and P. S. Chan

80

li

I

I

l

l

5~ "0

. .,.-, w,

e~o

do

II

,.-.,

,.A

b-, [.....,

Log-gamma order statistics and linear estimation K=$

~:=3 ~.5 5 4.5 4

-I

••

~.5 5 4.5

#

.e • • ° -2"+'*

3.5• *

3.5

,..5+

2.5 0

-0.5

0.5

1

1.5

.. ~

U

0

2.5

2

1

2

Expected Values of Order Statistics

Expected Values of Order Statistics

1<=10

K=4 ~.5

J.5 5

5

4.5

4.5

4

4

3.5

3.5

3

3

2.5

2.5 0

1

2

-1

Expected Values of Order Statistics

as

f

0

5 a..5

+41, • +..Ira,

4

• *

""~

•

,~

~'~ 5 4.5

i

3.5

0

3 2.5

4

~ 0

2

~=12

~+5

3.5 3 2+5

1

Expected Values of Order Statistics

K=6

-1

81

1

2

-1

Expected Values of Order Statistics

..i

., ,

1r=.9923 0

1

2

Expected Values of Order Statistics F i g . 2. Q - Q P l o t s .

Now, upon using the asymptotic normality of a linear function of order statistics (see David, 1981, p. 274), we immediately obtain approximate 95% confidence intervals for the expected lifetime and the standard deviation of the lifetime (for the choices of tc considered) to be [3.916,4.378], [3.920,4.375], [3.925,4.375], [3.925,4.3751, [3.926,4.3761, [3.928,4.374] and

82

N. Balakrishnan and P. S. Chan

[0.392,0.744], [0.388,0.732], [0.384,0.722], [0.383,0.717], [0.383,0.713], [0.382,0.712], respectively. Now, suppose the last two observations were censored, that is, the life-test was stopped as soon as the 21st ball bearing failed. As before, we can compute the BLUEs based on this censored sample by using the coefficients given in Table 4. The results are summarized in the following table. K

/~*

a*

Expected lifetime

Std. dev. of the lifetime

3.0 4.0 6.0 8.0 10.0 12.0

3.310(0.216) 2.815(0.285) 1.898(0.423) 1.053(0.554) 0.261(0.679) -0.490(0.799)

0.910(0.158) 1.065(0.182) 1.323(0.223) 1.539(0.258) 1.730(0.289) 1.902(0.317)

4.150(0.119) 4.153(0.118) 4.155(0.118) 4.156(0.117) 4.156(0.117) 4.157(0.118)

0.572(0.099) 0.567(0.097) 0.563(0.095) 0.562(0.094) 0.561(0.094) 0.561(0.093)

In this case, we obtain approximate 95% confidence intervals for the expected lifetime and the standard deviation of lifetime (for the choices of K considered) to be [3.917,4.383], [3.922,4.384], [3.924,4.386], [3.927,4.385], [3.927,4.385], [3.926,4.388] and [0.378,0.766], [0.377,0.757],[0.377,0.7491, [0.378,0.746], [0.377,0.745], [0.379,0.743], respectively. The above given estimates, their standard errors and the confidence intervals based on them, for the case of the right-censored sample are observed to be very close to those computed based on the complete sample.

Acknowledgements The first author thanks the Natural Sciences and Engineering Research Council of Canada for funding this research.

References Abramowitz, M. and 1. A. Stegun (Eds.) (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York. Arnold, B. C. and N. Balakrishnan (1989). Relations, Bounds and Approximations For Order Statistics. Lecture Notes in Statistics No. 53, Springer-Verlag, New York. Bain, L. J. (1978). Statistical Analysis of Reliability and L(fe-Testing. Marcel Dekker, New York. Balakrishnan, N. and P. S. Chan (1992a). Order statistics from extreme value distribution, I: Tables of means, variances and covariances. Commun. Statist. Simul. Comput. 21(4), 1199-1217.

Log-gamma order statistics and linear estimation

83

Balakrishnan, N. and P. S. Chan (1992b). Order statistics from extreme value distribution, II: Best linear unbiased estimates and some other uses. Commun. Statist. Simul. Comput. 21(4), 12191246. Balakrishnan N. and A. C. Cohen (1991). Order Statistics and Inference: Estimation Methods. Academic Press, San Diego. Cohen, A. C. and B. J. Whitten (1988). Parameter Estimation in Reliability and Life Span Models. Marcel Dekker, New York. David, H. A. (1981). Order Statistics. Second edition. John Wiley & Sons, New York. DiCiccio, T. J. (1987). Approximate inference for the generalized gamma distribution. Technometrics 29, 32-39. Gupta, S. S. (1960). Order statistics from the gamma distribution. Technometrics 2, 243 262. Gupta, S. S. (1962). Gamma distribution. In Contributions to Order Statistics (Eds., A. E. Sarhan and B. G. Greenberg) 431~450. John Wiley & Sons, New York. Lawless, J. F. (1980). Inference in the generalized gamma and log-gamma distributions. Technometrics 22, 67-82. Lawless, J. F. (1982). Statistical Models & Methods For Lifetime Data. John Wiley & Sons, New York. Lieblein, J. and H. E. Salzer (1957). Table of the first moment of ranked extremes. J. Res. Nat. Bur. Stand. 59, 203-206. Lieblein, J. and M. Zelen (1956)~ Statistical investigation of the fatigue life of deep groove ball bearings. J. Res. Nat. Bur. Stand. 57, 273 316. Lloyd, E. H. (1952). Least-squares estimators of location and scale parameters using order statistics. Biometrika 39, 88-95. Mann, N. R., R. E. Schafer and N. D. Singpurwalla (1974). Methods .for Statistical Analysis of Reliability and Life Data. John Wiley & Sons, New York. Prentice, R. L. (1974). A log-gamma model and its maximum likelihood estimation. Biometrika 61, 539-544. Schneider, B. E. (1978). Trigamma function. Algorithm AS121. Appl. Statist. 27, 97 99. White, J. S. (1969). The moments of log-Weibull order statistics. Technometrics 11, 373-386. Young, D. H. and S. T. Bakir (1987). Bias correction for a generalized log-gamma regression model. Technometrics 29, 183 191.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 17 © 1998 Elsevier Science B.V. All rights reserved.

A

Recurrence Relations for Single and Product Moments of Order Statistics from a Generalized Logistic Distribution with Applications to Inference and Generalizations to Double Truncation

N. Balakrishnan and Rita Aggarwala

I. G e n e r a l i z e d

logistic

distribution

1. I n t r o d u c t i o n

Let XI, X2,... ,X, be a r a n d o m sample o f size n f r o m the generalized logistic population with cumulative distribution function l l+(1-kx)

l/k'

F(x) = l + e1 - x '

-oc<x
when

k>O,

l<x<ec -~

when

k
-oo<x
when

k=O

(1.1) ,

and with probability density function (1 - ]~)(1/k)

1

E1 + (1 - kx)l/k] 2'

-ec < x < 1 1

e x (1 +e-X-2')

--oo < x <

oc

when

k > 0,

when

k < 0,

when

k = 0 .

(1.2)

Plots o f the density function for selected values of k are given in Fig. l. 1. Only positive values o f k are plotted, since the corresponding plots for - k can be obtained simply by reflection across the vertical axis. Let XI:, <_X2:, <_ ... <_X,:n denote the order statistics obtained by arranging • (i) the n X~'s in increasing order o f magnitude. Let us use ~r:, to denote the single m o m e n t s E(Xri:,) for 1 < r < n and i > 1, and #r,s:, to denote the p r o d u c t m o m e n t s E(Xr:,Xs:,) for 1 _< r < s < n. Let us further denote Var(X~:n) by a ..... 85

86

N. Balakrishnan and R. Aggarwala

-k=

0.4

=0.3

k= 0.1 -/-//// I"

\ \ ~-~- k=0.2

15

f(x

05

-6

-2

10U

~. X

Fig. 1.1 Pdf's of generalized logistic distribution.

and CoV(Xr:,,

Xs:,)

by a ...... For simplicity, we shall also use the notation #r:, for

In Section I of this paper, we establish several recurrence relations satisfied by the single moments/~!i~ and the product moments//r,s:,. These recurrence relations will enable one to compute all the single and product moments of order statistics for all sample sizes in a simple recursive manner. If we let the shape parameter k --+ 0, the recurrence relations reduce to the corresponding results for the logistic distribution established by Shah (1966, 1970); see also Balakrishnan (1992). By starting with the values of E(X) =/-/1:1, E(X2) =//12:1, and //1,2:2 = //2:1, for example, we have determined the means, variances and covariances of order statistics and have tabulated them for samples of size up to 8 for k = 0.1 (0.1)0.4. These quantities have then been used to determine the best linear unbiased estimators (BLUE's) of the location and scale parameters of the generalized logistic distribution, and the necessary tables of coefficients and the variances and covariance of the BLUE's have been tabulated for sample sizes 5(5)20 for k = 0.1(0.1)0.4. Next, we discuss maximum likelihood estimation for the twoparameter and the three-parameter models based on Type-lI right censored samples. Finally, for the generalized logistic distribution, we have presented an example to illustrate these methods of inference.

Recurrence relations for single and product moments

87

Zelterman and Balakrishnan (1992) discussed four types of generalizations of the logistic distribution. The generalized logistic distribution in (1.1), however, is not one of those types; for this reason, the distribution in (1.1) may be referred to as "Type V Generalized Logistic Distribution". This distribution has been proposed, discussed and applied by Hosking (1986, 1990). From (1.1) and (1.2), we observe easily that the hazard function for this family of distributions is given by

h(x) - 1~f(x) ff-(x)- {

(1 - kx)[1 + (1

- kx)I/k I '

1 +e -x'

-oc<x<¼

when

k>0,

±k < x < e c

when

k<0,

-ec<x
when

k=0

.

(1.3) It is clear from the expression of h (x) in (1.3) that the family of generalized logistic distributions in (1.1) is an IFR (Increasing Failure Rate) family for k _> 0 and a D F R (Decreasing Failure Rate) family for k < 0. Hence, the generalized logistic family discussed in this paper will be quite useful as a life span model; see Cohen and Whitten (1988) for an excellent treatment of many such life span models. Further, the first four moments of the generalized logistic random variable are as follows: E(X) = ~l [ l _ f l ( l _ k , l + k ) ]

1E

= ~ 1

sin~z) ' Ikl < 1

(1.4)

E(x 2) = 1 [1 - 2/3(1 - k, 1 + k) + / 3 ( 1 - 2k, 1 + 2k)] 1[ = k~ 1

2k~ 2k~ .] sin(krc~ + sin(2k~)J'

(1.5)

Ikl <

E ( x 3) = 1 [1 - 3fl(1 - k, 1 + k) + 3/3(1 - 2k, 1 + 2 k ) -/3(1

- 3k, 1 + 3k)]

1 I 3krc 6krc 3k~ .] k 3 k1 - sin(krc) ~-Sin(Zkzc) sin(3krc)J'

E(X 4) =

1

[1 - 4fl(1 - k, 1 + k )

1

[k[ <

(1.6)

+ 6/3(1 - 2k, 1 + 2 k )

- 4/3(1 - 3k, 1 + 3k) + / 3 ( 1 - 4k, 1 + 4k)] 1 z

- -

k4 1

4kTt 12k~z 12kTr 4kzt ] sin(krc) t sin(ZkTr) sin(3krc) ~ sin~-k~z)J'

Ikl < 1 (1.7)

88

N. Balakrishnan and R. Aggarwala Table 1.1 Values of coefficients of skewness and kurtosis for selected values of k k

skewness, ~

kurtosis,/~2

.02 .04 .06 .08 .10 .12 .14 .16 .18 .20 .22 .24

-0.17461 -0.35216 -0.53582 -0.72919 -0.93667 -1.16392 -1.41853 -1.71133 -2.05842 -2.48528 -3.03497 -3.78713

4.27530 4,50848 4,92294 5.56385 6.51021 7.89982 9.98345 13.25416 18.80437 29.55619 56.47212 199.73337

As a result, the coefficients of skewness and kurtosis (x/~l, fi2) may be calculated for any value of k in the specified regions. Selected values of these two measures are presented in Table 1.1. Note that only positive values of k are considered in Table 1.1, since V//?l (X; k) = - V//~l (X; - k ) and/~2(X; k) =/~2 (X; -k). A parametric plot of kurtosis versus skewness for the range -0.20 < k < 0.20 is given in Fig. 1.2. Fig. 1.3 displays plots of skewness versus and kurtosis versus k for the same range. These plots may be useful in the initial determination of the value of k for a given sample. The chosen range for these plots does not extend to (-0.25, 0.25), as the values for kurtosis tend to increase drastically due to its poles at k = -0.25 and k = 0.25. From (1.1) and (1.2), we observe that the characterizing differential equation for the distribution is (1 - k x ) f ( x ) = F(x)[1 - F ( x ) ] .

(1.8)

As Shah (1966, 1970) exploited the differential equation f ( x ) = F ( x ) [ 1 - F(x)] (case when k = 0 in (1.8)) of the standard logistic distribution in order to derive several recurrence relations for single and product moments of order statistics, we shall use the differential equation in (1.8) in the following sections to make similar developments for the generalized logistic distribution in (1.1). 2. Recurrence relations for single moments

The density function of Xr:, (1 < r < n) is given by [David (1981, p. 9), Arnold, Balakrishnan and Nagaraja (1992, p. 10)] f~:n(x) =

(r

1

n~ 1)!(n

1

F)]

[F(x)] r 1[1 - F ( x ) ] n - r f ( x ) .

(2.1)

Recurrence relations for single and product moments

89

30"

2

25"

20

kurtosis

10

L

I

l

,

,

,

l

l

J

,

,

I skewness

Fig. 1.2 Parametric plot of kurtosis vs. skewness.

30-

25"

20

15

10

~'~ -0.2

__ skewness

-0 I. 1

~

Fig. 1.3 Skewness and kurtosis plots.

2

N. Balakrishnan and R. Aggarwala

90

In this section, we establish some recurrence relations for the single moments of order statistics. It is of interest to point out here that similar results are available for many other distributions; for example, see Arnold and Balakrishnan (1989) and the article by Balakrishnan and Sultan (1998) in the companion volume. THEOREM 2.1. For n >_ 1 and i = 0, 1,2,...,

(i+,)

l'n+l ~ •

{

1 -}

k(i + 1)} #(i+,) l:n

H

i + Iv(i)

- - - -

(2.2)

l:n '

H

PROOF. For n _> 1 and i = 0, 1,2,..., let us consider from (2.1) z(i) , (i+1) = n f (x i - kxi+l)[1 - F(x)]n-l f(x) dx l:, - tC#l:n

= n fxx xiF(x)[1 - F(x)]" dx upon using (1.8). Integrating now by parts treating x i for integration and the rest of the integrand for differentiation, we obtain #(01:,_ k'#l:n(i+l) = i+ln In f xi+l [ 1 - F(x)]"-lf(x)dx

H

--i~-I

[ (i+l)

[#~:"

(i+l) 7

-#':~+'J

(2.3)

"

The relation in (2.2) follows simply by rewriting (2.3).

Q.E.D.

THEOREM 2.2. For 1 < r < n and i = 0, 1,2,..., (i+1)

~r+l:.+l

(iq-1)(nq- 1) [fl(i) __kll!i+nl)]

(i+1)

= ~r:.+l +

r(n --;T--f;

(2.4)

L r:.

PROOF. From (2.1), let us consider for 1 < r < n and i = 0, 1,2,..., #(i) (i+1) = ( r - 1 )n!! ( n - r ) [ fx (xi-kxi+l)[F(x)]~-I ,.:.- -. x~:. × [1 - F ( x ) ] " - V ( x )

=

(r -

n!

1 ) ! ( n - r)~

dx

~ xi[F(x)]r[1 - X(x)] "-r+~ dx

upon using (1.8). Integrating now by parts treating x i for integration and the rest of the integrand for differentiation, we obtain

Recurrence relations for single and product moments

/A(i)

, (i+l) r:n - kt%n

=

n!

l)[(n-

(r -

r)!(i +

91

I)

r

r ( n - r + 1) [#(i+1) -- (i Jr- ~)(-'H~7) [ r+l:n+l

(i+0? --/tr:n+lJ

(2.5)

•

The relation in (2.4) follows upon simply rewritirig (2.5).

Q.E.D.

REMARK 2.1. Letting the shape parameter k --+ 0 in Theorems 2.1 and 2.2, we deduce the recurrence relations #(i+1)

. (i+l)

i + 1. (i)

l:n+l = #l:n

n

/'tl:n~

n_>l, i=0,1,2,...

,

(2.6)

and (i+1) ]gr+hn+l

=

]/(i+l) r:n+l -[

(i + l)(n_+_l)u(i) r ( n - r + 1) ~'r:"'

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

(2.7) for the single moments of order statistics from the standard logistic distribution. These relations were originally derived by Shah (1970); see also Balakrishnan (1992). REMARK 2.2. The relations established in Theorems 2.1 and 2.2 will enable one to compute the single moments of all order statistics for all sample sizes in a simple recursive manner. For example, by starting with the values of/q:l = E(X) and /~(2) 1:1 = E(X2), one can use the recurrence relations in (2.2) and (2.4) in order to determine the first two single moments (or the means and variances) of order statistics for all sample sizes. This may be done for any choice of the shape parameter k that is of interest. The recursive computational algorithm is explained in detail in Section 4.

3. Recurrence relations for product moments

The joint density function of Xr:n and X~:n(1 _< r < s _< n) is given by [David (1981, p. 10), Arnold, Balakrishnan and Nagaraja (1992, p. 16)] fr,s:,(x,y)

=

(r -

n! 1)!(s - r - 1)!(n - s)! [ F ( x ) ] r - I [F(y) - F(x)] . . . . 1

× {1 - F ( y ) ] " - S f ( x ) f C v ) ,

x
.

(3.1)

92

N. Balakrishnan and R. Aggarwala

In this section, we establish some recurrence relations for the product moments of order statistics. THEOREM

3.1.

For

1 < r < n _

]lr'r+l:n+l

n + 1 n -- r +

1, 1 +

1

#r,~+l:.---n -- r

r , (2) ___1 n -~ 1 t~r+l:n+l n --

]. r #r:n ~

"

(3.2) PROOF. For 1 < r < n - 1, let us consider from (3.1) l~r:n -- k~r,r+l:n ~-

E [Xr:nX;+l:. o -- kX :oXr+ :n]

"'

= (r- 1)[(n- r-

1)!

//

( x - kxy)[f(x)] r-1

x
x [1 - F(y)] n r - l f ( x ) f ( y ) d y d x

= (r-1)!(n-r-

1)!

x[F(x)]~-~f(x)II(x)d~

(3.3)

where, upon using (1.8),

Ii(x) = f F(y)[1 - F(y)]" ~dy

D

Integrating by parts treating dy for integration and the rest of the integrand for differentiation, we get

11 (x)

- xF(x)[1 - F(x)] n-~ + (n - r) ~ y[l - F(y)]"-r-lf(y) dy - (n - r + 1) f f y[1 - Y(y)]"-~f(y) dy .

(3.4)

Upon substituting the expression of 11 (x) in (3.4) into (3.3) and simplifying the resulting expression, we obtain

r(n - r) , (2) #r:n -- kllr,r+l:n --

(/,/

-n Jr i- t~r+l:n+l -]-

(~

- r)(.

-,

n+ 1

- r)l~r'r+l:n

+ 1) [2r'r+l:n+l

"

The recurrence relation in (3.2) is derived simply by rewriting the above equation. Q.E.D.

Recurrence relations for single and produet moments

93

THEOREM 3.2. F o r 1 _< r < s _< n and s - r _> 2,

•

n-s+

n - s + 2

-#~,s ~:,

1

(3.5)

}

n-s+l

#~:~

PROOF. For 1 < r < s _< n and s - r >_ 2, let us consider from (3.1) #r:n -- k#r,s:n = t[Xr:nXOs..n - kXr:nXs:n]

_

n,

- (r - 1)!(s - r -

1)!(n - s)l

(x - kxy)[F(x)] r l x
x IF(y) - F(x)] s-r-1 [1 - F ( y ) ] n - S f ( x ) f ( y ) -_ ( r -

1 ) ! ( s - 7n-!

dydx

1 ) ! ( n - s ) ! fx x [ F ( x ) ] r - l f ( x ) h ( x ) d x

(3.6)

where, u p o n using (1.8), /2(X)

=

~y {F(y)

--

F(x)] . . . . iF(y){1 - F(y)] "-'+1 dy

B

Integrating now by parts treating dy for integration and the rest of the integrand for differentiation, we get I2(x) = - (s - r - 1) f y [ F ( y ) - F(x)]S-r-2[1 - F(y)]"-s+~ f ( y ) d y

+ (s - r - 1) f y [ F ( y ) - F(x)]S-r-2[1 - F ( y ) ] " - s + 2 f ( y ) d y - (n - s + 2) f y I F ( y ) - F(x)] s-r 111 - F(y)]" * + ' f ( y ) d y + (n - s + 1) ~ y [ F ( y ) - F(x)]* ~-1 [1 - F ( y ) ] " - ' f ( y )

dy .

(3.7)

U p o n substituting the expression of I2(x) in (3.7) into (3.6) and simplifying the resulting expression, we obtain ( n - s + 1 ) ( n - s + 2) #r,s-l:n+] n+l ( n - s + 1 ) ( n - s + 2) n+l # ..... + l + ( n - s + l ) p r , s : n .

#r:n - k# ..... = - (n - s + 1)# .... l:n -}

The recurrence relation in (3.5) is derived simply by rewriting the above equation. Q.E.D.

94

N. Balakrishnan and R. Aggarwala

THEOREM 3.3. F o r 1 < r < n - 1,

#r+l,r+2:n+l

r~

1 - r IJr'r+l:n-- n + 1 ~r+l:n+lJ

i- r/*~+1:" +

•

(3.8)

PROOF. F o r 1 < r < n - 1, let us consider from (3.1) ju,.+ 1:,, -

k,U~,r+ 1:,, =

E

[Y°nY~+ l:n - kX~:nX,.+ 1:, ]

= (F- 1)[(n- r-

1)[

(Y- kxy)[f(x)]r-1 x
x [1 - F(y)] "-r lf(x)f(y) dxdy = (r-

1)!(n-r-

1)!

y[1 -F(y)]n-r-lf(y)Jl(y)dy (3.9)

where, u p o n using (1.8), Jl Cv) = ~ [F(x)]~[1 - F(x)] dx

w

Integrating by parts treating dx for integration and the rest of the integrand for differentiation, we get J1 Cv)

y[F(y)]r[1 - F(y)] - r [ x[F(x)] r if(x) dx dx

f

(3.10)

+ (r + 1) Jx x[F(x)]rf(x) dx

m

U p o n substituting the expression of J1 (Y) in (3.10) into (3.9) and simplifying the resulting expression, we obtain

r(n - r) , (2) Pr+l:n -- k#r,r+l:n -- n + i t~r+l:n+l - r#r,r+l:n +

]2r+l,r+2:n+l

•

The recurrence relation in (3.8) follows simply by rewriting the above equation. Q.E.D. COROLLARY 3.1. Setting r = n - 1 in (3.8), we obtain the relation #,,,+1:n+i

.+1{, n

[

771--1/~":" +

1 --

/~, 1....

1,2,

n ~- 1 --,:,+lj', n > 2 .

(3.11)

THEOREM 3.4. F o r l _ < r < s _ < n a n d s - r _ > 2 , #r+l,s+l:n-kl = [Ar+2,s+l:n+l A v ~

T

7 ~s:n ~-

1 --

].l..... -- ['lrWl,s:n

"

(3.12)

Recurrence relations for single and product moments

95

PROOF. F o r 1 < r < s _< n a n d s - r > 2, let us consider f r o m (3.1) /~s:, - kpr,s:, = E [X°,Xs:n - kXr:nXs:n]

"

= (r - 1)!(s - r - 1)!(n - s)!

ff

(Y - kxy)[F(x)]r-'

x
x [F(y) - F(x)] s-r-1 [1 - F(y)]'-'f(x)f(y)

"!

= (r - 1)!(s - r - 1)!(n - s)!

f y[1 -

dx dy

F(y)]" Sf(y)J2(y ) dy

Y

(3.13)

where, u p o n using (1.8), J2(y) = fx [F(x)]~[1 - F(x)][F(y) - F(x)] s-~-I dx

D

I n t e g r a t i n g by parts treating dx for integration a n d the rest o f the i n t e g r a n d for differentiation, we get J2(Y) = - r j[x" x[F(x)] ~ ' [F(y) - F(x)]S-r-l f(x) dx + (r + 1) /

x[F(x)]~[F(y) - F(x)] * ~ if(x) dx

dx

(3.14)

+ (s - r - 1) Jx x[F(x)]~[F(Y) - F(x)]S-r-2f(x) dx - (s - r - 1) fx x[F(x)]r+] [F(y) - F(x)]'-r-2f(x) dx

i

U p o n substituting the expression o f J2 (Y) in (3.14) into (3.13) a n d simplifying the resulting expression, we o b t a i n

#~:, - k#r .... =

- - rl~r,s:n q -

r(r + 1) ~-~-llr+l,s+l:n+l r(r + 1)

-}- rl~r+l's:n

rl -I- 1

~r+2,s+l:n+l

"

T h e recurrence relation in (3.12) follows simply by rewriting the a b o v e equation. Q.E.D. COROLLARY 3.2.

Setting s = n in (3.12), we o b t a i n the relation

#r+l,n+l:n+l = ]2r+2,n+l:n+l -~- ~ ' 1

]~n:n + 1 -1
JAr,n:n -- ]2r+l,n: n

.

,

(3.15)

96

N. Balakrishnan and R. Aggarwala

THEOREM 3.5.

For 1
12r,r+2:n+l = 12r,r+1:n+l -- r(12r+1,r+2:n+l

n4-1

4- gl --

r

-- 12(2) r+l:n+lJ

(3.16)

(12r:n - k12r,r+l:n) •

PROOF. F r o m (3.3), we have for 1 < r < n - 1,

xIF(x)]r-lf(x)I~

12r:, - k12r,~+|:n = (r _ 1 ) ! ( ; , - - r -- 1)!

(x) dx

(3.17)

where

I1 (x) = fy F(y)[1 - F(y)l" r dy

A

U p o n integrating by parts treating dy for integration and the rest o f the integrand for differentiation, and splitting the integral into two by writing F(y) as F(x) + IF(y) - X(x)], we get

11(x) = -- xF(x)[1 - F(x)] n-r - ~ yll - F(y)]n-r f(y) dy + ( n - - r ) jy'yIF(y)--F(x)][1--F(y)] n-~ if(y)dy 4- (n - r)F(x) fyy y[1 - F(y)] n-r I f ( y ) dy

(3.18)

m

U p o n substituting the expression ofI1 (x) in (3.18) into (3.17) and simplifying the resulting expression, we obtain

r(n - r) ,(2)

12r:n-- k12r,r+1:n-

n 4- 1 r(n

--

~r+l:n+l

n- r n- r n + 1 12r,r+l:n+l 4- ~-f12r,r+Z:n+l

~)

4- - -

n4-1

12r+ l,r+2:n+ 1

which, when rewritten, yields the recurrence relation in (3.16). COROLLARY 3.3.

Q.E.D

U p o n setting n = 2 (and r = 1) in (3.16), we obtain the relation • (2)

t21,3:3 = 121,2:3 -- /22,3:3 4- /'t2:3 4- 3(121:2 -- k121,2:2)

•

(3,19)

THEOREM 3.6. F o r 1 _< r < s < n and s - r >_ 2, }tr's÷l:n÷l :

}tr's:n÷l

F S --'~7 (12r+l,s--l:n÷l -- 12r+1,s:n+1)

(3,20)

n4-1 +(n-s+1)(s-r)(12r:"-k12

..... ) .

Recurrence relations for single and product moments

97

Table 4. l Means of order statistics, E(X,:n), for generalized logistic distribution

PROOF.

i

n

k = 0.1

k = 0.2

k = 0.3

k = 0.4

1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8

1 2 2 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8

-0.16641 -1.18305 0.85023 -1.74220 -0.06474 1.30772 -2.13361 -0.56798 0.43849 1.59746 -2.43695 -0.92025 -0.03958 0.75721 1.80753 -2.68569 -1.19325 -0.37423 0.29507 0.98828 1.97138 -2.89711 -1.41712 -0.63359 -0.02843 0.53769 1.16851 2.10519 -3.08136 -1.60740 -0.84626 -0.27914 0.22228 0.72694 1.31571 2.21797

-0.34480 -1.41376 0.72416 -2.05513 -0.13100 1.15175 -2.52547 -0.64411 0.38210 1.40830 -2.90175 -1.02038 -0.07969 0.68996 1.58788 -3.21782 -1.32140 -0.41834 0.25895 0.90546 1.72437 -3.49174 -1.57425 -0.68926 -0.05712 0.49600 1.06924 1.83355 -3.73437 -1.79340 -0.91683 -0.30997 0.19574 0.67616 1.20027 1.92402

-0.54989 -1.71486 0.61508 -2.47208 -0.20040 1.02282 -3.05263 -0.73046 0.32966 1.25387 -3.53157 -1.13684 -0.12089 0.63003 1.40983 -3.94347 -1.47210 -0.46631 0.22453 0.83278 1.52524 -4.30731 -l.76043 -0.75129 -0.08635 0.45769 0.98281 1.61564 -4.63476 -2.01511 -0.99636 -0.34282 0.17013 0.63023 1.10034 1.68926

-0.80327 -2.12457 0.51804 -3.04949 -0.27474 0.91443 -3.78942 -0.82969 0.28021 1.12584 -4.41836 -1.27365 -0.16375 0.57618 1.26326 -4.97183 -1.65102 -0.51892 0.19141 0.76856 1.36220 -5.46995 -1.98310 -0.82081 -0.11640 0.42227 0.90708 1.43805 -5.92538 -2.28197 -1.08648 -0.37804 0.14524 0.58849 1.01327 1.49873

From

#r:~ -

(3.6), we have for l_2,

k # ..... =

(r -

1)!(s -

r -

1 ) ! ( n - s)!

x[F(x)]r-af(x)h(x)

dx

, (3.21)

where

I 2 ( x ) = f y I F ( y ) - F(x)] s-r 1 F ( y ) [ 1 - F ( y ) ] n - ' + l d y .

N. Balakrishnan and R. Aggarwala

98

Table 4.2 Variances and covariances, ~rl,/:n, for generalized logistic distribution i

j

n

k=O.1

k=0.2

k-0.3

k=0.4

1 1 1 2 1 1 1 2 2 3 1 1 1 1 2 2 2 3 3 4 1 1 1 1 1 2 2 2 2 3 3 3 4 4 5 1 1 1 1 1 i 2 2 2 2 2 3 3 3

l 1 2 2 l 2 3 2 3 3 1 2 3 4 2 3 4 3 4 4 1 2 3 4 5 2 3 4 5 3 4 5 4 5 5 1 2 3 4 5 6 2 3 4 5 6 3 4 5

1 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6

3.54009 3.21455 1.03356 1.79852 3.22336 1.04485 0.55362 1.32104 0.73478 1.40937 3.28505 1.06283 0.58503 0.36921 1.19989 0.68084 0.43740 0.93569 0.61883 1.23146 3.35729 1.08387 0.60343 0.39789 0.27466 1.15580 0.65767 0.43910 0.30559 0.80070 0.54420 0.38336 0.77173 0.55482 1.12576 3.42971 1.10535 0.61859 0.41378 0.29890 0.21788 1.13905 0.64835 0.43784 0.31824 0.23298 0.74209 0.50749 0.37197

4.46581 5.10946 1.I4267 1.53681 5.71999 1,34670 0,59750 1.42022 0,66110 1,04662 6.26143 1,49483 0.71809 0.39491 1.44100 0.71372 0.39942 0.87291 0.50235 0.84125 6.74599 1.62022 0.79929 0.47839 0.29245 1.49155 0.75234 0.45590 0.28093 0.83424 0.51443 0.32074 0.66174 0.42064 0.72488 7.18577 1.73146 0.86542 0.53337 0.35537 0.23138 1.55009 0.78834 0.49052 0.32881 0.21499 0.83079 0.52342 0.35375

6.94236 9.75063 1.35715 1.41980 12.10242 1.84310 0.68042 1.60667 0.62167 0.82761 14.18588 2.22352 0.92185 0.44289 1,80771 0.77263 0.37751 0.84371 0.42332 0.60870 16.08437 2.55631 1.10391 0.59832 0.32549 2.00411 0.88499 0.48550 0.26615 0.89382 0.49868 0.27644 0.58475 0.33005 0.49307 17.84484 2.85917 1.26044 0.71330 0.43831 0.25637 2.19232 0.98360 0.56187 0,34729 0.20396 0,95327 0.55122 0.34340

15.81242 26.71650 1.74585 1.41663 36.54763 2.70557 0.82155 1.92143 0.61055 0.69285 45.74617 3.53490 1.24534 0.52306 2.38204 0.86465 0.36902 0.84489 0.36962 0.46340 54.49984 4.30274 1.59998 0.78270 0.38027 2.82015 1.07213 0.53059 0.25965 0.98575 0.49570 0.24510 0.53199 0.26743 0.35184 62.91348 5.03126 1.92369 0.99537 0.56382 0.29761 3.24177 1.26138 0.65859 0.37514 0.19877 1.12247 0.59293 0.34024

Recurrence relations for single and product moments

99

Table 4.2 (Contd.) i

j

n

k=0.1

k-0.2

k=0.3

k=0.4

3 4 4 4 5 5 6 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 5 5 5 6 6 7 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2

6 4 5 6 5 6 6 1 2 3 4 5 6 7 2 3 4 5 6 7 3 4 5 6 7 4 5 6 7 5 6 7 6 7 7 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8

6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

0.27398 0.63533 0.47142 0.35041 0.67975 0.51319 1.05388 3.49933 1.12628 0.63213 0.42579 0.31232 0.23838 0.18024 1.13450 0.64561 0.43823 0.32303 0.24741 0.18754 0.71193 0.48793 0.36195 0.27847 0.21181 0.57303 0.42873 0.33191 0.25367 0.54469 0.42561 0.32766 0.62007 0.48341 1.00085 3.56536 1.14633 0.64455 0.43594 0.32219 0.24980 0.19781 0.15355 1.13611 0.64630 0.43994 0.32649 0.25385 0.20144 0.15662

0.23266 0.60833 0.41604 0.27603 0.54913 0.36970 0.64826 7.58979 1.83248 0.92303 0.57732 0.39642 0.28145 0.19107 1.61016 0.82268 0.51855 0.35780 0.25489 0.17347 0.84047 0.53490 0.37139 0.26574 0.18145 0.58954 0.41276 0.29713 0.20382 0.49130 0.35681 0.24646 0.47837 0.33434 0.59312 7.96462 1.92557 0.97487 0.61517 0.42892 0.31394 0.23245 0.16257 1.66948 0.85547 0.54334 0.38038 0.27919 0.20715 0.14510

0.20279 0.59574 0.37534 0.22349 0.45593 0.27517 0.42058 19,49694 3.14047 1.40176 0.81102 0.52073 0.34417 0.21109 2.37227 1.07430 0.62632 0.40405 0.26792 0.16471 1.01505 0.59741 0.38773 0.25819 0.15922 0.61824 0.40448 0.27089 0.16779 0.45202 0.30526 0.19030 0.37871 0.23865 0.37035 21.06088 3.40504 1.53250 0.89864 0.59017 0.40770 0.28261 0.17923 2.54463 1.15941 0.68426 0.45117 0.31251 0.21706 0.13786

0.18121 0.59674 0.34603 0.18570 0.38858 0.21111 0.28575 71.05382 5.73102 2.22871 1.18728 0.71178 0.43820 0.24404 3.65018 1.44019 0.77291 0.46546 0.28743 0.16043 1.25583 0.68014 0.41193 0.25538 0.14295 0.66112 0.40344 0.25145 0.14132 0.42410 0.26637 0.15059 0.30722 0.17538 0.24190 78.96685 6.40817 2.52060 1.36693 0.84404 0.55007 0.35734 0.20661 4.04755 1.61182 0.87957 0.54517 0.35620 0.23182 0.13422

N. Balakrishnan and R. Aggarwala

100 Table 4.2 (Contd.) i

j

n

k=0.1

k=0.2

k=0.3

k=0.4

3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 6 6 6 7 7 8

3 4 5 6 7 8 4 5 6 7 8 5 6 7 8 6 7 8 7 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

0.69517 0.47691 0.35572 0.27757 0.22086 0.17207 0.53883 0.40451 0.31711 0.25322 0.19783 0.48152 0.37990 0.30485 0.23911 0.48709 0.39377 0.31073 0.57774 0.46072 0.95953

0.85592 0.54789 0.38550 0.28394 0.21122 0.14825 0.58455 0.41392 0.30627 0.22861 0.16089 0.46665 0.34741 0.26056 0.18407 0.41954 0.31687 0.22514 0.42931 0.30803 0.55104

1.07680 0.64042 0.42433 0.29491 0.20534 0.13067 0.64520 0.43013 0.30025 0.20974 0.13381 0.45973 0.32280 0.22650 0.14502 0.36801 0.25993 0.16731 0.32702 0.21239 0.33319

1.38618 0.76211 0.47456 0.31105 0.20291 0.11769 0.72489 0.45401 0.29880 0.19551 0.11366 0.46044 0.30468 0.20018 0.11676 0.32861 0.21720 0.12730 0.25498 0.15065 0.21057

Upon integrating by parts treating dy for integration and the rest of the integrand for differentiation, and splitting the integrals into two by writing F(y) as F(x) + IF(y) -F(x)], we get

I2(x) = - (s - r) fxxY[F(y ) - F(x)]" r 1[1 _ F(y)]" s + l f ( y ) d y -- (s -- r -- 1 ) F ( x ) f y

+(n-s+l)

y[F(y) - F(x)]S-r-211 - F(y)]n-s+lf(y) dy

fyy{F(y)-F(x)]"

rI1-F(y)]"

~f(y)dy

(3.22)

+ (n - s + 1)F(x) fy y[F(y) - F(x)]' ~-1 [1 - F(y)]'-'f(y) dy

B

Upon substituting the expression of/2(x) in (3.22) into (3.21) and simplifying the resulting expression, we obtain -

=

(s -

r)(n

-

s +

1) (#r,,+l:°+,

-

n+l r(n - s + 1) -}-

~1-@--1

(#r4-1,s+l:n+l -- #r÷l,s:n--1)

which, when rewritten, yields the recurrence relation in (3.20).

Q.E.D.

Recurrence relations for single and product moments

101

Table 5.1 Coefficients for observed order statistics in BLUE's for generalized logistic distribution

07 n=5 k=0.1

01 k=0.2

k=0.3

k=0.4

k=0.1

k=0.2

k=0.3

k=0.4

.07486 .23328 .30752 .27178 •11257

.04985 .21701 .31589 .30212 .11513

.02744 .19840 .32719 .34389 .10309

.01054 .17613 .33854 .39800 .07679

-.12093 -.16055 -.07005 .07676 .27477

-.07036 -.17684 -.13730 .00675 .37774

-.03522 -.17932 -.19949 -.08201 .49604

-.01274 -.17095 -.25513 -.18900 .62782

n = 10 k=0.1 .01937 .06451 .10293 .13161 .14887 .15363 .14502 .12218 .08421 .02767

k=0.2 .01168 .05401 .09363 .12611 .14861 .15903 .15544 .13569 .09678 .01901

k=0.3 .00562 .04310 .08338 ,11979 .14829 .16569 ,16885 .15394 .11502 -.00370

k=0.4 .00185 •03270 .07245 .11223 .14701 .17259 .18461 .17743 .14157 -.04245

k=0.1 -.04482 -.07419 -.07547 -.06378 -.04250 -.01412 .01914 .05500 .09083 .14992

k=0.2 -.02052 -.05909 -.07399 -.07564 -.06536 -.04400 -.01215 .03000 .08365 .23711

k=0.3 -.00797 -.04451 -.06831 -.08169 -.08345 -.07240 -.04697 -.00441 .06205 .34766

k=0.4 -.00224 -.03222 -.06083 -.08361 -.09718 -.09845 -.08369 -.04699 .02472 .48049

n = 15 k=0.1 .00884 .02940 .04850 .06530 .07928 .09013 .09760 .10150 .10165 .09789 .09004 .07790 .06124 .03968 .01104

k=0.2 .00508 .02345 .04185 .05908 .07431 .08698 .09664 .10292 .10546 .10387 .09773 .08651 .06948 .04533 .00131

k=0.3 .00228 .01756 .03491 .05236 .06876 .08335 .09549 .10460 .11011 .11139 .10772 .09814 .08118 .05405 -.02191

k-0.4 .00070 .01241 .02820 .04539 .06263 .07900 .09368 .10594 .11500 .12001 .11990 .11322 .09766 .06842 -.06216

k=0.1 -.02566 -.04369 -.04885 -.04901 -.04539 -.03869 -.02947 -.01817 -.00519 .00907 .02422 .03982 .05544 .07074 .10483

k=0.2 -.01041 -.03032 -.04043 -.04628 -.04840 -.04708 -.04252 -.03489 -.02427 -.01074 .00569 .02519 .04829 .07692 .17926

k=0.3 -.00355 -.01992 -.03197 -.04145 -.04812 -.05179 -.05226 -.04932 -.04269 -.03203 -.01679 .00398 .03224 .07324 .28045

k = 0.4 -.00088 -.01266 -.02475 -.03617 -.04611 -.05398 -.05925 -.06139 -.05979 -.05367 -.04194 -.02282 .00723 .05727 .40891

n - 20 k=0.1 .00510 .01677 .02795 .03833 .04770 .05589 .06280 .06837 .07251 .07518

k-0.2 .00284 .01296 .02332 .03348 .04306 .05181 .05957 .06618 .07153 .07549

k=0.3 .00122 .00931 .01866 •02842 .03808 .04734 •05594 .06367 .07036 .07582

k-0.4 .00035 .00628 .01439 .02348 .03299 .04255 .05185 .06064 .06870 .07579

k-0.1 -.01742 -.02976 -.03431 -.03625 -.03624 -.03464 -.03169 -.02758 -.02246 -.01647

k=0.2 -.00652 -.01893 -.02572 -.03049 -.03357 -.03512 -.03524 -.03400 -.03145 -.02764

k-0.3 -.00204 -.01137 -.01847 -.02456 -.02959 -.03350 -.03623 -.03773 -.03793 -.03676

k=0.4 -.00046 -.00660 -.01305 -.01948 -.02555 -.03105 -.03579 -.03964 -.04242 -.04399

N. Balakrishnan and R. Aggarwala

102 Table 5.1 (Contd.)

o~

oT n = 20 k=0.1 .07631 .07586 .07376 .06997 .06442 .05703 .04772 .03636 .02274 .00523

k=0.2

k=0.3

k=0.4

k=0.1

k=0.2

k=0.3

k=0.4

.07794 .07879 .07790 .07513 .07035 .06334 .05384 .04143 .02530 -.00425

.07987 .08235 .08305 .08173 .07813 .07187 .06243 .04893 .02952 -.02669

.08168 .08613 .08886 .08954 .08777 .08298 .07435 .06041 .03782 -.06656

-.00973 -.00237 .00551 .01377 .02230 .03097 .03966 .04827 .05695 .08150

-.02261 -.01637 -.00895 -.00032 .00953 .02072 .03349 .04845 .06749 .14725

-.03415 -.03001 -.02421 -.01658 -.00687 .00536 .02086 .04129 .07112 .24137

-.04416 -.04274 -.O3945 -.03399 -.02585 -.01428 .00203 .02587 .06483 .36579

Table 5.2 Variances and covariance of BLUE's for generalized logistic distribution k 5

10

15

20

Var(oT)

Var(0~)

Cov(0;,o;_)

0.63269 0.64600 0.66815 0.69936 0.30861 0.31196 0.31714 0.32380 0.20393 0.20553 0.20790 0.21082 0.15226 0.15323 0.15464 0.15631

0.17883 0.20481 0.25010 0.31783 0.07901 0.08603 0.09879 0.11845 0.05063 0.05413 0.06083 0.07163 0.03724 0.03941 0.04377 0.05104

-0.08046 -0.16286 -0.24942 -0.34292 -0.03739 -0.07447 -0.11123 -0.14805 -0.02426 -0.04813 -0.07144 -0.09434 -0.01794 -0.03553 -0.05258 -0.06919

~

0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

o~ -

o~

COROLLARY 3.4. S e t t i n g r = 1 a n d s = n i n (3.20), w e o b t a i n t h e r e l a t i o n

1 /21,n+l:n+l = ]/'l,n:n+l - - n~-'-i-(122'n+l:n+l -- #2,n:n+l) (3.23) n+l

-~

n--1

,(]AI:n --k[Al,n:n],/

l~l )> 3

m

REMARK 3.1. L e t t i n g t h e s h a p e p a r a m e t e r k - + 0 i n T h e o r e m s 3 . 1 - 3 . 6 , w e d e d u c e the recurrence relations for the product moments of order statistics from the l o g i s t i c d i s t r i b u t i o n e s t a b l i s h e d b y S h a h (1966); see a l s o B a l a k r i s h n a n (1992).

Recurrence relationsfor single and product moments

103

Table 5.3 Coefficients for observed order statistics in BLUE's based on right-censored samples for generalized logistic distribution (sample size n = 20, r = number observed, k = 0.1) ,

,

r

0t

02

5

-0.19604 -0.27905 -0.26271 -0.21836 1.95617 -0.02280 -0.02639 -0.01702 -0.00432 -0.01014 0.02553 0.04119 0.05667 0.07155 0.86546 0.00145 0.01092 0.02162 0.03207 0.04188 0.05081 0.05871 0.06543 0.07091 0.07499 0.07764 0.07874 0.07820 0.07592 0.26073

-0.13848 -0.21007 -0.21422 -0.19832 0.76109 -0.05057 -0.08158 -0.08893 -0.08877 -0.08329 -0.07368 -0.06079 -0.04525 -0.02763 0.60050 -0.02744 -0.04591 -0.05190 -0.05379 -0.05266 -0.04913 -0.04357 -0.03635 -0.02765 -0.01779 -0.00692 0.00472 0.01691 0.02943 0.36204

10

15

Var(<)

Var(0_;)

Cov(< ,0_;)

0.86691

0.27064

0.36583

0.20775

0.10373

0.03760

0.15567

0.05780

-0.01048

0~

~

0~

REMARK 3.2. The relations established in Theorems 3.1-3.6 are complete in the sense that they will enable one to compute the product moments of order statistics for all sample sizes in a simple recursive manner. This may be done for any choice o f the shape parameter k that is of interest. The recursive computational algorithm is explained in the next section in detail.

4. Recursive computational algorithm Starting with the values of #H = E(X) a n d ,~L:l (2) = E(X2), relations (2.2) and (2.4) . (2) can be used to determine /~r:n and #~:n for r = 1 , 2 , . . . ,n and for n = 2, 3 , 4 , . . . . From these values, variances o f all order statistics can be readily computed.

104

N. Balakrishnan and R. Aggarwala

By starting with the fact that/.1,2:2 = /121:1(see Arnold and Balakrishnan, 1989), and /*2,3:3 can be determined from relations (3.2) and (3.11), respectively; /.l~3:3 can then be determined from (3.19). For the sample of size 4,/.1,2:4 and/*2,3:4 can be determined from (3.2),/*3,4:4 from (3.11),/.1~3:4 from (3.5),/*2,4:4 from (3.15), and finally/.1~4:4 from (3.23). This process may be followed similarly to determine /*r,s:n for 1 _< r < s _< n and for n = 5, 6,.... From these values, one can readily compute all the covariances of order statistics. Table 4.1 gives the means of order statistics for selected values ofk = 0.1 (0.1)0.4 up to sample size n = 8. Table 4.2 gives the values of variances and covariances for the same choices of n and k. Only positive values for k are considered, since E(X,-:n;k) = -E(X~_/+l:n; - k ) and Cov(X/:n,Xj:~;k) = Cov(Xn_j+l:n,Xn_i+l:n;- k ) , 1 < i _<j _< n, where Cov(X/:.,X/:.) = Var(X/:.). 121,2:3

5. Best linear unbiased estimators

With the use of the recursive algorithms to compute means, variances and covariances of order statistics from the generalized logistic distribution, we can develop best linear unbiased estimators of location (01) and scale (02) parameters for a given value of k when available samples are either complete or censored either from the left, the right or both, given prior knowledge of which order statistics were observed. Using the results from Balakrishnan and Cohen (1991) and David (1981), we have

07 = - . ' r r ,

(5.1)

= I'FY

where Y is the vector of observed order statistics,/, is the corresponding vector of means of order statistics for the standard (location 0, scale 1) distribution, Z is the corresponding variance-covariance matrix for the standard distribution, 1 is a vector of ones of appropriate dimension, and finally, F = E-l(1/~' A- PI')E-1

and

A = (l'Z-11)(/{E-1/*) - (l'E

1,/1)2

From this, one also obtains var(0~) = 02/*'£-1/*A '

var(0~) = 02

1'2-11 A (5.2)

cov(0 , 0;) =

a2 /~'I2-11

S

Table 5.1 gives coefficients for the observed order statistics obtained using (5.1) for the BLUE's of the location and scale parameters for selected values of k when the entire sample of size n is observed. Table 5.2 gives corresponding variances and covariance of the estimators obtained using (5.2). A short table of coefficients, variances and covariance of BLUE's is also given for k = 0.1 and n = 20

Recurrencerelationsfor singleandproductmoments

105

for right censored samples, where the number observed is r = 5, 10, and 15 (Table 5.3).

6. Maximum likelihood estimation We consider in this section two- and three-parameter maximum likelihood estimation. For the two-parameter case, we assume that the shape parameter k is a known quantity, and we estimate the location (0a) and scale (02) parameters using the maximum likelihood method. For the three-parameter case, the shape parameter k, as well as the scale and location parameters are assumed to be unknown. In both cases, for a right censored sample of size r from n independent random variables from the generalized logistic distribution with location parameter 01, scale parameter 02, and shape parameter k, the likelihood function to be maximized is given by Lx~:°,x2:,,...,xr:,(xl ,x2,...

{

,xr)

[,-

l) r

ri=, E'-

,}2

O(

0~{1 + [ 1 - k

{xr-O''~]¼\~ rrV t 02)jj 1i=1{

(6.l)

and its logarithm is given by in L = constant - r In 02 +

( ~ - l ) ~ln[1-k(Xi-01"~] i=1

t

o2

)J

-2£1n/=1 1+ 1- t , ~ J J J (6.2)

- ( n - r ) l n { l + f 1-k(xr-Ol')]\ 02 j[l} . It should be noted here that maximization of L(.) is subject to the constraint when k > 0 and Xl _> 01 + 02/k when k < 0. Notice that in both cases, at the boundary xr (or xi) = 01 + 02/k, the likelihood function takes the value 0. Thus, the maximum likelihood estimates must be subject to the strict inequalities xr < 01 + 02/k when k > 0 and Xl > 01 + 02/k when k < 0. In the two-parameter case, upon differentiation of in L with respect to 01 and 02, the maximum likelihood estimates for 01 and 02 are obtained by simultaneously solving the following two equations:

xr <_Ol + 02/k

106

N. Balakrishnan and R. Aggarwala

r

Oln L 001

,=, ,+ [,_<~)], (6.3)

k

(n--r) 1 [1 -k(~)]~ 1(~22)~---0

1+[,-<~)] ~ and Oln L 002

o~ ~

-' ~1~1

i=,

~(~)

1 ÷ [l_k(~)] l[1-k(~)]~-l[k(~)]

-

(n -

¼

t-(T) 1~.@) =0 . (6.4)

r) k

Upon simplification, these equations become

(~_)

k

2~ I1 - k(~/15 I (6.5)

-~

(n -- r)

, =0

and

(6.6) +

kfx_o,~

( n - r)

, = o.

Recurrencerelationsfor singleandproductmoments

107

These two equations must be solved numerically for 01 and 02. In the three-parameter case, in addition to solving these two equations, we must simultaneously solve the following equation:

£[

01nL

1 ln l k2 i=l

°q~

k(Xi-Ol~] \

.,-0,

- I £

02 /J

02_2

i=1 1 - k ( ~ )

X'o20'

-- (~y) ln[ ] -k(xr-Ol~ ~ 02 /] -- ( - ~ ) , I

[1 - k(~)]k 02 JJ 1 {-kl-~ln[ 1-k(xi-Ol~]\ '=' 1+ [1 -k(~)] ~ r

-2~-~

k [1 - k ( @ ) ]

1 ÷ [1 -k(~2<)]¼

x {-~--~ln[1-k(Xr-Ol'~]

1

(~2<)

}

for 01, 02, and k. Using (6.5) and (6.6), this likelihood equation simplifies to:

"[

,=1

+2£

-

1-

k(Xi-Ol~ \

02 }

1

-(n-r)ln

[1-k(Xr-Ol~] \

02 JJ

[1-k(~-°']]~ \ o= )J , l n [ 1 - k ( X i - 0 1 ) ] ,=,l+[l_k(~)]~ k, 02 JJ

(n -

r)[1

- k(~)]¼

(6.8)

ln[1-k(xr-Ol'~]

Thus, in the three-parameter case, one must solve equations (6.5), (6.6), and (6.8) simultaneously for 01, 02 and k. REMARK 6.1. Notice that, as k ~ 0, the left hand side of (6.8) approaches 0. Thus, k = 0 is always a solution to the three equations (6.5), (6.6) and (6.8). However, for any particular sample, the probability is 0 that the maximum likelihood estimate of k is 0. One should be wary of this when using computer algorithms to solve (6.5), (6.6) and (6.8) simultaneously.

N. Balakrishnan and R. Aggarwala

108

7. Numerical example A sample of size n = 20 was generated using A P L with k = 0.1 , 01 = 50 and 02 = 10, using the probability t r a n s f o r m method, that is, setting

F(x; 01; 0 2 )

1 =

, =

U

,

a uniform (0, 1) r a n d o m variable. The ordered sample obtained was: 11.98487, 21.04888, 26.85949, 30.21810, 33.59806, 35.34005, 40.67467, 45.92994, 47.88610, 48.10389, 49.12268, 54.51988, 55.83412, 59.00270, 59.64955, 62.58630, 63.72846, 63.88126, 69.40010, 72.72817 . A calculation of the correlation between the ordered sample values Yi:n and the quantiles F -1 [i/(20 + 1)], i = 1 , 2 , . . . , 20, was made assuming k = 0.1 and found to be 0.99255. Similar calculations assuming k = 0.05, k = 0.15, k = 0.20 and k = 0.25 were obtained, giving the results 0.98924, 0.99363, 0.99255, and 0.98940 respectively, indicating that perhaps any value of k between 0.1 and 0.2 describes the data well. Figs. 7.1 and 7.2 show quantile plots for k = 0.1 and k = 0.2. Using 5000 samples each time, p-values for the significance of correlations between Y/:n and F-l[i/(20+ 1)1 were calculated to test the values of k = 0.1, k = 0.2, k = 0.3 and k = 0.4 and were found to be 0.9648, 0.9686, 0.7638 and 0.4894, respectively. N o n e of these were significant, which was to be expected, considering the nature of the test. However, the p-values for k = 0.1 and k = 0.2 were much higher than the others, confirming the region of best choice for k. The B L U E ' s obtained using the full sample were as follows: (SE = standard error, obtained using Table 5.2) Using k = 0.1, 0 T = 49.54979,

SE(0~) = 3.72811,

0~ = 9.55430,

SE(0~) = 1.84375

SE(0~) = 3.78815,

0~ = 9.67733,

SE(0~) = 1.92114

and using k -- 0.2, 0~ = 50.33276,

In both cases, the estimates of 01 and 02 obtained are very close to the true values. Next, the B L U E ' s were calculated for right-censored samples using the coefficients given in Table 5.3, with k = 0.1. The results were as follows: (r = n u m b e r observed) for r = 5: for r = 10:

0~ 0~ 0~ 02

= = = =

43.84540, 7.74302, 49.16279, 10.21530,

SE(0~) SE(0~) SE(0T) SE(0~)

= = = =

7.20936, 4.02812 4.65610, 3.29010

109

Recurrence relations for single and product moments (2=2 OBSERVATIONS) 80.000 75.789 71.579 67.368 63.158 58.947 54.737 50.526

-2

46.316 42.105 Y 37.895 33.684 29.474 25.263 21.053 16.842 12.632 8.4211 4.2105 -0.17764E-14

-4.000

-2.000

0.000

2.000

4.000

F-1[i/(n+l)]

Fig. 7.1Quantile plot ofgenerated values, ~ ffomgeneralizedlogistic distribution with quantiles from generalized logistic distribution(k=0.1).

for r = 15:

0~ = 49.90012, 0~ = 10.75206,

SE(0~) = 4.24223, SE(0~) = 2.58488

for r = 20:

0~ = 49.54979, 0~ = 9.55430,

SE(0~) = 3.72811, SE(0~) = 1.84375

As expected, for small values o f r, the standard errors increase drastically. The r = 20 line is, o f course, the full sample, re-displayed for comparison. Again, the estimates are all close to the true values o f 01 and 02.

110

N. Balakrishnan and R. Aggarwala

(2=2 OBSERVATIONS) 80.000 75.789 71.579 67.368 63.158

.2

58.947 54.737 50.526

-2

46.316 42.105 Y 37.895 33.684 29.474 25.263 21.053 16.842 12.632 8.4211 4.2105 -0.17764E-14

-6.000

-3.000

0.000

3.000

6.000

F-1[i/(n+l)]

Fig. 7.2 Quantile plotofgenerated values, ~ from generalizedlogistic distribution with quantiles from generalizedlogistic distribution(k=0.2).

U s i n g M a p l e V Release 3, m a x i m u m l i k e l i h o o d estimates were also o b t a i n e d for the full s a m p l e as follows: U s i n g k = 0.1,

01 = 49.44142

and

0 2 - - 9.30316 .

U s i n g k = 0.2,

01 = 50.18144

and

02 =- 9.28734 .

A t w o - p a r a m e t e r m a x i m u m l i k e l i h o o d analysis was c o n d u c t e d for k = 0.1, a n d r = 5(5)20. The results are as follows:

Recurrence relations for single and product moments

for r = 5:

01 = 41.02149

02 = 6.38566

for r = 10: for r = 15: for r = 20:

01 = 48.25653 O1 = 49.58711 01 = 49.44142

02 = 9.41583 02 = 10.21531 02 = 9.30316

111

In this case, we had to ensure that these were the best possibl e estimates in the permissible region with respect to maximization of the likelihood function. In fact, they turned out to be the only permissible estimates. One may also obtain approximate standard errors of these estimates. However, due to the non-regularity of the two-parameter distribution, this should be done via a simulation study, rather than by the method of approximation of the information matrix. Thus, 1000 random samples of size 20 from the generalized logistic distribution with shape parameter k = 0.1 were generated, and the biases, variances, and covariance of the m a x i m u m likelihood estimates were obtained as follows: bias(O~) o2 5 10 15 20

-0.383334 -0.093311 -0.037775 -0.020462

bias(b2) o2 -0.174934 -0.066467 -0.040353 -0.021671

Var(b,) o~

Var(02) o~

o2

0.634252 0.193798 0.148690 0.144961

0.185203 0.088359 0.054835 0.035865

0.218841 0.028468 -0.008588 -0.015321

F o r comparison purposes, 1000 random samples of size 20 from the generalized logistic distribution with shape parameter k = 0.2 were generated, and the biases, variances, and covariance of the m a x i m u m likelihood estimates for the full sample were obtained as follows: r

20

bias(O~)

02

-0.024896

bias(02)

02

-0.034524

Var(O~)

~

0.159742

Var(02)

0~

0.034619

Cov(O~,O2)

0~

-0.035709

In obtaining these values, we ensured that each estimate from the simulation was a permissible one. In correcting for the bias of the m a x i m u m likelihood estimates, since these are only simulated and not exact values, we must first determine whether or not the calculated bias is significantly different from zero. Since the estimated bias is just an average of 1000 m a x i m u m likelihood estimates, a simple z-test will suffice. Thus, to test H0: (estimated bias of 0i)/02 = 0 for i -- 1,2, the following p-values are obtained: For k = 0.1: r: i = 1: i = 2:

5 0.000 0.000

10 0.000 0.000

15 0.002 0.000

20 0.089 0.000

112

N. Balakrishnan and R. Aggarwala

and for k = 0.2 (r = 20 only): i = 1: i = 2:

0.049 0.000

All of these values are at least somewhat significant, even if dependence between the estimates of 01 and 02 is considered, and Type-I errors are summed using a Bonferroni argument. As such, we have corrected for the bias of all m a x i m u m likelihood estimates of the original simulated sample. The resulting estimates, corrected for bias, and their variances are given by the following: For k = 0.1:

5 10 15 20

43.9883 49.1977 49.9892 49.6360

7.2504 4.5149 4.0965 3.6127

7.7396 10.0862 10.6449 9.5092

3.8227

9.6194

4.0369 3.2116 2.5975 1.8408

and for k = 0,2: r

01*)

20

50.4209

1.8538

All of these values compare very well with the best linear unbiased estimates presented earlier. Using these unbiased estimates of 02, we may also now obtain standard errors and an estimate of the covariance of the original (not corrected for bias) maxim u m likelihood estimates: For k = 0.1:

and for

r

01

SE(01)

02

5 10 15 20

41.02149 48.25653 49.58711 49.44142

6.1638 4.4402 4.1047 3.6205

6.38566 9.41583 10.21531 9.30316

SE(02) Coy(01,02) 3.3308 2.9981 2.4927 1.8009

13.1089 2.8961 -0.9731 -1.3854

k = 0.2:

F

01

SE(01)

02

sg(02)

Cov(01,02)

20

50.18144

3.8447

9.28734

1.7898

-3.3043

Recurrence relations for single and product moments

113

Incidentally, the values which are obtained when using the method of approximating the information matrix, that is, substituting the original m a x i m u m likelihood estimates (not corrected for bias) into the matrix of negatives of second derivatives of the log-likelihood and inverting, agree quite well with these variances and covariance for larger values of r, and are given by the following: For k = 0.1:

r 5 10 15 20

V/,~(l,1) v/Z(2,2) 5.04440 4.14645 4.03852 3.71267

2.90077 2.77845 2.30928 1.7619t

I(1,2) 10.04024 2.07025 -1.29874 -1.80124

and for k = 0.2: r 20

@I(1,1)I~(2,2~ 3.70088

1.80608

-[(1,2) --3.14516

Next, we consider a three-parameter analysis. If the shape parameter k is assumed to be unknown, best linear unbiased estimates for the three parameters can not be obtained. However, m a x i m u m likelihood estimation is still possible. The m a x i m u m likelihood estimates for r = 10, 15 and 20 were as follows: for r = 10: for r = 15: for r = 20:

O1 =49.06638 01 = 48.82512 01 = 50.51735

02 = 14.03007 02 = 11.13237 02 = 9.40326

k = -0.23045 k = -0.05129 k = 0.24861

In light of the close agreement between simulated results and the approximated information matrix for the two-parameter case, the approximate information matrix for this three-parameter case was also obtained, with the following results (3 rd parameter is k):

r

/r(1,1)

I(2,2)

I(3,3)

I(1,2)

/r(l,3)

-1(2,3)

10 15 20

40.08580 21.10314 15.60632

66.89431 10.44796 3.81140

0.13555 0.07841 0.03745

26.90281 -1.45586 -3.01869

-0.69739 0.37938 0.25180

-2.74171 -0.62508 0.12295

A simulation study was also conducted for the full sample case. 500 samples of size 20 from the generalized logistic distribution with shape parameter k = 0.1 were simulated, and t h e following values were obtained:

114

N. Balakrishnan and R. Aggarwala

bias(O,)

bias(b2)

02

02

bias(/~)

Var(O, ) 0.169703

-0.032107

-0.031482

-0.005367

Cov(0~,~2)

Cov (0~,~)

Cov (&~)

-0.015203

0.011412

0.004379

0~

~2

o~

Var(02)

Var/i~j

o~

0.037129

0.023955

02

For comparison purposes, 500 samples of size 20 from the generalized logistic distribution with shape parameter k = 0.2 were simulated, and the following values for the full sample case were obtained: bias(Ol)

bias(02)

02

02

bias(/~)

Var(b~) 0.146539

-0.026821

-0.050732

-0.025533

Coy(O, ,62) 0~

Coy (O~,~) T

(&~) Cov 02

-0.023482

0.006827

0.012748

0~

Var(b2)

Var(/})

0~

0.043898

0.022385

Again, in this case, in order to correct for bias, we first tested whether the bias was significantly different from zero. The p-values obtained for the test H0: bias (parameter/) = 0, i = 1,2, 3, where parameter 1 = 01, parameter2 = 02, parameter3 = k, were as follows: For k = 0.1: i: p-val:

1 0.081

2 0.000

3 0.438

1 0.117

2 0.000

3 0.000

and for k = 0.2: i: p-val:

We have chosen here to correct estimates of 01 and 02 for bias in the simulation with k = 0.1, and to correct estimates of 02 and k for bias in the simulation with k = 0.2. Thus, the unbiased estimates and their standard errors are given in the following table ( k s i m = value of k used for corresponding simulation):

sim 0.1 0.2

0 )SEE ) l 50.8291 50.5174

3.9882 3.7920

9.7089 9.9058

1.9316 2.1864

0.2486 0.2741

0.1548 0.1496

Finally, we are able to approximate the variances and covariances of the original maximum likelihood estimators using the unbiased estimate for 02:

Recurrence relations for single and product moments

ksim V~ar(01) V~ar(02) Var(l~) 0.1 0.2

15.9967 14.3791

3.4999 4.3075

I 15

Cov(~)l,02)

Cov(~)l,k)

Cov(02,/¢)

-1.4331 -2.3042

0.1108 0.0676

0.0425 0,1263

0.0240 0.0224

For the most part, these agree well with the values given above for the approximate information matrix (r = 20). Although best linear unbiased estimates can not be obtained for the threeparameter analysis, the method of moment estimation, where expected and observed sample mean, variance and skewness are equated, is possible in this case when the full sample is observed. The estimates obtained using the method of moments for the full sample were as follows: for r = 20:

01 = 48.38003

02 = 9.16606

f~ = 0.05126 .

A simulation was conducted, and 1000 samples of size 20 with shape parameter k = 0.1 were generated. Again, for comparison purposes, 1000 samples of size 20 with shape parameter k = 0.2 were also generated. The approximate biases, variances and covariances of the moment estimators were obtained as follows:

ksim

bias(0~) 02

0.1 0.2

-0.097042 -0.188451

ksim 0.1 0.2

COY(01,02) 0~ -0.017226 -0.052578

bias(b2) 02

a;asri~

Var(0,)

-0.023411 0.045116

-0.056100 -0.113374

0.148594 0.190247

Cov(01,k) 02

Cov(02,k) 02

0.001375 0.000397

0.004774 0.009752

0~

Var(02)

0~

0.042713 0.079860

Var(/~) 0.004154 0.003862

Once again, in order to correct for bias, we first tested whether the biases were significantly different from zero. The p-values obtained for the test H0: bias ( p a r a m e t e r i ) = 0 , i - l , 2 , 3 , where p a r a m e t e r l = 0 1 , p a r a m e t e r 2 = 0 2 , parameter 3 = k, were as follows: For k = 0.1: i:

1

2

3

p-val: 0.000 0.000 0.000 and for k = 0.2: i:

1

2

3

p-val: 0.000 0.000 0.000 All of these values are highly significant, and as such, all three parameters may be corrected for bias:

116

N. Balakrishnan and R. Aggarwala

0.1 0.2

49.2908 50.0328

3.5813 3.6572

9.38579 8.77038

1.9863 2.3715

0.1074 0.1646

0.0645 0.0621

For the most part, these estimates are comparable to those obtained using the maximum likelihood method. Therefore, due to the comparable ease of computation for the method of moments, one may prefer to use this method for obtaining estimates of parameters in the case when all three parameters of the generalized logistic distribution are unknown.

II. D o u b l y truncated generalized logistic distribution 8. Introduction

Let X1, X2,...,Am be a random sample of size n from the doubly truncated generalized logistic population with cumulative distribution function 1

(e

kx)l/e],

(1

Q)[1 +

/

Qj < x < P ,

<¼

1

F(x) = 1 (P-Q)[l+e

when k > O

J

x],

lc < Q1 < x < P1

when k < O, (8.1)

QI < x < P l

when k = O ,

Q1 < x %P1 <¼

when k > O

and with probability density function

( P - Q){ [1 + (1

k~c)llkl2}'

-

1

f(x) = I

< Q1 < x < Pj

when k < O, (8.2)

e-X

(P - Q)[(1 + e-X)2] '

QI < x < PI

when k = 0 ,

where 1 - P is the proportion of right truncation and Q is the proportion of left truncation of the generalized logistic distribution with cumulative distribution function as given by (1.1): 1 1+

(1

G(x)= 1 - 1 + e -x'

kx?l/k,

--oc<x<~

when k > O

1 ~<x
when k < O ,

-oo<x
when k = O

]

.

117

Recurrence relations for single and product moments

Thus, P = G(PI) and Q = G(QI). Let Xl:n < X2:n _<.-- _<,l;n:, denote the order statistics obtained by arranging the n ~.'s in increasing order of magnitude. Let us use #~f) to denote the single moments E(X/:,) for 1 < r < n and i > 1, and #~,s:, to denote the product moments E(X~:,Xs:n) for 1 _< r < s _< n. Let us further denote Var(Xr:n) by a~.... and Cov(Xr:,,Xs:,) by a ...... For simplicity, we shall also use #~:, for #~,I. In Section II of this paper, we establish several recurrence relations satisfied by the single moments #~i)~and the product moments #r,s:, from the doubly truncated generalized logistic distribution. These relations will enable one to compute all the single and product moments of order statistics for all sample sizes in a simple recursive manner. If we let the shape parameter k -+ 0, the recurrence relations reduce to the corresponding results for the doubly truncated logistic distribution established by Balakrishnan and Kocherlakota (1986). By starting with the values F~(X) :/21:1, E(X 2) : ~(2) 1:1 and /~1,2:2=/~2:1, one can determine the means, variances and covariances of all order statistics for all sample sizes through this recursive computational procedure. Work of this nature has been carried out by Joshi (1978, 1979, 1982) for truncated exponential distributions and by Balakrishnan and Malik (1987) and Ali and Khan (1987) for truncated log-logistic distributions. From (8.1) and (8.2), we observe that the characterizing differential equation for the doubly truncated generalized logistic population is (1 - l~c)f(x) = g ( x ) - (P - Q)F2(x)

(8.3)

= (1 - P + Q ) F ( x ) + (P - Q ) F ( x ) [ I - F(x)] .

(8.4)

As Balakrishnan and Kocherlakota (1986) exploited these equations for the standard doubly truncated logistic distribution (case when k = 0) in order to derive several recurrence relations for the single and the product moments of order statistics, we shall use (8.3) and (8.4) in the following sections to establish similar results for the doubly truncated generalized logistic distribution in (8.1). These generalized recurrence relations are then shown to be complete in the sense that they will enable one to compute all the single and product moments of order statistics for all sample sizes in a simple recursive manner. 9. Recurrence relations for single moments

The density function of Xr:n (1 < r < n) is given by [David (1981, p. 9), Arnold, Balakrishnan and Nagaraja (1992, p. 10)] n!

= ( r - 1)!(n- r)!

IF(x)] r-l[1 - F(x)]"-~f(x),

Q1 <_ x <_ t:)1.

(9.1) In this section, we establish some recurrence relations for the isingle moments of order statistics from the doubly truncated generalized logistic distribution.

N. Balakrishnan and R. Aggarwala

118

THEOREM 9.1. F o r 1 < r < n - 1, n > 2 a n d i = 0, 1 , 2 , . . . , #(i4.1)

H(/4,1) n+l [ i+1 (#(,) _k#!{+l) ) r4-1:n4,1 ~--- t~r:n4-1 -]- ~ Q Y(YlT ;--~ 1) \ r:n

(9.2) + .1 - P .+ Q {. n-r+l

. (/+1) 'ar+l:n'}J (g+l)~] ~G:n --

and for n > 1 and i = 0, 1,2,..., /~(/+1) n+l:n+l

.(,+1) n+l /~n:n+l -]- ~

(#(i)_k/~(ni:+l)) \ n:n

[i~1 - -

(9.3) +(1-P+Q)(tt!i:+l)-p{+l)]

.

PROOF. For 1 < r < n - 1 and i = 0, 1 , 2 , . . . , let us consider from (9.1) and (8.4) (1 - e -~- Q)

/l~i! -- k~t!in+l) = (r - l ) [ ( n - y)[

xi[f(x)]r[l - F(x)] n-r dx I

Integrating now by parts treating x g for integration and the rest of the integrands for differentiation, we obtain r(1 - P + Q) r (g+l) r'r:n ,,(,/_

i+

[~r+ 1:n -

1

(i+1)] J

(9.4) r(P.

Q)(n - r + 1) [. (i+1)

_/z(i+,)]

The relation in (9.2) follows simply by rewriting (9.4). Relation (9.3) is obtained by setting r = n in the above proof and simplifying. Q.E.D. THEOREM 9.2.

For n >_ 2 and i = 0, 1 , 2 , . . . ,

#(i÷1) (1 . . . . (i+l) - - - , (g+l) - r" + ~d)#1.,,-1 - (1 - 2P + 2~)#l:n l:n+l =

i + 1 {. (i) n

\#1:~-k~'i:.

(9.5)

. (i+1)~ . ) ,

and for i = 0, 1 , 2 , . . . , #(g+l)l:2 = (1

-- P + Q)p(g+l)

--(i+

_

,..,.-.~\ (,+1)

(1 - 2P + z~d)#H

1 ) ( , u l i { - k • (i+l)'~ 'Ul:l

) '

(9.6)

Recurrence relationsfor single and product moments

119

PROOF. For n _> 1 and i = 0, 1,2,..., let us consider from (9.1) and (8.4)

l:n--g#l:n /~(i/ , (i+1) : r / < { ( l - P - I - Q )

fo P' xi[f(x)][1-F(x)] n lax i

Integrating by parts treating x i for integration and the rest of the integrands for differentiation, and then writing F(x) as 1 - [1 - F(x)], we obtain k~(i)

.(i+1)_

l:n -- kt~l:n

n

i; 1

{(l_p+Q)[,(i+l

)

.(i+1)1

Lt~l:n-I -/Xl:n

]

The relation in (9.5) follows simply by rewriting (9.7). Relation (9.6) is obtained by setting n = 1 in the above proof and simplifying. Q.E.D. REMARK 9.1. Letting the shape parameter k ~ 0 in (9.2)-(9.5), we deduce the recurrence relations established by Balakrishnan and Kocherlakota (1986) for the single moments of order statistics from the doubly truncated logistic distribution. Furthermore, letting P -+ 1 and Q ~ 0, we deduce the recurrence relations for the generalized logistic distribution, established in Section I. REMARK 9.2. The recurrence relations established in (9.2)-(9.5) will enable one to compute all the single moments of all order statistics for all sample sizes in a simple recursive manner. This is explained in detail in Section 11. 10. Recurrence relations for product moments

The joint density function of X/:~ and Xj:n (1 < i < j _< n) is given by [David (1981, p. 10), Arnold, Balakrishnan and Nagaraja (1992, p. 16)] n! fi,j:n(x,y) = (i - 1 ) ! ( j - i - 1 ) ! ( n - j ) ! [F(x)]i-I[F(Y) - F(x)]J-i=l x [1-F(y)]"-Jf(x)f(y),

Q1 <_x
•

(10.1)

In this section, we establish several recurrence relations for the product moments of order statistics, from the doubly truncated generalized logistic distribution. THEOREM 10.1. For 1 _< i < j _< n and j - i _> 2, n+l

]Ai+lff+l:n+l ~- /Ai+2,j+l:n+ 1 -~" ( i + 1 ) ( P - Q) )< I#i,j:n -- ~i+l,j:n ~- ~ (~2j:n- k]2i,j:n)]

(10.2)

N. Balakrishnan and R. Aggarwala

120 and for 1 < i < n - 1, • (2)

].li+l,i+2:n+l

z

# i + 2 : n + l ~-

x

[

n+l (i + 1)(P - Q)

1

, (2) q_ 7 (/~i+lin _ kl~i,i+l:n ) #i,i+l:n -- /*i+l:n

]

(10.3)

PROOF. F o r 1 _< i < j _< n a n d j - i _> 2, let us consider from (10.1) and (8.3)

~+:. - k~i,;:. = E (x;:0 xj:. - kX~:~X;:n) =

n_[ (i- 1)!(j'-i-

[P'y[1-

F(y)]" J f ( y ) I i ( y ) d y

1)!(n-J)[aOt (10.4)

where 11 (y) =

/(

IF(x)] i IF(y) - F(x)] j-i-I dx

1

- (P - Q) fQi[F(x)]i+l [F(y) - F(x)] j-*-I dx

m

Integrating by parts treating dy for integration and the rest of the integrands for differentiation, we obtain I1 (y) z

x(j

-- i -- 1)[f(x)]i[f(y)

-- f(x)lJ-i-2f(x)

dx

1

- (P-

Q)

/j

x(j-i-

1)[F(x)]i+~[F(y) - F ( x ) ] ;

i 2f(x)dx

1

+ (P - O)

£

x(i + 1)[F(x)]~[FCv) - F(x)]J<-~f(x) dx 1

U p o n substituting this expression o f h (y) into (10.4) and simplifying the resulting expression, we obtain

Pj:, - klxi,y:n = i[#i+lj:, - #i,j:,] -~

i ( i + 1 ) ( P - Q) n+l

× I/~i+ld+l:n+l - #i+2,j+l:n+l] • The recurrence relation in (10.2) is derived simply by rewriting the above equation. Relation (10.3) is obtained by setting j = n in the above p r o o f and simplifying. Q.E.D.

,

Recurrence relations for single and product moments THEOREM

10.2. For 1 < i < j _ < n - l a n d

121

j-i>_2,

[

n+l ~-2-)(-P - Q) (1 - 2P + 2Q) (#id_,: n - #ij:,)

#i,j:n+l ~ / ' / i d - l : n + l -~ (n - j

n(1 - P + Q) ;-)71 1 n-j+l

(10.5)

-

(/~,:. - kl~ia:.)] ;

and for 1 < i < n - 2 , n+l ]Ai'n:n+l =]'li'n-I:n+l q 2(P Z Q)[(1 - 2P + 2Q)(#i,n_l: n -- ]Ai,n:n)

-n(1 - P + Q)(#i,,

l:n-1 -- Pl~i:n

1) -

(10.6)

(lli:n -- kl'li,n:n)]

•

PROOF. For 1 < i < j < n - 1 a n d j - i ___2, let us consider from (10.1) and (8.4)

0 _

n!

- (i - 1)!(j - i - 1)!(n - j)!

f0 °' xIF(x)]i-lf(x)Jl(X)

dx ,

1

(10.7) where Jl(x) = (1 - P + Q )

~xx P I IF(y) - F ( x ) ] j i-1[1 - F ( y ) ] n - J F ( y ) d y

+ (P - Q)

IF(y) - F(x)] j:i 111 - F(y)]" J+lF(y) dy .

Writing F(y) as 1 - [1 - F(y)] in the above integrals and then integrating by parts treating dy for integration and the rest of the integrands for differentiation, we obtain an expression for J1 (x) which, when substituted into (10.7) and simplified, gives I~i:, - k#id:, = (1 - P + O){n[pi,j:,_ 1 - #i,j l:n-l] + (n - j + 1)[#i,j_,:. - #i,j:n]} + (P - Q) { (n - j + 1)I#id:, - #i,j ,:~] (n - j +

1)(n-j+2) n + 1

} [l'li,j-l:n+l - #i,j:n+l]

•

The recurrence relation in (10.5) is derived simply by rewriting the above equation. Relation (10.6) is obtained by setting j = n in the above proof and simplifying. Q.E.D.

N. Balakrishnan and R. Aggarwala

122

THEOREM 10.3. For 1 < i < n - 2, n+l [ ( . (2) "~ #i,i+l:n+l = (n - i + I)(P - Q) (1 - P + Q) npi,i+l:._ 1 -- t#i+l:. ) - (1 - 2 P + 2Q)Yi,i+l:n i n-i+l

n - 1 i (#/:.

__

kui,i+l:n) 1

#(2) . i+1:.+1 ,

(10.8)

and for n >_ 2,

I'ln-l.n:n+l -- 2(p÷-l-Q) I(l -- g ÷ Q)(l'lPl#n-l:n-, -- ( n

1]# (2,~/n:nl

-- (1 -- 2P + 2Q)#,,_l,n: n - (#,, l:n -- k#n-, .... )] n - l , (2) -

PROOF.For

t~n:n+l

•

1 < i < n - 2, let us consider from (10.1) and (8.4)

#i:n-k#i,i+l:n

= E(X/:ng/°l:n - k~i:nX/+l:n)

n!

= (i - 1)!(n -- i - 1)!

£P~x{F(x)]i-lf(x)J2(x)

dx,

(10.10)

I

where

J2(x)

= (1 - P + Q) fx PI [1 - F(y)]" i-iF(y) dy + (P - Q)

f"

[1 -

dy.

x

Writing F(y) as 1 - [1 - F(y)] in the above integrals and then integrating by parts treating dy for integration and the rest of the integrands for differentiation, we obtain an expression for J~ (x) which, when substituted into (10.10) and simplified by combining integrands containing ( 1 - P + Q)x 2 and then combining integrands containing (P - Q)x 2, gives 1" . (2) + n#i,i+l:n_ 1 -- (n -- i)#i,i+l:n] #i:n -- k#i,i+l:,, = (1 - P + Q) [-t#i+l:~ I-i #(2) + (P - O)(n - i) kn + 1 i+l:n+l ÷

n-i+l n+ 1

#i,i+l:n

] #i,i+l:n+l

"

Recurrence relations for single and product moments

123

The recurrence relation in (10.8) is derived simply by rewriting the above equation. Relation (10.9) is derived by setting i = n - 1 in the above proof and simplifying. Q.E.D. THEOREM 10.4. For 1 < i < j < n " 1 and j - i > 2, i /zi,jwl:n+l z /zi~j:n+l ~ - ~ / . (/zi+l,j:n+l - /zi+l,jWl:n+l ) F/ -]- 1

(1 -- P ~- Q)

~ ( n - j + 1)(P- Q)

1/ +7--_

+

(/zi+l,j:n -- /zi+ld+l:n)

- k/z/j:,)}

(10.11)

and f o r l < i < n - 2 ,

_ i (/zi+l:n+l t` (2) -- /zi+l,i+2:n+l) /zi,i+2:n+l = /zi,i+l:n+l ~n+l -~ ( l ' l - i ) ( P - a )

{(1 - P +Q)[i(/zl2+)l:n- /zi+l'i+2:n)

+ (/zi,i+l:n-/zi,i+2:n)]-{-(/zi:n-k/zi,i+l:n)}

,

(10.12)

PROOF. Following the proof of Theorem 10.2, writing F(y) as F(x) + [F(y) - F(x)l in Ji (X) and then integrating by parts treating dy for integration and the rest of the integrands for differentiation, we obtain an expression for Jl(x) which, when substituted into (10.7) and simplified, gives, for 1 <_i<j<_n-1 and j - i _ > 2, /zi:n -- k/zi,j:n = (1 - P + Q){i[/zi+l,j+l:n -- /zi+lj:n] ~- ( J - i)[/zi,j+l:n -- /zi,j:n]} q-

(P- Q)(n-j+ n+l

1)

{ i[/zi+l,j+l:n+l - /zi+l,j:n+l ]

+(J -- i) [/zi,j+l:n+l - /zid:n+l] } "

The recurrence relation in (10.11) is derived simply by rewriting the above equation. Relation (10.12) is obtained by setting j = i + 1 in the above proof and simplifying. Q.E.D. THEOREM 10.5. For 1 < i < n - 2, i /zi,n+l:n+l = /zi,n:n+l -~" l'l --~ (/Zi+I .... +1 -- /zi+l,n+l:n+l ) +P~--Qn + 1 L[1- n P_ 7+ Q +

n-i

(i/Zi+I. . . .

-- nPl/zi:n

(/zi:n -- k/zi,n:n

;

1

-}-/zi,n:n)

(10.13)

N. Balakrishnan and R. Aggarwala

124 and for n _< 2,

#.-1,.+1:.+1 = # . - 1 .... +~ + (n

--

1') /~#n:n+l (2)

÷ J (1 r, 1 - P ÷ ÷ ~ n_ Q

- #.#+1:.+1)

Q)[(n - 1]# . ,(2) , : . - - n P l U n - l : n - l ÷ #n-l,n:n]

÷ (#n-am -- k#n-l,n:n)} •

(10.14)

PROOF. For 1 < i < n - 2, let us consider from (10.1) and (8.4) #~:, - k # , .... = E ( X ~ : , X ; ° : , - ~ . : , X ~ : , )

n! /)[ = ( i - 1 ) ! ( n - i - 1)! x[F(x)]i-lf(x)Jg(x) dx ,

(10.15)

1

where J3(x) = (1 - P + Q) ~x PI IF(y) - F(x)] n-i 1F(y) dy + (P - Q)

fx"

IF(y) - F(x)]" i-1 [1 - F(y)]F(y) dy .

Writing F(y) as F(x) ÷ IF(y) - F(x)l in the above integrals and then integrating by parts treating dy for integration and the rest of the integrands for differentiation, we obtain an expression for J3(x) which, when substituted into (10.15) and simplified by combining integrands containing P1, gives #i:, - kgi .... = (1 - P ÷ Q)[nPl#i:n-, - i#i+l .... - (n - i)#/,n:,] P-Q

÷ - n - ~ T {i[#i+l,n+l:n+l - #i+l,n:n+l]

+

i)[#,,,,+,:.+1-

#,,.:.+,] }

The recurrence relation in (10.13) follows readily upon simplifying the above equation. Relation (10.14) is obtained by setting i = n - 1 in the above proof and simplifying. Q.E.D. REMARK 10.1. Letting the shape parameter k -+ 0 in the above results, we deduce the recurrence relations for the product moments of order statistics from the doubly truncated logistic distribution established by Balakrishnan and Kocherlakota (1986). Furthermore, letting P --+ 1 and Q --+ 0, we deduce the recurrence relations for product moments of order statistics from the generalized logistic distribution, which have been established in Section I. REMARK 10.2. The recurrence relations established in this section are complete in the sense that they will enable one to compute all the product moments of all order statistics for all sample sizes in a simple recursive manner. This can be done

Recurrence relations for single and product moments

125

for any choice of the shape parameter k; the recursive computational algorithm is explained in Section 11 in detail.

11. Recursive algorithm Starting with the value of #1:1 = E(X), we can use (9.6) to obtain//1:2 and (9.5) to obtain/21:3,/21:4 . . . . ,/21:n. Then, using (9.3), we can obtain/22:2 and using (9.2), we can obtain/22:3. Again, using (9.3), we can obtain/23:3 and using (9.2), we obtain /22:4 and/23:4. Proceeding in this way, we can obtain all the first-order moments/2r:n ~- (2) = E ( X 2 ) , w e for r = 1 , 2 , . . . , n and n = 2, 3, 4,.... Starting with the value oi/21:a proceed exactly on similar lines to obtain all the second-order moments./2~2) for r = 1,2,... ,n and n = 2, 3,4,.... From these values, variances of all order statistics can be computed. By starting with the fact that/21,2:2 ~---/221:1(see Arnold and Balakrishnan, 1989), /22,3:3,/23,4:4,''" ~/2n l . . . . can be determined from (10.3). Then/21,2:3 can be determined using (10.9), and /21,2:4 /21,2:5,''',/21,2:n from (10.8). /21,3:3 c a n then be determined using (10.14) with n = 2 and/22,4:4,/23,5:5,''',/2n-2,n:n can be found using (10:2). /21,3:4 can then be found using (10.5), and /21,3:5, /21,3:6,--.,/21,3:n can be determined using (10.5). /22,3:4 is then determined using (10.9) and /22,3:5, /22,3:6,''',/22,3:n f r o m (10.8). Next, /2~,4:4 can be determined using (10.13) and following the steps above, we may similarly determine/2r,s:, for 1 _< r < s _< n and n = 5, 6,.... From these values, covariances of order statistics can be readily computed. Thus, by starting just with the values of E(X) and E(X2), we may compute the means, variances and covariances of order statistics for all sample sizes in a simple recursive manner. This may be done for any value of the shape parameter k and the truncation parameters P and Q.

Acknowledgements The authors thank Dr. Gail Wolkowicz for her help in the computation of maximum likelihood estimates, and the Natural Sciences and Engineering Research Council of Canada for funding this research.

References Ali, M. M. and A. H. Khan (1987). On order statistics from the log-logistic distribution. J. Statist. Plann. Infer. 17, 103 108. Arnold, B. C. and N. Balakrishnan (1989). Relations, Bounds and Approximations for Order Statistics. Lecture Notes in Statistics No. 53, Springer-Verlag, New York. Arnold, B. C., N. Balakrishnan and H. N. Nagaraja (1992). A First Course in Order Statistics. John Wiley & Sons, New York.

126

N. Balakrishnan and R. Aggarwala

Balakrishnan, N. (Ed.) (1992). Handbook of the Logistic Distribution. Marcel Dekker, New York. Balakrishnan, N. and A. C. Cohen (1991). Order Statistics and Inference: Estimation Methods. Academic Press, San Diego. Balakrishnan, N. and S. Kocherlakota (1986). On the moments of order statistics from the doubly truncated logistic distribution. J. Statist. Plann. Infer. 13, 117-129. Balakrishnan, N. and H. J. Malik (1987). Moments of order statistics from truncated log-logistic distribution. J. Statist. Plann. Infer. 17, 251~67. Balakrishnan, N. and K. S. Sultan (1998). Recurrence relations and identities for moments of order statistics. In the companion volume. Cohen, A. C. and B. J. Whitten (1988). Parameter Estimation for Reliability and Life Span Models. Marcel Dekker, New York.. David, H. A. (1981). Order Statistics, Second edition. John Wiley & Sons, New York. Hosking, J. R. M. (1986). The theory of probability weighted moments. I B M Research Report PC 12210. Hosking, J. R. M. (1990). L-moments: Analysis and estimation of distributions using linear combinations of order statistics. J. Roy. Statist. Soc. Series B 52, 105-124. Joshi, P. C. (1978). Recurrence relations between moments of order statistics from exponential and truncated exponential distributions. Sankhya, Set. B 39, 362-371. Joshi, P. C. (1979). A note on the moments of order statistics from doubly truncated exponential distribution. Ann. Inst. Statist. Math. 31, 321-324. Joshi, P. C. (1982). A note on the mixed moments of order statistics from exponential and truncated exponential distributions. J. Statist. Plann. Infer. 6, 13-16. Shah, B. K. (1966). On the bivariate moments of order statistics from a logistic distribution. Ann. Math. Statist. 37, 1002-1010. Shah, B. K. (1970). Note on moments of logistic order statistics. Ann. Math. Statist. 41, 2151~152. Zelterman, D. and N. Balakrishnan (1992). Univariate generalized distributions. In Handbook of the Logistic Distribution (Ed., N. Balakrishnan), pp. 209-221. Marcel Dekker, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 17 © 1998 Elsevier Science B.V. All rights reserved

D

Order Statistics from the Type III Generalized Logistic Distribution and Applications

N. Balakrishnan and S. K. Lee

1. Introduction

In the past 150 years, the theory, methodology and applications of the logistic distribution have been studied in detail by numerous authors. The logistic function was first used as a growth function by Verhulst (1838, 1845) in demographic studies and was given its present name by Reed and Berkson (1929). Due to its simplicity and that it has a shape that resembles the normal distribution, the logistic function has been very popular since the end of the nineteenth century. Pearl and Reed (1920, 1924), Pearl, Reed and Kish (1940), Schultz (1930) and more recently Oliver (1982) all applied the logistic model as a growth model in human populations and biological organisms. Some applications of the logistic function in bioassy problems were discussed by Berkson (1944, 1951, 1953), Wilson and Worcester (1943), and Finney (1947, 1952). Other applications and significant developments concerning the logistic distribution can be found in the book by Balakrishnan (1992). In the past 50 years, several different forms of generalizations of the logistic distribution have been proposed in the literature. These distributions are referred to as "the generalized logistic distributions". Balakrishnan and Leung (1988) defined three types of generalized logistic distributions by compounding an extreme-value distribution of the double exponential type, a reduced log-Weibull distribution and an exponential-gamma distribution with a gamma distribution (denoted by Types I, II and III), respectively. It is of interest to mention here that the Type I generalized logistic distribution was first derived by Dubey (1969). The Type II generalized logistic distribution is simply negative of the Type I distribution. The Type III generalized logistic distribution was first proposed by Gumbel (1944). He derived the Type III distribution as the limiting distribution of the m t h midrange ~ of a large sample from an unlimited symmetric continuous distribution with zero mean. The Type III distribution was also studied by Davidson (1980) who established two other characterizations of the distribution as a I The m th midrange of a sample of size n is defined as (Xm:~+X,~--m+l:n)/2. 127

128

N. Balakrishnan and S. K. Lee

difference of two independent and identically distributed r a n d o m variables from some distributions. Prentice (1976) proposed the Type IV generalized logistic distribution, with probability density function (p.d.f.) F(p + q)

e -qy

g(Y)=r(p)r(q)(l+e y)P+q'

-oc
p,q>O

,

(1.1)

as an alternative to the usual logistic model for binomial regression data. It is readily observed that Types I, II and IH are all special cases of the Type IV distribution. The density function in (1.1) corresponds to the density function of Types I, II and III distribution when q = 1, p = 1, and p = q, respectively. Zelterman and Balakrishnan (1992) have reviewed the developments on these four generalized forms of the logistic distribution. In this paper, we first obtain the Type III generalized logistic distribution by compounding an exponential-gamma distribution with a g a m m a distribution as in Balakrishnan and Leung (1988). Then we present a reparametrized model which has a standard normal distribution as the limiting distribution as the shape parameter tends to infinity. This reparametrized model, in addition to being useful as a life-time model, can also serve as meaningful alternative to the normal distribution for robustness studies concerning some classical procedures based on normality. In Section 3, we study the order statistics and moments from this reparametrized distribution. Tables of means, variances and covariances of order statistics are presented for sample size n = 20 and shape parameter = 0.5(0.5)3(1)6(2)12. Following steps similar to those of G u p t a (1960), we derive explicit formulae for the single moments of order statistics for positive integer values of the shape parameter. In Sections 4 and 5, we study the best linear unbiased estimators (BLUEs) and the m a x i m u m likelihood estimators (MLEs) of the location and scale parameters, respectively. The coefficients of the BLUEs, the variance and covariance factors of the BLUEs and simulated values of the bias, variances, covariance and mean square errors of the MLEs for sample size n = 20 and various choices of e are also presented. Finally, we compute the relative efficiency between the BLUEs and the MLEs, make some comparative comments, and present two examples which illustrate both these methods of estimation.

2. Type III generalized logistic distribution Let Y be a random variable with a p.d.f, of an exponential-gamma distribution given by

f(Y I 2)

=

e -'~e Ye-~y,

- o c < y < oc,

c~ > 0,

2 > 0 .

(2.1)

Let us now assume that the parameter 2 in (2.1) has a g a m m a distribution with density function

Order statistics from Type III generalized logistic

9(2)--

e-,t)t~ 1 F(~) '

2>0,

129

c~>0 .

(2.2)

Then, we obtain the density function of the compounding distribution based on (2.1) and (2.2) as f(y) =

/0

=

f ( y [ 2)g(2) d2

e ~Y

[oo22~_~e ~(a+e--v)d2

1

e -~y

=B(c~,c~)(l+e-y) 2~'

-oo
c~>0 ,

(2.3)

where B(., .) is the complete beta function and c~is the shape parameter. Balakrishnan and Leung (1988) referred to this as the Type I I I generalized logistic distribution. Gumbel (1944) derived this Type Ill distribution as the limiting distribution of the mth midrange of a large sample from an unlimited symmetric continuous distribution with zero mean. Davidson (1980) showed that the Type lII distribution can also be expressed as the distribution of the difference of two independent and identically distributed random variables from the generalized extreme value distribution. From the density function in (2.3), it is clear that the distribution is symmetric about zero. When the shape parameter e = 1, (2.3) corresponds to the usual logistic density function, and for positive integer values of e (2.3) is simply the density function of the sample median in a random sample of size 2e - 1 from the usual logistic distribution. The corresponding c.d.f, of the distribution in (2.3) can be obtained as 1

F(y) = B(e, e)

fY e-at dt, oo (1 + e-t) 2~

-oc
c~>0

(2.4)

Making the transformation u = l@e-,, we get F(y)

-

1

[P(Y) u~_I(I_u) ~ l d u

=Ip(y)(C~,~),

- o c < y < oo,

c~ > 0,

(2.5)

where Ip(y)(., .) is the incomplete beta ratio and p(y) - - l + e 1 Y" For integer e, integrating Eq. (2.5) by parts repeatedly, treating u ~-~ for integration and the rest of the integrand for differentiation, we get F(y) = Z

2~

1 {p(y)}r{q(y)}Z~-,-r

e-y where q(y) = 1 - p(y) - l+e Y"

(2.6)

130

N. Balakrishnan and S. K, Lee

Davidson (1980) showed that the moment generating function (m.g.f.) of the Type III generalized logistic distribution is My(t) =

+ t)r(

{F(~)} 2

- t)

,

I tl<e

.

(2.7)

As mentioned by Balakrishnan and Leung (1988), the mean and the coefficient of skewness (V/~l) of the distribution are equal to 0. Variance and the coefficient of kurtosis (fi2) can be obtained from the m.g.f, in (2.7) as follows: Var(Y) = E(Y 2) = Mi~(t)lt= 0 = 2O'(e )

(2.8)

and E ( Y 4) f l 2 ( y ) -- { V a r ( y ) } 2

=

(2.9)

,

where O(z) is the psi (or digamma) function defined by 0(z) = d In F(z)/dz, tk'(z) and O"'(z)are its first and third derivatives, respectively. These two measures have been computed for various values of e and are presented in Table 1. From Table 1, we can see that the Type III generalized logistic distribution is a family of symmetric distributions with coefficient of kurtosis smaller than that of the logistic (when c~ > 1). The value of fi2 decreases with increasing values of c~; and close to value 3 as ~ gets large. This indicates that Table 1 Values of variance and coefficient of kurtosis

0.5 1.0 1.5 2.0 2.5 3.0 4.0 5.0 6.0 8.0 10.0 12.0

Var(Y)

fi2

9.869604 3.289868 1.869604 1.289868 0.980716 0.789868 0.567646 0.442646 0.362646 0.266274 0.210333 0.173804

5.000000 4.200000 3.806250 3.593763 3.465596 3.381282 3.278475 3.218723 3.179874 3.132556 3.104879 3.086739

Order statistics from Type Ill generalized logistic

131

the distribution has thicker tails than the normal distribution for small ~ and close to the normal distribution when ~ is large. Therefore, we will consider a reparametrized model of the Type Ill generalized logistic distribution from now on. Let Y be a random variable from the Type III generalized logistic distribution with density as given in (2.3) with mean 0 and variance 2~'(c~). From Abramowitz and Stegun (1965, pp. 259), we have

1

~

B2n

2nz 2----~

(z) ~-, In z - 2z - ~ = lnz -

1

1

1 - - + 1. . . + - 12z2 120z4 - 252z6

2z

where B2n are the Bernoulli numbers. Taking the limit as the shape parameter of the distribution ~--+cc, it immediately follows that (since ~ ' ( ~ ) ~ I/a) Var(Y) ~ 2/~. Now consider the random variable X = (2/~)-1/2y. Then the p.d.f, and c.d.f. of X are given by

,2/5N~

e(x) = B'~v'--',~)~,(1

e-,/~ + e-X/57-~)2='

- o o < x < 001

~>0

(2.10)

and

ix G(x) = ~ L ~

e V~t (1 + e - V / ~ ' ) 2~ dt

1 rP(xX) ~ 1 - B(~,~J ° u-(1-

u)~ 1 du

= Ip(xx)(~, ~)

(2.11)

where p(.) is as defined before and K = X/~-/e. Further, the characteristic function of X can easily be found as

~o(t) =

{F(~)} 2

From Abramowitz and Stegun (1965, p. 257), we can express the gamma function as i

l[

1

1

F(z) ~ e-ZzZ-~(2~z)~L1 + i~z + 288z ~

139 51840z 3

571 2488320z4 ~

] (2.13)

Upon substituting (2.13) into (2.12) and taking the limit as e ---+ oo, we get

132

N. Balakrishnan and S. K. Lee

" ( ) " 0~'2\/2-:11 lijnooqo(t) = li+m [( 1 + it/V~ ) ~+lt@L/2)l/2 I 1 - - it/v/~ a-,t(/, _ [!im(l+it/(2/o:)l/2~it(~/2)'/2 [lim(1 -- k~ ~ \ l - - ~ J

k~+~

(it)2~-1/2 ] 2~ J

x [ l i m ( 1 - ( i t ) 2 ~ ~] = e (it)a-l.e

002

2

(it)2 e-~-

(2.14)

which is the characteristic function of the standard normal distribution. In Figure 1, we have plotted the density of X in (2.10) against the standard normal density for various choices of ~. It is seen that this family of distributions is very close to the normal distribution and hence could be used in robustness studies of some classical procedures based on normality; they could also serve as useful and meaningful alternatives to the normal distribution while measuring the performance of some goodness-of-fit tests for normality.

3. Order statistics and moments Let X1,X2,... ,Xn be a random sample of size n from the reparametrized distribution mentioned above with p.d.f, f(x) and c.d.f. F(x) as given in (2.10) and (2.11), respectively. Let Xl:n _<X2:, _<... _< Xn:~ be the order statistics obtained from the above random sample. Then the density function of the ith order statistic, X/:n(1 < i _< n), is given by [David (1981, p. 9); Arnold, Balakrishnan and Nagaraja (1992, p. 10)] n!

J}:,(x) -- ( i - 1)[(n- i)[

{F(x)}i-l{1 - F ( x ) } ° - i f ( x ) ,

-oo < x < (3.1)

and the joint density function of the order statistics, X/:, and Xj:, (1 _< i < j _< n), is given by [David (1981, p. 10); Arnold, Balakrishnan and Nagaraja (1992, p. 16)1

fi,j:n(X,y) = (i-

n! 1)! 0" - - i - - 1)!(n

-- j)! {F(x)}i-I {F(Y) -F(x)}J-i 1 x { 1 - F C y ) } n Jf(x)f(y), -oo<x
Order statistics from Type III generalized logistic

133

,,,','(0,1)

4N(0,1) 0,4.

03

0.2

0.1

-2

0

2

-4

-2

0

2

x

x

N(0,1)

o

-2

2

0

2

x

x

0.4-

O.4

0.2

0

4

-2

0

2

2

O

2

x

x

Fig. 1. Density plots of Eq. (2.10) against N(0,1) density.

Cov(X/:n,Xj:n), b y i a i j : , (1 _< i < j < n). Note that 6i,i: n Var(X,.:,) and we will use Pi:, instead of #}.~ for convenience. Upon using the expression o f f ( x ) and F ( x ) in (2.10) and (2.11), respectively, we may compute the single moments of order statistics from (3.1) as =

• (k) =#i:n

5

00

xkfi:n(x) dx,

1 < i < n ,

(3.3)

N. Balakrishnan

134

S. K. Lee

and

..N(0,1)

4,v(o,1) o.4 I O.3

0.2

0,1

-2

o x

2

./N(O.I)

t

0 !4

-2

0 x

2

.N(0J)

0.4 0.3 0.2

O-4 x

x

., N(oj)

014

-2

/N(O,I)

6 ×

x

Fig. 1. (Contd.). and the product moments of order statistics from (3.2) as

#i,j:,

[l

xyf~j:, (x,y)dx dy,

1 < i < j <_ n

(3.4)

oo<x
directly by using numerical integration methods since explicit algebraic expressions cannot be derived from Eqs. (3.3) and (3.4) in general. Following steps

Order statistics from Type III generalized logistic

135

similar to those of Gupta (1960), however, exact explicit formulae may be derived for the single moments of order statistics for positive integer values of cc For positive integer values of c~,upon using the expression ofF(x) in Eq. (2.6), we can write the moment generating function of Xn:n as

Mn:n(t) = E(e tx":") - n x / ~ ° ~ f et x I ~ T j ( 2 ~ ; 1 ) { B(oq c~) a-oo × q

P

(V/~X)} ~

(1 ÷ e-X/~-~x)2~ dx n--1

-B(~,~)~

12~ ( 2 ~ 7 1) ur(1-u)2~-l-r L r=c~

x u~+g@St 1(1 - u) ~-x/~'-' du nV(2@

re(n-l)

{r(~)}2r(nm + 1)j=

,)

×F[nm-@+j+v~t

) +1] ,

(3.5)

'

n

m blr ( 1 - u) m-r] where aj(m,n) denotes the coefficient of uJ(1 - u) mn j in [2rm~ (r) with m = 2~ - 1. The single moments of the largest order statistic, Xn:n, may be easily derived from Eq. (3.5). For example, from (3.5) we immediately obtain

d ~'n:n= ~M.:n(t)r,=o

nv/7~r(2~) =

m(n-l)

{r(~)I2r(,m+ 1) j=o~(n-1) Z aj(m,n-

1)F(~+j)

x F[nm - (~ +j) + 1]{O[c~+ j ] - tp[nm - (c~+j) + 1]}

(3.6)

and d2 =

m(n-l)

_

nc~r(2.)

1) ~

- 2{r(~)}zr( nm+ × r[nm - (c~ + j ) +

a j ( m , n - 1)r(~+j)

j=~in-ll

1]{@[~ +j] -

O[nm - (~ + j ) +

+ ¢'[~+j] + ~'[nm- (~+j) + 1]} ; thence, the variance of X.:n is given by

1]) 2 (3.7)

N. Balakrishnan and S. K. Lee

136

{7n,n: n .~-

(3.8)

#(2) _ (#n:n)2 ?/:n

As a result, the first two single moments of X,:, can be easily computed from Eqs. (3.6) and (3.7) at least for small values of n. Furthermore, upon using the recurrence relation [David (1981, p. 48); Arnold, Balakrishnan and Nagaraja (1992, p. 112)] that n

/z!k) = ~--'~(--1)r--i(~)

(;

--1"~# (k)

(3.9)

the single moments of all other order statistics may be computed in a simple recursive manner. Also, the coefficients a j ( m , n - 1) involved in Eqs. (3.5)-(3.7) can be easily generated in a recursive way as follows. First of all, we note that ao(m,O)=l ,

aj(m,O)=O

and

aj(m, 1)= (m~ , j _ > l . \J,/

(3.10) Next, for n > 2 we have

aj(m, n) = Coefficient of ud(1 - u) m~ j in ~ ( m r ) u r ( 1 - u ) m Coeff. of ur(1 -- U)m-r in

=

ur(1

-

-

~

11,)m-r

× [Coeff. of uj r(1 - u) m(~-l)-~-r) in

{r=~(7)bit(I--hi)m--r} n-l] r=c~

=~(m)aj-~(m,n-1)

•

(3.11)

Thus, by starting with the values of aj(m, 1) given in (3.10), we can compute the coefficients aj(m, n) for any value of n by repeated application of the recurrence relation in (3.11). Means, variances and covariances of all order statistics have been computed for sample size n = 20 and for choices of e = 0.5~0.5)3(1)6(2)12 by using Eqs. (3.3) and (3.4). Since the distribution is symmetric around zero, we have

Xi:n d -X,-i+l:,

(3.12)

Order statistics from Type 1II generalized logistic

137

and (X~:n,Xj:n) _a (-Xn j+~:,,,-X,,_~+~:,,) .

(3.13)

These results reduce the amount of computation involved in the evaluation of moments of order statistics. From (3.12) and (3.13), we readily get z

n-i+l:n

and

#i,j:~ =/~-j+~,n-i+~:n •

(3.15)

The IMSL routine D Q D A G was employed to perform the single integration in the computation of the single moments of order statistics, while the double integration involved in the computation of the product moments was performed by IMSL routine DTWODQ. Values of means, and variances and covariances of all order statistics are presented in Table 2 and Table 3, respectively; all these values were computed in double precision and rounded off to five decimal places. In order to check the correctness of these values, we used the following known identities: Table 2 Means of order statistics (#i:,) for sample size n = 20 n

i

~=0.5

~--1.0

~=1.5

7=2.0

~=2.5

~=3,0

20

1l 12 13 14 15 16 17 18 19 20

0.07943 0.24027 0.40727 0.58547 0.78165 1.00595 1.27552 1.62408 2.13524 3.14255

0.07071 0.21356 0.36087 0.51628 0.68464 0.87320 1.09417 1.37147 1.76431 2.50863

0.06777 0.20456 0.34529 0.49316 0.65239 0.82929 1.03435 1.28797 1.64012 2.28745

0.06630 0.20008 0.33755 0.48170 0.63647 0.80768 1.00500 1.24709 1.57928 2.17765

0.06542 0.19741 0.33294 0.47488 0.62701 0.79487 0.98765 1.22298 1.54345 2.11270

0.06484 0.19564 0.32989 0.47037 0.62075 0.78641 0.97621 1.20712 1.51991 2.06999

n

i

c~= 4.0

7 = 5.0

~ = 6.0

c~= 8.0

~ = 10.0

~ = 12.0

20

11 12 13 14 15 16 17 18 I9 20

0.06412 0.19344 0.32609 0.46477 0.61300 0.77594 0.96208 1.18755 1.49093 2.01744

0,06369 0,19213 0,32383 0.46144 0.60839 0.76972 0.95369 1.17597 1.47381 1.98645

0.06341 0.19126 0.32233 0.45923 0.60533 0.76560 0.94814 1.16831 1.46251 1.96604

0.06305 0.19017 0.32047 0.45648 0.60153 0.76049 0.94126 1.15883 1.44853 1.94083

0.06284 0.18952 0.31935 0.45483 0.59927 0.75744 0.93716 1.15318 1.44022 1.92588

0.06270 0.18909 0.31861 0.45374 0.59776 0.75541 0.93444 1.14943 1.43471 1.91599

t Missing values can be found by the symmetry relation #i:~ = -#n-i+l:n.

138

N. Balakrishnan and S. K. Lee

Table 3 Variances and covariances of order statistics (~tj:n) for n = 20 n

i

j

~=0.5

~ = 1.0

7 = 1.5

~=2.0

~=2.5

~=3.0

20

1 1 1 1 1 1 1 1 1 1 1 1 l 1 1 1 1 1 1 1 2 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 3

1.60641 0.60709 0.35907 0.25018 0.18992 0.15212 0.12648 0.10815 0.09454 0.08416 0.07608 0.06971 0.06463 0.06058 0.05734 0.05477 0.05276 0.05124 0.05015 0.04945 0.61272 0.36290

0.84810 0.33539 0.20612 0.14842 0.11588 0.09501 0.08051 0.06984 0.06166 0.05520 0.04996 0.04563 0.04200 0.03889 0.03622 0.03389 0.03184 0.03002 0.02840 0.02695 0.34949 0.21560

0.62091 0.25628 0.16160 0.11845 0.09371 0.07760 0.06624 0.05777 0.05119 0.04592 0.04158 0.03794 0.03483 0.03212 0.02973 0.02759 0.02564 0.02383 0.02210 0.02035 0,27459 0,17396

0.51786 0,22057 0.14134 0.10467 0.08340 0.06943 0.05949 0,05203 0~04619 0,04148 0,03757 0.03426 0.03141 0.02891 0.02667 0.02465 0.02277 0.02100 0.01924 0.01737 0.24103 0.15522

0.46061 0.20067 0.12995 0.09688 0.07754 0.06475 0.05562 0.04872 0.04330 0.03890 0.03524 0.03213 0.02943 0.02705 0.02492 0.02297 0.02115 0.01940 0.01765 0.01570 0.22234 0.14472

0.42466 0.18810 0.12272 0.09190 0.07378 0.06175 0.05312 0.04658 0.04143 0.03724 0.03374 0.03075 0.02816 0.02586 0.02379 0.02189 0.02011 0.01838 0.01663 0.01465 0.21053 0.13805

n

i

j

~ = 4.0

~ = 5.0

e¢ = 6.0

e¢ = 8.0

¢~= 10,0

~ = 12.0

20

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2

l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 3

0.38243 0.17322 0.11411 0.08594 0.06926 0.05812 0.05009 0.04399 0.03916 0.03522 0.03191 0.02908 0.02661 0.02441 0.02243 0.02059 0.01886 0.01717 0.01543 0.01341 0.19651 0,13009

0.35864 0.16475 0.10918 0.08251 0.06665 0.05602 0.04834 0,04249 0,03784 0.03404 0.03085 0.02811 0.02571 0.02358 0.02164 0.01984 0.01814 0.01647 0.01474 0.01271 0.18850 0.12552

0.34344 0.15931 0.10599 0.08029 0.06495 0.05466 0.04720 0.04151 0.03698 0.03327 0.03015 0.02747 0.02513 0.02303 0.02112 0.01935 0.01767 0.01602 0.01429 0.01226 0.18334 0.12256

0.32521 0.15272 0.10213 0.07759 0.06289 0.05299 0.04581 0.04031 0.03593 0.03233 0.02931 0.02670 0.02441 0.02236 0.02049 0.01876 0.01710 0,01547 0.01376 0.01171 0.17707 0.11896

0.31469 0.14890 0.09987 0.07601 0.06168 0.05201 0.04499 0.03960 0.03531 0.03178 0.02881 0.02624 0.02399 0.02197 0.02012 0.01841 0.01677 0.01515 0.01344 0.01139 0.17342 0.11685

0.30784 0.14639 0.09839 0.07497 0.06088 0.05137 0.04445 0.03914 0.03490 0.03142 0.02848 0.02594 0.02371 0.02171 0.01988 0.01818 0.01655 0.01493 0.01324 0.01119 0.17103 0.11547

t Missing values can be found by the symmetry relation aij:,, = ¢7n-j+l,n i+l:n.

Order statistics from Type 1II generalized logistic

139

Table 3 (Contd.) n

i

j

~=0.5

~=1.0

~=1.5

~=2.0

~=2.5

~=3.0

20

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3

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

0.25305 0.19221 0.15402 0.12810 0.10956 0.09580 0.08529 0.07712 0.07067 0.06553 0.06142 0,05814 0,05554 0,05351 0.05197 0.05086 0.36880 0.25750 0.19576 0.15696 0.13061 0.11175

0.15557 0.12162 0.09981 0.08462 0.07345 0.06488 0.05810 0.05260 0.04805 0.04423 0.04097 0.03815 0.03570 0.03355 0.03164 0.02993 0.22603 0.16345 0.12796 0.10512 0.08919 0.07745

0.12784 0.10128 0.08396 0.07172 0.06259 0.05549 0.04979 0.04510 0.04116 0.03779 0.03485 0.03227 0.02995 0.02783 0.02587 0.02400 0.18600 0.13700 0.10870 0.09020 0.07711 0.06732

0.11525 0.09197 0.07665 0.06573 0.05751 0.05108 0.04588 0.04157 0.03792 0.03477 0.03201 0.02954 0.02730 0.02523 0.02326 0.02132 0.16796 0.12500 0.09990 0.08333 0.07151 0.06261

0.10816 0.08670 0.07249 0.06230 0.05461 0.04855 0.04364 0.03955 0.03606 0.03304 0.03038 0.02798 0.02580 0.02375 0.02179 0.01983 0.15782 0.11822 0.09490 0,07942 0.06831 0.05990

0.10363 0.08333 0.06981 0.06010 0.05273 0.04692 0.04219 0.03823 0.03486 0.03193 0.02933 0.02698 0.02483 0.02281 0,02086 0.01887 0,15137 0.11389 0.09170 0.07690 0.06624 0.05815

n

i

j

ct = 4 . 0

~ =

ct =

c~ =

~ =

~ =

20

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3

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

0.09821 0.07927 0.06659 0.05744 0.05047 0.04494 0.04043 0.03665 0.03340 0.03057 0.02805 0.02577 0.02367 0.02168 0.01974 0.01774 0.14364 0.10868 0.08784 0.07385 0.06374 0.05603

0.09509 0.07692 0.06473 0.05589 0.04915 0.04379 0.03941 0.03572 0.03256 0.02979 0.02732 0.02507 0.02300 0.02102 0.01909 0.01709 0.13920 0.10567 0.08560 0.07209 0.06229 0.05480

5.0

6.0

0.09306 0.07540 0.06351 0.05488 0.04829 0.04304 0.03874 0.03512 0.03200 0.02927 0.02683 0.02462 0.02256 0.02060 0.01867 0.01667 0.13631 0.10372 0.08414 0.07093 0.06133 0.05399

8.0

0.09059 0.07353 0.06202 0.05365 0.04723 0.04212 0.03792 0.03437 0.03132 0.02864 0.02624 0.02406 0.02202 0.02008 0.01816 0.01615 0.13279 0.10132 0.08235 0.06951 0.06017 0.05299

10.0

0.08914 0.07244 0.06114 0.05292 0.04661 0.04157 0.03743 0.03394 0.03092 0.02827 0.02589 0.02373 0.02170 0.01977 0.01786 0.01585 0.13072 0.09992 0.08130 0.06868 0.05948 0.05241

t Missing values can be found by the symmetry relation crid:, = ~r, j+l,n-i+l:n.

12,0

0.08818 0.07172 0.06057 0.05244 0.04620 0.04121 0.03711 0.03365 0.03066 0.02802 0.02566 0.02351 0.02150 0.01957 0.01766 0.01566 0.12937 0.09899 0.08060 0.06813 0.05902 0.05202

140

N. Balakrishnan and S. K. Lee

Table 3 (Contd.) n

i

j

c~= 0.5

c~= 1.0

c~= 1.5

c~= 2.0

c~-- 2.5

~ = 3.0

20

3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4

9 10 I1 12 13 14 15 16 17 18 4 5 6 7 8 9 10 1l 12 13 14 15

0.09774 0.08705 0.07872 0.07215 0.06692 0.06273 0.05938 0.05673 0.05466 0.05309 0.26366 0.20067 0.16104 0.13410 0.11480 0.10045 0.08949 0.08096 0.07422 0.06885 0.06455 0.06112

0.06844 0.06131 0.05552 0.05074 0.04671 0.04327 0.04031 0.03772 0.03545 0.03343 0.17221 0.13502 0.11103 0.09428 0.08192 0.07242 0.06490 0.05880 0.05374 0.04948 0.04585 0.04272

0.05971 0.05360 0.04856 0.04433 0.04071 0.03756 0.03477 0.03228 0.03000 0.02789 0.14648 0.11640 0.09668 0.08271 0.07226 0.06411 0.05757 0.05218 0.04764 0.04376 0.04038 0.03739

0.05563 0.04998 0.04530 0.04133 0.03790 0.03490 0.03221 0.02977 0.02752 0.02538 0.13476 0.10785 0.09005 0.07733 0.06774 0.06021 0.05412 0.04906 0.04478 0.04107 0.03782 0.03492

0.05328 0.04790 0.04342 0.03960 0.03629 0.03337 0.03075 0.02835 0.02611 0.02396 0.12812 0.10299 0.08627 0.07425 0.06515 0.05797 0.05213 0.04727 0.04312 0.03953 0.03635 0.03350

0.05176 0.04656 0.04220 0.03849 0.03525 0.03239 0.02980 0.02743 0.02520 0.02305 0.12387 0.09987 0,08382 0,07226 0,06346 0.05651 0.05084 0.04610 0.04205 0.03853 0.03540 0.03258

n

i

j

c~= 4.0

c~= 5.0

c~= 6.0

~ = 8.0

c~= 10.0

c~ = 12.0

20

3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4

9 10 11 12 13 14 15 16 17 18 4 5 6 7 8 9 10 11 12 13 14 15

0.04992 0.04492 0.04073 0.03713 0.03399 0.03120 0.02866 0.02633 0.02411 0.02196 0.11874 0.09609 0.08086 0.06984 0.06142 0.05474 0.04927 0.04468 0,04075 0.03731 0.03424 0.03147

0.04884 0.04396 0.03986 0.03634 0.03325 0.03050 0.02800 0.02568 0.02348 0.02133 0.11577 0.09389 0.07914 0.06842 0.06022 0.05370 0.04835 0.04385 0.03998 0.03659 0.03357 0.03082

0.04814 0.04334 0.03930 0.03582 0.03277 0.03004 0.02756 0.02526 0,02307 0.02092 0.11383 0.09246 0.07801 0.06750 0.05944 0.05302 0.04775 0.04330 0.03948 0.03612 0.03313 0.03040

0.04727 0.04257 0.03860 0.03518 0.03218 0.02949 0.02703 0.02475 0.02257 0.02041 0.11146 0.09070 0.07662 0.06636 0.05848 0.05218 0.04700 0.04263 0.03886 0.03555 0.03258 0.02987

0.04676 0.04211 0.03819 0.03480 0.03183 0.02916 0.02672 0.02444 0.02227 0.02012 0.11006 0.08966 0.07581 0.06569 0.05791 0.05169 0.04656 0.04223 0.03850 0.03521 0.03226 0.02956

0.04642 0.04181 0.03792 0.03456 0.03159 0.02894 0.02651 0.02424 0.02208 0.01993 0.10914 0.08898 0.07527 0.06524 0.05753 0.05136 0.04627 0.04197 0.03825 0.03498 0.03205 0.02936

t Missing values can be found by the symmetry relation cri,j:n = a,, /+l,,-i+~ ....

Order statistics from Type III generalized logistic

141

Table 3 (Contd.) n

i

j

e = 0.5

c~ = 1.0

c~= 1.5

c~= 2.0

c~ = 2.5

c~= 3.0

20

4 4 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6

16 17 5 6 7 8 9 10 11 12 13 14 15 16 6 7 8 9 10 11 12 13

0.05840 0.05627 0.20713 0.16641 0.13869 0.11881 0.10402 0.09271 0208390 0.07695 0.07140 0.06696 0.06341 0.06059 0.17324 0.14453 0.12392 0.10857 0.09683 0.08767 0.08043 0.07466

0.03999 0.03758 0.14291 0.11766 0.09999 0.08694 0.07690 0.06894 0.06248 0.05712 0.05261 0.04876 0.04544 0.04254 0.12513 0.10644 0.09261 0.08197 0.07352 0.06665 0.06096 0.05616

0.03471 0.03227 0.12460 0.10361 0.08870 0.07754 0.06883 0.06183 0.05606 0.05120 0.04703 0.04341 0.04020 0.03733 0.11115 0.09524 0.08331 0.07399 0.06649 0.06030 0.05509 0.05062

0.03228 0.02984 0.11616 0.09709 0.08343 0.07313 0.06503 0.05847 0.05302 0.04840 0.04441 0.04090 0.03776 0.03491 0.10463 0.08998 0.07891 0.07021 0.06315 0.05729 0.05231 0.04800

0.03089 0.02845 0.11134 0.09335 0.08040 0.07058 0.06283 0.05653 0.05127 0.04678 0.04289 0.03945 0.03636 0.03353 0.10087 0.08695 0.07637 0.06802 0.06121 0.05553 0.05069 0.04648

0.02999 0.02756 0.10823 0.09093 0.07843 0.06893 0.06140 0.05526 0.05012 0.04573 0.04190 0.03851 0.03545 0.03263 0.09844 0.08498 0.07472 0.06659 0.05995 0.05439 0.04964 0.04549

n

i

j

c~= 4.0

c~= 5.0

c~ = 6.0

c~= 8.0

~ = 10.0

c~= 12.0

20

4 4 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6

16 17 5 6 7 8 9 10 11 12 13 14 15 16 6 7 8 9 10 11 12 13

0.02891 0.02648 0.10447 0.08799 0.07604 0.06691 0.05966 0.05372 0.04873 0.04444 0.04070 0.03736 0.03434 0.03155 0.09548 0.08257 0.07269 0.06484 0.05840 0.05299 0.04835 0.04428

0.02827 0.02586 0.10227 0.08628 0.07464 0.06573 0.05863 0.05281 0.04791 0.04369 0.03999 0.03669 0.03369 0.03091 0.09375 0.08116 0.07151 0.06381 0.05749 0.05217 0.04759 0.04357

0.02786 0.02545 0.10084 0.08515 0.07372 0.06496 0.05796 0.05221 0.04737 0.04319 0.03953 0.03625 0.03327 0.03050 0.09261 0.08023 0.07072 0.06314 0.05689 0.05162 0.04709 0.04310

0.02735 0.02495 0.09907 0.08377 0.07259 0.06400 0.05713 0.05148 0.04670 0.04258 0.03896 0.03571 0.03275 0.02999 0.09121 0.07909 0.06976 0.06230 0.05615 0.05095 0.04647 0.04252

0.02705 0.02465 0.09803 0.08295 0.07192 0.06343 0.05664 0.05104 0.04631 0.04222 0.03862 0.03539 0.03244 0.02969 0.09038 0.07841 0.06919 0.06180 0.05571 0.05056 0.04610 0.04218

0.02685 0.02446 0.09735 0.08242 0.07148 0.06306 0.05632 0.05075 0.04605 0.04198 0.03839 0.03518 0.03223 0.02949 0.08983 0.07797 0.06881 0.06148 0.05542 0.05029 0.04586 0.04195

t Missing values can be found by the symmetry relation aiff:n

=

~n-j+l,n-i+l:n.

142

N. B a ~ k r ~ h n a n and S. K. L e e

Table 3 (Contd.) n

i

j

c~= 0.5

c~= 1.0

~ = 1.5

c~= 2.0

c~= 2.5

c~= 3.0

20

6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 10 10

14 15 7 8 9 10 11 12 13 14 8 9 10 11 12 13 9 10 11 12 10 11

0.07004 0.06634 0.15186 0.13034 0.11429 0.10200 0.09241 0.08482 0.07877 0.07392 0.13833 0.12141 0.10845 0.09832 0.09030 0.08390 0.13027 0.11648 0.10569 0.09714 0,12650 0.11490

0.05207 0.04853 0.11379 0.09909 0.08776 0.07875 0.07143 0.06535 0.06023 0.05585 0.10655 0.09443 0.08479 0.07694 0.07043 0.06493 0.10221 0.09184 0.08338 0.07636 0.10017 0.09100

0.04673 0.04329 0.10249 0.08971 0.07972 0.07167 0.06503 0.05943 0.05462 0.05044 0,09691 0.08618 0.07752 0.07036 0,06432 0.05914 0.09356 0.08420 0.07647 0.06993 0.09197 0.08357

0.04422 0.04084 0.09715 0.08526 0.07589 0.06829 0.06197 0.05660 0.05196 0.04787 0.09232 0.08223 0.07403 0.06720 0.06140 0.05638 0.08940 0.08053 0.07314 0.06685 0.08802 0.07998

0.04276 0.03942 0.09406 0,08267 0.07366 0.06632 0.06019 0.05496 0.05041 0.04638 0.08965 0.07992 0.07199 0.06536 0.05969 0.05477 0.08697 0.07838 0.07119 0.06504 0.08571 0.07788

0.04182 0.03850 0.09205 0.08098 0.07221 0.06504 0.05902 0.05388 0.04939 0.04541 0.08790 0.07842 0.07066 0.06415 0.05857 0.05371 0.08538 0.07697 0.06991 0.06386 0.08419 0.07651

n

i

j

~=4.0

~=5.0

~=6.0

~=8.0

~=10.0

~=12.0

20

6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 10 10

14 15 7 8 9 10 11 12 13 14 8 9 10 1l 12 13 9 10 11 12 10 11

0.04066 0.03738 0.08959 0.07892 0.07043 0.06346 0.05760 0.05256 0.04815 0.04422 0.08576 0.07656 0.06902 0.06266 0.05720 0.05242 0.08343 0.07524 0.06834 0.06240 0.08233 0.07481

0.03998 0.03672 0.08815 0.07771 0.06938 0.06253 0.05675 0.05178 0.04742 0.04353 0.08449 0.07547 0.06805 0.06179 0.05639 0.05166 0.08227 0.07422 0.06741 0.06155 0.08122 0.07381

0.03954 0.03629 0.08720 0.07691 0.06868 0.06191 0.05620 0.05127 0.04694 0.04307 0.08366 0.07475 0.06741 0.06121 0.05586 0.05115 0.08151 0.07354 0.06680 0.06098 0.08050 0.07315

0.03899 0.03576 0.08603 0.07592 0.06783 0.06116 0.05551 0.05064 0.04635 0.04250 0.08264 0.07386 0.06662 0.06049 0.05520 0.05053 0.08057 0.07270 0.06604 0.06028 0.07959 0.07233

0.03866 0.03544 0.08533 0.07533 0.06732 0.06070 0.05510 0.05026 0.04599 0.04216 0.08202 0.07334 0.06615 0.06007 0.05481 0.05016 0.08001 0.07221 0.06559 0.05986 0.07906 0.07184

0.03844 0.03523 0.08487 0.07495 0.06698 0.06041 0.05483 0.05001 0.04576 0.04194 0.08162 0.07298 0.06584 0.05979 0.05455 0.04992 0.07964 0.07188 0.06529 0.05959 0.07870 0.07152

t Missing values can be found by the symmetry relation ~ri,j:. = an_jWI,n_i÷l: n,

Order statistics from Type III generalized logistic

143

~-~-'~#i:. = nE(X) i=1 =

n,ul: 1

and n

o-ij:, = n Var(X) i=1 j = l :

nol,l: 1 .

4. BLUEs of location and scale parameters Let Y be a random variable with p.d.f. 2X/~

e-~2-~(Y-~)/~

g(y; #, a) = aB(c~, c~) (1 + e-X/T~(y-~)/,,)2~' '

- o c < y , # < oc, c~,a > 0 (4.1)

where #, a and ~ are the location, scale and shape parameters, respectively. Then, Y is the location-scale form of the reparametrized model presented in Section 2. The corresponding c.d.f, is found as

G(y;#,o)--B(~,~f0

- -

u~ l ( 1 - u ) ~ - l d u

where p(.) is as defined earlier. Now, let 111,I12,..., Yn be a random sample of size n from the distribution with p.d.f, and c.d.f, as given in (4.1) and (4.2), respectively, where the shape parameter e is assumed to be known. Further, let Yl:n < Y2:~-< " " -< Y, .... be a Type-II right-censored sample obtained from the above random sample. The best linear unbiased estimators (BLUEs) of # and a are then given by [David (1981, pp. 128132); Arnold, Balakrishnan and Nagaraja (1992, p. 172)] n-F

#* = -/~'FY = d Y =

~aiYi:,,

(4.3)

i=1

and n--r

•* = I ' / ' Y = b'Y= ~-~biYi:, , i=1

(4.4)

144

N . B a l a k r i s h n a n a n d S. K . L e e

where a and b are the vectors of coefficients for the BLUEs of the location and scale parameters, respectively,

/'=X1 [r 1=

1(1/{ _/d,)12-1],

{1 , 1 , . . . , 1 ]'(n-r)×l,

#=

A = (ltX-11)(/t'.Y-lp) -- (/Rt~Y'-ll) 2,

[#l:n,#2:n,''',#n-r:n(nl'

r)×l

and o.l,l:n

O'l,2:n

"' "

o.i,n-r:n

O-l,2:n

o.2.2:n

"''

ff2,n-r:n

.

i

o.l,n-r:n

0"2,n-r:n

12=

.

.

"""

~Tn-r,n-r:n

(n-r) ×(n-r)

Further, we have the variances and covariance of the BLUEs #* and o.* as Var(#*)

(4.5)

= o.2/R'12 1/[///~ ,

Var(o.*) = o-21'12

I1/A

(4.6)

,

and Cov(/~*, a*) = - a 2 # ' .r - l l / A

.

(4.7)

After computing the means, variances and covariances of order statistics for sample size n = 20 from the standard reparametrized distribution as described in Section 3, the coefficients of the BLUEs o f # and o- were computed from (4.3) and (4.4). All the computations were carried out in double precision and the IMSL routine D L I N D S was used for finding the inverse of the variance-covariance matrix. In Tables 4 and 5, we have presented the coefficients ai and b i of the best linear unbiased estimators #* and a*, respectively, for sample size n = 20, r = 0, 2, 4 and e = 1.0, 2.0(2)10.0. Values of these coefficients have been rounded off to five decimal places. In order to check the accuracy of the tabulated coefficients, we verified the conditions n-r

n-r

Z ai = 1, Z ai#i:n = O i=1

(4.8)

i=1

and n

r

n--r

~bi=O,

Zbi#i:,= l ,

i=I

i=1

(4.9)

based on the fact that the estimators #* and a* are unbiased for # and a, respectively. The values of Var(#*)/~ 2, Var(a*)/a 2 and Cov(#*, cr*)/a 2 have been computed and are presented in Table 6 for sample size n = 2 0 , r = 0 , 2 , 4 and c~= 1.0,2.0(2)10.0. These values have also been rounded off to five decimal places.

Order statistics from Type III generalized logistic

145

Table 4 Coefficients for the B L U E of p for n = 20 r

i

~=1.0

~=2.0

~=4.0

~=6.0

~=8.0

~=10.0

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 ll 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10

0.00685 0.02012 0.03221 0.04291 0.05214 0.05986 0.06605 0.07070 0.07381 0.07536 0.07536 0.07381 0.07070 0.06605 0.05986 0.05214 0.04291 0.03221 0.02012 0.00685 0.00574 0.01894 0.03108 0.04187 0.05123 0.05911 0.06548 0,07032 0,07363 0,07539 0.07559 0.07423 0.07131 0.06683 0.06078 0.05317 0.04401 0.06129 0.00231 0.01540 0.02771 0.03885 0.04866 0.05708 0.06404 0.06951 0.07345 0.07584

0.01838 0.03445 0.04297 0.04893 0.05337 0.05673 0.05923 0.06102 0.06217 0.06274 0.06274 0.06217 0.06102 0.05923 0.05673 0.05337 0.04893 0.04297 0.03445 0.01838 0.01549 0.03178 0.04067 0.04701 0.05181 0.05553 0.05840 0.06054 0.06205 0.06296 0.06330 0.06307 0.06224 0.06077 0.05857 0.05549 0,05127 0.09904 0,00907 0,02602 0,03580 0.04302 0.04870 0.05327 0.05697 0.05993 0.06224 0.06393

0.03070 0.04276 0.04731 0.05023 0.05228 0.05378 0.05486 0.05562 0.05611 0.05635 0.05635 0.05611 0.05562 0.05486 0.05378 0.05228 0.05023 0.04731 0.04276 0.03070 0.02573 0.03884 0.04415 0.04770 0.05033 0.05235 0.05393 0.05517 0.05611 0.05680 0.05724 0.05744 0.05739 0.05707 0.05641 0.05534 0.05368 0.12433 0.01660 0.03181 0.03860 0.04338 0.04710 0.05012 0.05265 0.05479 0.05660 0.05813

0.03630 0.04537 0.04841 0.05032 0.05165 0.05260 0.05329 0.05377 0.05407 0.05422 0.05422 0.05407 0.05377 0.05329 0.05260 0.05165 0.05032 0.04841 0.04537 0.03630 0.03033 0.04097 0.04494 0.04759 0.04956 0.05110 0.05234 0.05333 0.05414 0.05478 0.05526 0.05559 0.05576 0.05576 0.05556 0.05510 0.05425 0.13364 0.02003 0.03354 0.03920 0.04319 0.04633 0.04892 0.05113 0.05305 0.05474 0.05624

0.03941 0.04661 0.04889 0.05031 0.05128 0.05198 0.05248 0.05283 0.05305 0.05316 0.05316 0.05305 0.05283 0.05248 0.05198 0.05128 0.05031 0.04889 0.04661 0.03941 0.03287 0.04196 0.04526 0.04747 0.04913 0.05045 0.05152 0.05242 0.05316 0.05377 0.05428 0.05467 0.05495 0.05511 0.05512 0.05494 0.05448 0.13844 0.02194 0.03435 0.03943 0.04305 0.04590 0.04829 0.05036 0.05219 0.05382 0.05531

0.04138 0.04733 0.04915 0.05028 0.05105 0.05160 0.05199 0.05227 0.05244 0.05252 0.05252 0.05244 0.05227 0.05199 0.05160 0.05105 0.05028 0.04915 0.04733 0.04138 0.03448 0.04252 0.04542 0.04738 0.04886 0.05005 0.05103 0.05187 0.05257 0.05318 0.05369 0.05412 0.05446 0.05471 0.05485 0.05484 0.05461 0.14136 0.02314 0.03480 0.03955 0.04294 0.04564 0.04791 0.04989 0.05167 0.05328 0.05475

2

4

N. Balakrishnan and S. K. Lee

146

Table 4 (Contd.) r

i

~=1.0

~=2.0

ct=4.0

~=6.0

~=8.0

~=10.0

11 12 13 14 15 16

0.07667 0.07590 0.07352 0.06950 0.06383 0.16775

0.06503 0.06553 0.06541 0.06461 0.06301 0.21744

0.05939 0.06040 0.06115 0.06160 0.06169 0.24601

0.05756 0.05873 0.05972 0.06054 0.06113 0.25594

0.05666 0.05789 0.05901 0.06000 0.06083 0.26097

0.05612 0.05740 0.05858 0.05967 0.06065 0.26399

Table 5 Coefficients for the BLUE of ~r for n = 20 r

i

~ = 1.0

c~ = 2.0

c~~ 4.0

~ = 6.0

~ = 8.0

~ = 10.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

-0.05686 -0.06110 -0.06034 -0.05660 -0.05076 -0.04338 -0.03483 -0.02545 -0.01549 -0.00520 0.00520 0.01549 0.02545 0.03483 0,04338 0.05076 0.05660 0.06034 0.06110 0.05686 -0.06308 -0.06766 -0.06671 -0.06245 -0.05588 -0.04761 -0.03807 -0.02762 -0.01654 -0,00511 0.00643 0.01782 0.02882 0.03915 0.04851 0.05653

-0.07491 -0.07176 -0.06409 -0.05580 -0.04731 -0.03875 -0.03014 -0.02153 -0.01292 -0.00431 0.00431 0.01292 0.02153 0.03014 0.03875 0.04731 0.05580 0.06409 0.07176 0.07491 -0.08433 -0.08044 -0.07159 -0,06210 -0.05241 -0.04266 -0.03289 -0.02312 -0.01337 -0.00363 0,00609 0.01580 0.02548 0.03512 0.04471 0.05417

-0,09085 -0.07550 -0.06356 -0.05334 -0.04409 -0.03545 -0.02722 -0.01927 -0.01149 -0.00382 0.00382 0.01149 0.01927 0.02722 0.03545 0.04409 0.05334 0.06356 0,07550 0,09085 -0A0350 -0,08548 -0,07161 -0.05978 -0.04909 -0.03911 -0.02962 -0.02046 -0.01152 -0.00270 0.00606 0.01485 0.02373 0.03280 0.04213 0.05185

-0.09751 -0.07616 -0.06294 -0.05223 -0.04284 -0.03426 -0.02620 -0.01849 -0.01101 -0.00366 0.00366 0.01101 0.01849 0.02620 0.03426 0.04284 0.05223 0.06294 0.07616 0.09751 -0.11159 -0.08656 -0.07114 -0,05868 -0.04777 -0.03781 -0.02846 -0.01953 -0.01087 -0.00237 0.00608 0.01456 0.02317 0.03201 0.04122 0.05096

-0.10109 -0.07637 -0.06255 -0.05162 -0.04218 -0.03364 -0.02568 -0.01810 -0.01077 -0.00358 0.00358 0.01077 0.01810 0.02568 0.03364 0.04218 0.05162 0.06255 0.07637 0.10109 -0.11596 -0.08696 -0.07082 -0.05808 -0.04708 -0.03714 -0.02787 -0.01906 -0.01055 -0.00220 0.00609 0.01442 0.02289 0.03162 0.04076 0.05049

-0.10331 -0.07645 -0.06229 -0.05125 -0.04178 -0.03327 -0.02537 -0.01787 -0.01063 -0,00353 0.00353 0.01063 0.01787 0.02537 0.03327 0.04178 0.05125 0.06229 0.07645 0.10331 -0.11868 -0.08716 -0.07060 -0.05770 -0.04666 -0.03673 -0.02752 -0.01878 -0.01035 -0.00209 0.00610 0.01434 0.02273 0.03139 0.04049 0.05021

Order statistics frorn Type III generalized logistic

147

Table 5 (Contd.) r

i

c~= 1.0

c~- 2.0

c~= 4.0

7 = 6.0

7 = 8.0

c~= 10.0

17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 t4 15 t6

0.06274 0.19071 -0.07233 -0.07723 -0.07580 -0.07061 -0,06281 -0.05310 -0.04198 -0.02985 -0.01705 -0.00391 0.00930 0.02228 0.03474 0.04634 0.05671 0.33532

0.06339 0.22178 -0.09747 -0.09224 -0.08157 -0.07026 -0.05879 -0.04729 -0.03582 -0.02438 -0.01300 -0.00166 0.00961 0.02082 0.03195 0.04295 0.05377 0.36336

0.06212 0.23933 -0.11992 -0.09812 -0.08160 -0.06756 -0.05490 -0.04312 -0.03193 -0.02115 -0.01065 -0.00032 0.00992 0.02016 0.03047 0.04093 0.05160 0.37619

0.06147 0.24532 -0.12935 -0.09937 -0.08105 -0.06628 -0.05336 -0.04158 -0.03055 -0.02003 -0.00984 0.00015 0.01005 0.01996 0.02999 0.04023 0.05081 0.38021

0.06111 0.24833 -0.13444 -0.09983 -0.08067 -0.06557 -0.05255 -0.04079 -0.02985 -0.01946 -0.00943 0.00039 0.01012 0.01987 0.02975 0.03989 0.05041 0.38219

0.06089 0.25014 -0.13761 -0.10006 -0.08041 -0.06513 -0.05205 -0.04031 -0.02943 -0.01912 -0.00919 0.00053 0.01016 0.01981 0.02961 0.03968 0.05017 0.38336

Table 6 Values of (1) Var(/~*)/rr2, (2) Var(rr*)/a2 and (3) Cov(g*, cr*)/rr2 for n = 20 e=l.0

~--2.0

(1) (2) (3)

0.07595 0.03660 0.00000

0.06282 0,03224 0,00000

(1) (2) (3)

0.07609 0.04065 0.00072

0,06323 0,03645 0.00130

(1) (2) (3)

0.07695 0.04673 0.00298

0.06468 0.04247 0.00424

e=4.0 r=0 0.05635 0.02962 0.00000 r=2 0.05705 0.03405 0.00174 r=4 0.05889 0.03999 0.00505

a=6.0

~=8.0

e=10.0

0.05421 0.02870 0.00000

0.05315 0.02824 0.00000

0.05252 0.02796 0.00000

0.05504 0.03322 0.00192

0.05405 0.03280 0.00201

0.05346 0.03255 0.00207

0.05703 0.03912 0.00534

0.05612 0.03869 0.00549

0.05557 0.03842 0.00558

5. MLEs of location and scale parameters C o n s i d e r a r a n d o m s a m p l e o f size n f r o m t h e r e p a r a m e t r i z e d d i s t r i b u t i o n w i t h p . d . f , g(y) a n d c.d.f. G(y) as g i v e n in (4.1) a n d (4.2), r e s p e c t i v e l y , w h e r e t h e s h a p e p a r a m e t e r e is a s s u m e d t o b e k n o w n . L e t Yl:n _< Y2:n _< " " <_ Y, . . . . b e a T y p e - I I right-censored sample obtained from the random sample. The likelihood function b a s e d o n s u c h a T y p e - I I r i g h t - c e n s o r e d s a m p l e is t h e n g i v e n b y

N. Balakrishnan and S. K. L e e

148

n[

n r

L = 7., [1 - G(ro_~:.)]~I] g(~:~) i=1 n

n!e =

r

i=~

n-r

rD"-r I~ U1 + e-V/~(Y':"-~)/'] 2~ i=1 tr

x

1

B(~,c~)ao

where p(.) is as defined before. We then obtain the likelihood estimating equations as n

OlnL _ ( n _ r ) _ 2

F

E e v&7~(yi:~ #)/° + ~?'0" h(y~-r:.) = 0 i=1 1 + e-V@-g(fi:,-u)/G

0~

(5.2)

and

OlnL O~

nmr

(. - r > + ~

~yi:. i=1

n--r

_ 2v/~ Z

yi:,,e V'T~O":"-~)/°

i=1 1 - { - e

r~y, . . . . h(y._r:.) = 0 ,

(5.3)

v @gcv~:" ~')/~

where h(t) = ~g(t) is the hazard function. Then, the maximum likelihood estimators/2 and ~-are obtained by solving (5.2) and (5.3) simultaneously. Since these two equations cannot be solved analytically, some numerical method must be employed. In order to study the performance of these estimators, we generated 5,000 pseudorandom samples of size n = 20 from the standard reparametrized population (with # = 0 and o- = 1) with choices ofc~ = 0.5(0.5)3(1)6(2)12. The IMSL routines DNEQNJ and D N E Q N F were employed to solve the maximum likelihood estimating equations. Both routines are based on the M I N P A C K subroutine HYBRDJ and HYBRD1, respectively, which use a modification of M. J. D. Powell's hybird algorithm. These algorithms are a variation of Newton's method, which take precaution to avoid large step size or increasing residuals. Thus we determined the bias, variances, covariance and mean square errors of the maximum likelihood estimators /2 and ir for r = 0,2,4. The values of Bias(/~)/cr, Bias(&)/~r, Var(/2)/a 2, Var(&)/cr 2, Cov(/2,&)/~r 2, MSE(/2)/~ 2 and MSE(6)/a 2 determined through this Monte Carlo process are presented in Table 7.

Order statistics from Type III generalized logistic

149

Table 7 Simulated values of (1) Bias(/0/a, (2) Bias(f-)/a, (3) Var(/t)/0- 2, (4) Var(f-)/a 2, (5) Cov(~, 6)/0 "2, (6) MSE(~)/a 2 and (7) MSE(&)/0-2 n

r

(1)

(2)

(3)

20

0 2 4

-0.00128 -0.00321 -0.00823

-0.02477 -0.02924 -0.03704

0.07287 0.07314 0.07403

20

0 2 4

-0.00503 -0.00783 -0.01261

-0.03186 -0.03726 -0.04341

20

0 2 4

0.00361 -0.00031 -0.00611

-0.03255 -0.03974 -0.04742

20

0 2 4

0.00381 0.00071 -0.00412

-0.03415 -0.03912 -0.04499

20

0 2 4

0.00077 -0.00217 -0.00786

-0.03562 -0.04007 -0.04761

20

0 2 4

0.00162 -0.00242 -0.00791

-0.03993 -0.04709 -0.05418

(4)

~=1.0 0.03434 0.03768 0.04310 ~=2.0 0.06178 0.03068 0.06197 0.03438 0.06324 0.03890 = 4.0 0.05800 0.02755 0.05858 0.03126 0.06064 0.03613 ~=6.0 0.05538 0.02725 0.05622 0.03107 0.05824 0.03635 ~=8.0 0.05477 0.02691 0.05544 0.03080 0.05722 0.03573 a = 10.0 0.05170 0.02576 0.05282 0.02960 0.05466 0.03434

(5)

(6)

(7)

-0.00015 0.07287 0 . 0 0 0 7 5 0.07315 0.00286 0.07409

0.03495 0.03853 0.04447

-0.00115 0.06181 -0.00034 0.06203 0 . 0 0 2 0 1 0.06339

0.03170 0.03577 0.04079

0.00052 0.05801 0 . 0 0 1 9 8 0.05858 0 . 0 0 5 1 3 0.06068

0.02861 0.03284 0.03838

0.00000 0.00170 0.00490

0.05540 0.05622 0.05825

0.02842 0.03260 0.03837

-0.00020 0.00135 0.00429

0.05477 0.05545 0.05728

0.02818 0.03240 0.03800

-0.00115 0.00085 0.00377

0.05170 0.05282 0.05472

0.02735 0.03182 0.03728

6. Comparison of the BLUEs with the MLEs I n o r d e r to c o m p a r e t h e p e r f o r m a n c e o f t h e B L U E s a n d M L E s , we d e f i n e t h e r e l a t i v e efficiency b e t w e e n t h e t w o m e t h o d s o f e s t i m a t i o n as f o l l o w s : Eft(#)

MSE(/z)

- V a r ( # * ~ x 100

(6.1)

and Eft(a)

-

MSE(6-) V a r ( a * ~ x 100 .

(6.2)

W e i n t e r p r e t t h e v a l u e o f E f t ( # ) as f o l l o w s : • if E f t ( # ) > 100, t h e n we c o n c l u d e t h a t the e s t i m a t i o n o f # b a s e d o n t h e B L U E s is m o r e efficient t h a n t h a t b a s e d o n t h e M L E s . • i f E f t ( # ) < 100, t h e n we c o n c l u d e t h a t the e s t i m a t i o n o f # b a s e d o n the M L E s is m o r e efficient t h a n t h a t b a s e d o n t h e B L U E s . T h e i n t e r p r e t a t i o n f o r E f t ( a ) f o l l o w s similarly.

N. Balakrishnan and S. K. Lee

150

We computed the values of Eff(/~) and Eft(a) for sample size n = 20 and r = 0, 2, 4 with choices of ~ = 1.0, 2.0(2)10. These values are presented in Table 8. As we observe from Table 8, Eff(/~) are quite close to 100 which indicates that /~* and ~ have about the same efficiency in estimating/~. On the other hand, 8 is slightly more efficient than a* in estimating a as indicated by the values of Eft(a).

7. Illustrative examples Now we shall present two examples in order to illustrate the practical importance of Tables 4-7. EXAMPLE 1 The following arranged sample of size 20 was simulated from the reparametrized distribution with location parameter /~ = 25, scale parameter a = 5 and shape parameter ~ = 2.0: 17.0914 17.3896 18.4054 23.2186 23.3792 24.4208 28.9735 29.9410 30.5459

18.4836 18.5006 20.6758 26.0103 27.1519 28.7569 30.6616 32.2781 33.5083

21.0104 28.8479

Assume that the data are from the reparametrized distribution with p.d.f. (4.1) and a = 1.0~2.0, 4.0. Q-Q plots of the data in Figure 2 indicate that these models all appear reasonably consonant with the data and hence they will be used here to illustrate both methods of estimation. We should mention here that the correlation values from the Q-Q plots are at least 0.96; and based on i,000 simulated samples, we found the p-values to be at least 0.33. From Tables 4-6, we obtained the necessary coefficients ai and bi for the B L U E s / : and a* and the values of Var(#*)/a 2 and Var(a*)/a 2 for the choices of mentioned above. Also, we obtained the values of Var(/~)/a 2 and Var(6)/a 2 from Table 7. By making use of these values and the numerical method described in Section 5, we computed the BLUEs and MLEs o f / z and a, and also their standard errors for complete as well as Type-II right-censored samples with r = 0, 2, 4, respectively. These findings are summarized in Table 9. Table 8 Relative efficiency between the BLUEs and the MLEs for n = 20 c~= 1.0

a = 2.0

Eft(#) Eft(a)

95.94 95.49

98.39 98.33

Eft(p) Eft(a)

96.14 94.78

98.10 98.13

Eft(#) Eft(a)

96.28 95.16

98.01 96.04

a = 4.0 r=0 102.95 96.59 r~2 102.68 96.45 r=4 103.04 95.97

a = 6.0

~ = 8.0

c~= 10.0

102.20 99.02

103.05 99.79

98.44 97.82

102.14 98.13

102.59 98.78

98.80 97.76

102.14 98.08

102.07 98.22

98.47 97.03

Order statistics from Type III generalized logistic Q--Q p ~ for F_xamp~I alpha-| .0

i:il ~

÷ + + +

÷

÷

-0.5

+

+

÷ +

÷

+ 4+

-1 -1,5

rho=0.3o98 p-value.0.3330

...2 -2.~

f

i

25 samp4equa.~.es

3o

Q - Q plot foe Ex,~Tg,e I alpha,.2..O + + + +

1 + + +

"- o.5

0 ÷ ÷ i-0.5

+

+

+

4.+ ÷ ÷

--I

+ rho,.0.9736 p--vahJe-0.4010

--1.5

~

3'o

sample quanmes

Q - Q l~Otfor Exemp~ 1 ~oha,,4.0

+ + + + + m o

+

0

+

+

+ + +

++

i-.0.5 + + + -I .5

+

d)<~0.9754 p-val~-0.3o40

Fig. 2. Q-Q plots for Example 1.

151

N. Balakrishnan and S. K. Lee

152

Table 9 The BLUEs and MLEs of # and a for Example 1

c¢

r=0 1.0 2.0 4.0 r=2 1.0 2.0 4.0 r=4 1.0 2.0 4.0

~*

S.E.(#*)

a*

S.E.(a*)

[l

25.0255 24.9811 24.9672

1,2916 1.2609 1.2358

4.6868 5,0307 5.2061

0.8966 0,9033 0.8960

25.0511 25.0257 25.0240

1.3417 1.3077 1.2828

4.8641 5.2007 5.3705

25.1902 25.2207 25.2509

1.4581 1,4273 1.4047

5.2565 5.6123 5.7885

S.E.([~)

&

S.E.(#)

25.0501 1.2373 25.0001 1.2203 24.9800 1.2235

4.5834 4.9097 5.0803

0.8494 0.8600 0.8432

0.9807 0.9929 0.9910

25.0662 1.2787 25.0350 1.2567 25.0245 1.2625

4.7280 5,0483 5.2163

0.9178 0.9360 0,9223

1.1363 1.1566 1.1576

25,1996 1.3731 25.2160 1.3537 25.2306 1.3693

5,0465 5,3830 5.5606

1.0477 1.0617 1.0570

F r o m T a b l e 9, we can see t h a t the B L U E s as well as the M L E s o f # a n d a are close to the true values o f 25 a n d 5 for c o m p l e t e as well as censored samples for all values o f c~ considered. Besides, we also notice t h a t the s t a n d a r d errors o f the M L E s are all slightly smaller t h a n those o f the B L U E s . EXAMPLE 2 The following a r r a n g e d s a m p l e o f size 20 was s i m u l a t e d f r o m the r e p a r a m e t r i z e d d i s t r i b u t i o n with l o c a t i o n p a r a m e t e r # = 75, scale p a r a m e t e r a = 10 a n d shape p a r a m e t e r ~ = 8.0: 56.6439 58.5165 62.1652 66.8842 67.0852 68,6479 70.8896 71.1141 71.7507 72.1389 72.1936 74.2165 77.7925 78.6317 79.5452 84.5838 86.6567 86.9632 88.7370 91.5514

A s s u m e t h a t the d a t a are f r o m the r e p a r a m e t r i z e d d i s t r i b u t i o n with p.d.f. (4.1) a n d ~ = 6.0, 8.0, 10.0. Once again, Q - Q plots o f the d a t a in F i g u r e 3 indicate t h a t these m o d e l s a p p e a r to fit the d a t a very well a n d the c o r r e l a t i o n values f r o m the Q - Q plots are at least 0.98. Based on 1,000 s i m u l a t e d samples, we f o u n d the pvalues to be at least 0.81, thus p r o v i d i n g strong evidence t o w a r d s the a p p r o p r i ateness o f the m o d e l s c o n s i d e r e d here. F o l l o w i n g the p r o c e d u r e as in E x a m p l e 1, we c o m p u t e d the B L U E s a n d M L E s o f # a n d a, a n d also their s t a n d a r d errors for c o m p l e t e s a m p l e as well as T y p e - I I right c e n s o r e d samples with r = 0, 2, 4 a n d choices o f e m e n t i o n e d above. These findings are s u m m a r i z e d in T a b l e 10. Once again, the B L U E s a n d the M L E s o f # a n d a are close to the true value o f 75 a n d 10 for c o m p l e t e as well as c e n s o r e d samples for all values o f e considered. Also, the s t a n d a r d errors o f the M L E s are all slightly smaller t h a n those o f the BLUEs.

Order statistics from Type III generalized logistic Q--Q ix~oHorExamp~:, ¢pha-s,o

1.

4¢4" + +

|

4, +

/ 4, ÷ o, +

-1

+ 4,

rho-0.9888

-13

p-value,.O.8190

+ 65

70

5 80 sample qu~tiles

Q-Q p~otfor Exmp~ 2 .lp~e.O

+ + + ÷ 0.~

o-

+

+

+

+

o .i.÷

~-0.5

4.

+ 4+

-I

4+

rho~,9888 p-vaJue.O.8190

-1.5 +

sample quan~es

Q - Q plot Ior Example 2 Itlpha.10.0 + ÷

i + 1

+ +

0.5 4, 4-

~

o +

~-o.5

4,

+

rho=0.9089

-1.5

p-value.O.8260 i

i

I

60

65

70

1

i

75 80 sample quan~le$

|

85

Fig. 3. Q-Q plots for Example 2.

i

90

153

154

N. Balakrishnan and S. K. Lee

Table 10 The BLUEs and MLEs of # and a for Example 2 c¢

r=0 6.0 8.0 10.0 r= 2 6.0 8.0 10.0 r=4 6.0 8.0 10.0

#*

S.E.(,u*)

74.3274 74.3299 74.3313

2.2760 2.2730 2.2712

74.5044 74.5143 74.5201 74.7030 74.7185 74.7277

a*

S.E.(~*)

~

S.E.(~)

&

S.E.(gr)

9.7753 9.8595 9.9106

1.6560 1.6569 1.6572

74.3187 74.3234 74.3261

2.2060 2.2160 2.1576

9.3739 9.4457 9.4890

1.5474 1.5495 1.5230

2.3924 2.3906 2.3895

10.1974 10.2829 10.3346

1.8586 1.8623 1.8645

74.4582 74.4685 74.4746

2.3048 2.3059 2.2608

9.7204 9.7933 9.8372

1.7134 1.7187 1.6925

2.5249 2.5252 2.5253

10.5728 10.6597 10.7124

2.0912 2.0967 2.0997

74.5983 74.6131 74.6218

2.4176 2.4140 2.3699

10.0177 10.0918 10.1366

1.9099 1.9076 1.8784

References Abramowitz, M. and I. A. Stegun (Eds.) (1965). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover Publications, New York. Arnold, B. C., N. Balakrishnan and H. N. Nagaraja (1992). A First Course in Order Statistics. John Wiley & Sons, New York. Balakrishnan, N. (Ed.) (1992). Handbook of the Logistic Distribution. Marcel Dekker, New York. Balakrishnan, N. and M. Y. Leung (1988). Order statistics from the Type I generalized logistic distribution. Comm. Statist. - Sim. Comput. 17(1), 25-50. Berkson, J. (1944). Application of the logistic function to bioassay. J. Amer. Statist. Assoc. 37, 357-365. Berkson, J. (1951). Why I prefer logits to probits. Biometrics 7, 32%339. Berkson, J. (1953). A statistically precise and relatively simple method of estimating the bio-assay with quantal response, based on the logistic function. J. Amer. Statist. Assoc. 48, 565 599. David, H. A. (1981). Order Statistics, Second edition. John Wiley & Sons, New York. Davidson, R. R. (1980). Some properties of a family of generalized logistic distributions. In Statistical Climatology, Developments in Atmospheric Science, 13 (Eds., S. Ikeda et al.). Elsevier, Amsterdam. Dubey, S. D. (1969). A new derivation of the logistic distribution. Nay. Res. Log. Quart. 16, 37-40. Finney, D. J. (1947). The principles of biological assay. J. Roy. Statist. Soe. Series B 9, 46-91. Finney, D. J. (1952). Statistical Methods in Biological Assay. Hafner, New York. Gumbel, E. J. (1944). Ranges and midranges. Ann. Math. Statist. 15, 414-422. Gupta, S. S. (1960). Order statistics from the gamma distribution. Technometries 2, 243~62. Oliver, F. R. (1982). Notes on the logistic curve for human populations. J. Roy. Statist. Soc. Series A 145, 359-363. Pearl, R. and L. J. Reed (1920). On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proc. Natl. Acad. Sci. 6, 275-288. Pearl, R. and L. J. Reed (1924). Studies in Human Biology. Williams and Wilkins, Baltimore. Pearl, R., L. J. Reed and J. F. Kish (1940). The logistic curve and the census count of 1940. Science 92, 486-488. Prentice, R. L. (1976). A generalization of the probit and logit methods for dose response curves. Biometrics 32, 761-768. Reed, L. J. and J. Berkson (1929). The application of the logistic function to experimental data. J. Phys. Chem. 33, 760-779. Schultz, H. (1930). The standard error of a forecast from a curve. J. Amer. Statist. Assoc. 25, 139 185.

Order statistics from Type III generalized logistie

155

Shah, B. K. (1966). On the bivariate moments of order statistics from a logistic distribution. Ann. Math. Statist. 37, 1002-1010. Verhulst, P. J. (1838). Notice sur la lois que la population suit dans sons accroissement. Corr. Math. et Physique 10, 113 121. Verhulst, P. J. (1845). Recherches math6matiques sur la loi claccroissement de la population. A caddmie de Bruxelles 18, 1-38. Wilson, E. B. and J. Worcester (1943). The determination of L. D. 50 and its sampling error in bioassay. Proc. Natl. Acad. Sei. 29, 79 85. Zelterman, D. and N. Balakrishnan (1992). Univariate generalized distributions. In Handbook o f the Logistic Distribution (Ed., N. Balakrishnan), pp. 209521. Marcel Dekker, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 17 © 1998 Elsevier Science B.V. All rights reserved.

l'1~)

Estimation of Scale Parameter Based on a Fixed Set of Order Statistics

Sanat

K. Sarkar

and

Wenjin

Wang

1. Introduction

Suppose that F ( x ) is a known absolutely continuous distribution function which is free of any unknown parameters. Then, for any real/~ and o(> 0), F ( ( x - # ) / a ) is called a location-scale distribution with the location parameter # and the scale parameter a, generated by the distribution function F(x). Let Cg(F) denote the family of all location-scale distributions generated by F ( x ) , that is Cg(F)={F((x-#)/o),

-oc<#<~,

o>0}

.

The density function f o f F is called the generating density function for the family Cg(F). Any parameter-free distribution can generate a family of location-scale distributions. Some important generating distribution functions are the following: 1. 2. 3. 4.

The standard exponential distribution: F ( x ) = 1 - e X(x > 0); The standard normal distribution ~b(x) with density function: O(x) = ~--~2e 4; The standard logistic distribution: L(x) - 1+~1 x,. The uniform distribution with density: f ( x ) = 1 if - ½ < x < ½ and = 0 otherwise; 5. The Weibull distribution with known shape parameter b(> 0): F(x) = 1 - e-X~ (x > 0); 6. The Gamma distribution with density function: g(x) = r~seX' ~ X(x> 0), where r is a known positive constant.

These generating distributions generate different families of location-scale distributions except that the standard exponential distribution is a special case of Weibull distribution with b = 1 and is also a special case of Gamma distribution with r = 1. These location-scale families have wide applications and have been frequently studied in the literature. Take, for example, the logistic distribution. It has been applied in studies of population growth, physicochemical phenomena, bio-assay, mental ability, survival data, agricultural production data (see references given by Hatter and Moore 1967; Balakrishnan 1992).

159

S. K. Sarkar and IV. Wang

160

The primary goal of this article is to present some new results on estimation of the scale parameter of a family of location-scale distributions based on a fixed set of order statistics. Let (1.1)

~J(1 ~ X2 ~ "'" ~ Yn

be the n observed order statistics from a random sample of size N ( > n) drawn from a population whose cumulative distribution function is a member of :g(F). These order statistics are any fixed set of order statistics selected from the sample, not necessarily the first n observations. They can be singly or doubly Type II censored observations, or any other Type II censored data. Throughout this chapter, we will use n to denote the number of observed order statistics and N to denote the sample size which, for simplicity, is not indicated in the notations. When n = N, the sample in (1.1) is a complete sample. Let Yi = (X~-/~)/o- (i = 1 , . . . , n). Then, Yi's are the standardized order statistics which can be considered as n observed order statistics for a sample of size N drawn from a population with distribution function F. Let ai denote the expected value of Yi, and qij denote the covariance between Yi and Yj (i ¢ j) or the variance of Y~ (i = j). It is clear that these ai's and qij's are not dependent on the location and scale parameters; they only depend on the form of F(x). Let X = (X1,...

,Xn)',

a = (al,...,a~)',

and Q

= (qij)

•

(1.2)

Then, we have E(x) = ,e +

Cov(X) =

2Q,

(1.3)

where e is an n-dimensional vector of unit entries. Model (1.3) can be used to obtain some estimators of the scale parameter. First note that the scale parameter is one of the regression coefficients in model (1.3). By applying the theory of linear model, one is able to obtain the best unbiased linear estimator of the scale parameter, as done by Lloyd (1952). This estimator as well as the simplified "unbiased nearly best" and "nearly unbiased, nearly best" linear estimators proposed by Blom (1958, 1962) will be reviewed in Section 2. Since the scale parameter a in model (1.3) is known to be positive, it is a natural requirement that any good estimator of o- should possess the same property. Is Lloyd's estimator always positive with probability one? In Leslie, Stephens and Fotopoulos (1986), the authors have noted that it is difficult to establish this positivity. They had to assume, like Sarkadi (1981), that it is always positive in order to prove the consistency of Shapiro-Wilk's (1965) W-statistic for testing normality which involves this estimator. In the text book on order statistics by Arnold, Balakrishnan, and Nagaraja (1992), the authors have made a conjecture based on empirical evidence that Lloyd's estimator is positive with probability one. Recently, Bai, Sarkar, and Wang (1996) gave an affirmative answer analytically to this conjecture for the first time. They proved that Lloyd's estimator is positive with probability one when the underlying generating density is log-concave, a property shared by many of the commonly used

Estimation of scale parameter based on a fixed set of order statistics

161

distributions. Similar result holds for Blom's L-estimators. Their results are stated in Section 3. Observing that in model (1.3) the scale parameter a appears not only as a regression coefficient but also as a common component of the variance-covariance matrix of X, in Section 4, we propose three non-linear estimators of the scale parameter which involve quadratic functions of the given set of order statistics. One is based on the positive square-root of the residual sum of squares obtained by fitting model (1.3). Shapiro and Wilk (1965) and Nelson and Hahn (1972, 1973) mentioned about such an estimator. They, however, did not offer any detailed investigation of this estimator either in those papers or in any subsequent publications. We attempt to study this estimator in this paper. In particular, we provide simplified and explicit representations of such an estimator for some specific distributions, such as uniform and exponential. The second estimator is the best linear combination of the best unbiased L-estimator and the positive square root of the aforementioned residual sum of squares. This combined estimator is an improvement over the best unbiased L-estimator in the sense of having smaller mean squared error. Through simulation, the amount of improvement is shown to be quite significant for several generating distributions. Constants needed to compute the combined estimator are calculated based on simulation for some generating distributions and presented in tables. These simulation results are also reported in Section 4. The third non-linear estimator proposed in Section 4 is determined by minimizing an objective function which is slightly different from, but more meaningful in the present context than, the one used in the usual weighted least squares approach for Lloyd's best unbiased L-estimator. In Section 5, we extend the positivity of the best unbiased L-estimator to censored scale regression model which involves several populations with location parameters being linear functions of some covariates.

2. Linear estimators

This section is devoted to L-estimators of the scale parameter. An L-estimator is an estimator which is linear in X. Estimators of this type appear to be very intuitive when the family of distributions under consideration depends on location and scale parameters. They are attractive in case of censored samples, and often provide quick measures of location and scale, especially when the number of observed order statistics is relatively small. In the literature, many L-estimators have been introduced to estimate the scale parameter in different situations. Results related to some important L-estimators are reviewed here. 2.1. T h e b e s t u n b i a s e d L - e s t i m a t o r

Lloyd (1952) first applied the least-squares theory originally developed by Aitken (1935) to the model (1.3) and obtained the best unbiased L-estimators of # and a.

S. K. Sarkar and W. Wang

162

According to Gauss-Markov's Theorem, the unique best linear unbiased estimators of # and a are the weighted least-squares (WLS) estimators given by (~t, 6-)'= ( A ' Q - ' A ) - I A ' Q - ' X

(2.1)

where A = (e, a) and X, a, Q are as given in (1.2). That is, d Q -I (ea' - ae')Q-1X

=

(2.2)

(etQ -le) ( a ' Q - ' a ) - ( e ' Q - ' a ) 2 ' e'Q -1 (ea t - ael)Q-1X

=

(2.3)

( e t Q - l e ) ( a t Q - ' a ) _ (etQ-la) 2 '

It can be shown that the variances of the best unbiased L-estimators are as follows, var(~) =

a'Q-la -0"2 (etQ-le) (a,Q-la) - (e,Q-la) 2 ' e'Q-% var(6) = G2 (e'Q le)(a'Q ' a ) - (e'Q 'a) 2

(2.4)

.

If the generating density function f(x) is symmetric about the origin and the sample is either complete or, if there is a censoring, the censoring is symmetric about the median, then it can be shown that e/Q-la = 0 (cf David 1981; Arnold, Balakrishnan, and Nagaraja 1992). In this symmetric case, the weighted leastsquares estimators given in (2.2) and (2.3) can be further simplified as /l = e t Q - 1 X / e ' Q - l e ,

6- = a ' Q - 1 X / a ' Q - l a ,

(2.5)

and the corresponding variances are 002

(;2

var(/~) -- e,Q_le ,

var(6-) - a,Q_la .

(2.6)

The best unbiased L-estimators based on the n fixed order statistics given in (2.2) and (2.3) can also be obtained directly by minimizing variances. Let us consider the estimation of 0-. Suppose that p'X is an unbiased estimator. The unbiasedness leads to the two conditions: p'e = 0 and p'a = 1, for p. The variance of p'X is E(ptX

_ 0.)2

pt(002Q + (e + 00a)(e + 0-a)')p

-

0-2 =

p1Qp 002 .

Thus, the best unbiased linear estimator is the one that minimizes p~Qp subject to the aforementioned conditions, or equivalently, the one obtained by minimizing p'Qp - 22p'e - 6(p'a - 1) with respect to p and using the conditions on p. By taking the first partial derivative with respect to p, and equating it to zero, we have

Estimation o f scale parameter based on a fixed set o f order statistics

Qp-2e-c~a=O

163

.

Hence p = Q-l(2e + 6a) ,

(2.7)

where 2e'Q-le+6e'Q-la=O,

2a'Q l e + S a ' Q l a = 1 ,

because of the conditions on p. Solving 2 and 6 from the last two equations and putting the solutions back into (2.7), we get the best choice of the coefficient vector p and hence the best unbiased L-estimator of a as is given in (2.3). Lloyd's estimators are highly efficient. For some location-scale families, such as those generated by 1 and 4 mentioned in Section I, they are in fact the uniformly minimum variance unbiased estimators (UMVUE) (see Arnold, Balakrishnan, and Nagaraja 1992). The processes of computing the best unbiased L-estimators are often easier than that for some other types of estimators, such as the maximum likelihood estimators, because for most generating distributions these latter estimators do not have closed-form and have to be obtained through numerical methods. Since the coefficient vector p depends only on the generating distribution function, not the parameters, tables are available for many commonly used location-scale families. See Sarhan and Greenberg (1962) and references given in David (1981), Balakrishnan and Cohen (1991). The scale estimator given in (2.3) is often used in the context of life testing and reliability (Lawless 1982; Harter 1988; David 1981; Arnold, Balakrishnan, and Nagaraja 1992; Nelson and Hahn 1972, 1973). Another important application of is in testing of distributional assumptions. For non-censored sample, Shapiro and Wilk (1965) compared this best unbiased L-estimator with the usual sample sum of squares about the sample mean and constructed the well-known W statistic for testing for normality, which is W=

(a'Q-la)262

(a,Q_2a) ~/N1 (Xi _ ~)2

'

where X~ is the ith order statistic in the sample (1.1) without censoring and J( is the sample average. The distribution of W statistic does not depend on the parameters and its percentage points can be obtained through simulations or approximations (cf Shapiro and Wilk 1968) or by use of some asymptotic results (cf. Leslie, Stephens and Fotopoulos 1986). Recently, Parrish (1992) gave new tables of coefficients and percentage points for the W test. The same technique has also been adapted to develop a test for testing exponentiality (Shapiro and Wilk 1972). The W test has been shown through extensive simulations to be very powerful against a wide class of alternative distributions (Shapiro, Wilk, and Chen 1968; Gan and Koehler 1990). The W statistic is also useful for identifying outliers (Barnett and Lewis 1984; Iglewicz and Hoaglin 1993).

164

s. K. Sarkar and W. Wang

2.2. N e a r l y best L-estimators

The estimator given in (2.3) requires full knowledge of a and Q, the expected vector and the variance-covariance matrix of the given set of order statistics. However, the explicit forms of the expectations and the covariances of order statistics are known only for a few generating distributions, such as the exponential, uniform, power, pareto, and for normal distribution when the sample size does not exceed 5 (see, for example, David 1981; Arnold, Balakrishnan, and Nagaraja 1992). For most generating distributions, we need extensive numerical calculations to obtain the exact values of the expectations and covariances of order statistics. This may be computationally laborious for large sample sizes. Due to this difficulty, Gupta (1952) suggested a very simple method which simply takes Q = I, the n x n unit matrix, in (2.3). The resulting estimator appears to be the usual (unweighted) least-squares estimator of the scale parameter in model (1.3). This estimator eliminates the need of the computation of the covariances between order statistics. It has been noted that this simplified linear estimator of a gives surprisingly good result, at least in the case of normal distribution (Chernoff and Lieberman 1954; Ali and Chan 1964). Unlike Gupta's method, Blom (1958, 1962) proposed an "unbiased nearly best" linear estimator of the scale parameter, denoted by O-approx. This estimator is obtained by replacing qij'S in (2.3) by their approximations, where qij is the (i,j) th entry of Q. To be more specific, suppose that the X/ in (1.1) is the qi th order statistics based on the random sample of size N and define pi = q i / ( N + 1), then we have for large N (David and Johnson 1954), ai ~ F - l ( p i )

and

pi(1 - pj) qij ~" ( N + 2 ) f ( F _ l ( p i ) ) f ( F _ i ( p j ) )

,

(2.8)

where F -1 (.) is the inverse of the generating distribution F and f is its density function. Then, O'approx is obtained by replacing the elements of Q in (2.3) by the approximations given in (2.8) (see, e.g., Balakrishnan and Cohen 1991; David 1981). This approximate estimator is also asymptotically normal and relatively easy to compute (see, Balakrishnan and Cohen 1991, p. 217 for calculation of Q-l). The positivity of this approximate estimator was also proved by Bai, Sarkar, and Wang (1996). Like Gupta's simplified estimator, the "unbiased nearly best" linear estimator still requires the exact expectations of the order statistics. However, if the unbiasedness is given up, one may also approximate asymptotically the expectations ai's by their approximations given in (2.8) and obtain a "nearly unbiased, nearly best" linear estimator of the scale parameter. 2.3. Other L-estimators

Many other L-estimators of the scale parameter have been introduced in the literature. For interested reader, we refer to the books written by David (1981), and by Balakrishnan and Cohen (1991).

Estimation of scale parameter based on a fixed set of order statistics

165

3. The positivity of the best unbiased L-estimator Being an estimator of a positive quantity, the best unbiased L-estimator 6 should also be positive. But, is it in general positive with probability one? The answer to this question is not apparently clear, because it is a usual weighted least-squares estimator of a regression parameter which, even if the regression parameter is known to be positive, may not turn out to be always positive. The alternate approach to obtain this best unbiased L-estimator, as shown in Section 2.1, does not also guarantee the positivity of the estimator. The positivity appears to depend on properties of the underlying generating distribution. It is easy to see that 6 is always positive when n = 2. Note that a necessary and sufficient condition for an L-statistic, p'X, to be an unbiased estimator of the scale parameter is that pie = 0 and p'a = 1. If n = 2, that is, only two observed order statistics (X1 _< X2) are available, then there exits a unique unbiased L-estimator of the scale parameter obtained by solving the equations Pl + p 2 = 0, and p l a l +p2a2 = 1, which is given by

x2-x~ a2 -- al This estimator, of course, is also the best unbiased L-estimator and clearly it is positive with probability one. In the case when n is greater than two, there exist numerous unbiased L-estimators of a. M a n y of them can be negative with probability greater than zero. For example, when n = 3, all unbiased L-estimators can be expressed as

1 - - [X3 - X l q-p2((a3 - al)X2 - (a2 - al)~3 - (a3 - a2)X1)] , a3 - al (3.1) where p2 can be any real number. Obviously, in this case, not all unbiased L-estimators are positive with probability one. The best choice for/)2 which guarantees that 6- has the minimum variance is related to the second moments of the order statistics X1 ,X2 and X3. The positivity of the best unbiased L-estimator obtained in this way is not clear. As mentioned before, it seems to depend on the properties of the generating density function. If it is assumed that the generating density function is symmetric about the origin and that a m o n g the three observed order statistics, X1 ~__22 ~ X3, X1 and)(3 are symmetrically placed with respect to the median X2, then the estimator given in (3.1) with p2 = 0 is the best unbiased L-estimator of a, which again is indeed positive with probability one. In general, we have THEOREM 1 (Bai, Sarkar and Wang, 1996). The best unbiased L-estimator of the scale parameter given in (2.3) is positive with probability one if at least one of the following conditions is satisfied:

166

s. K. Sarkar and W. Wang

1) The generating density function is log-concave; 2) n = 2; 3) n = 3 and the generating density function is symmetric about the origin and the censored sample is symmetric about the median. Under the same conditions stated in the theorem, Blom's "unbiased, nearly best" and "nearly unbiased, nearly best" linear estimators as given in Section 2.2 are also positive with probability one. This theorem is a special form of a more general result proved by Bai, Sarkar and Wang (1996) in the context described in Section 5. Note that the most important condition in the theorem is that the generating density function is log-concave. A non-negative real function f ( x ) , defined on an interval subset A on the real line, is said to be log-concave in x on A if f ( x - y ) f ( x ' - y') >_ f ( x - y ' ) f ( x ' - y) forall x < _ x t, y < yl, x , y , xr,y, E A

.

An equivalent definition is that f ( x ) satisfies the inequality f(~x+(1-e)x)>f~(x)fl-~(y),

for all 0 < c ~ < 1 and x , y E A

.

From the above definition, f ( x ) is a log-concave function on (-co, oe) if and only i f f ( x - y) is a TP2 (totally positive of order 2) function on the real plane. Karlin (1968) gave a comprehensive treatment of TP2 probability density functions, and Das Gupta and Sarkar (1982) obtained some general results providing interrelationships between TP2 and log-concave properties of certain probability densities. The log-concavity property is shared by many commonly used generating density functions, such as uniform, exponential, normal, gamma (with the shape parameter greater than one), beta (with the parameters greater than one), double exponential, logistic. 4. Nonlinear estimators

In this section, we introduce three non-linear estimators of the scale parameter a. 4.1. A n estimator based on residuals

Since, as mentioned in Section 1, the scale parameter a appears not only as a regression coefficient parameter but also as a parameter in the covariance matrix in model (1.3), it is possible to have an alternate estimator of a through the weighted residual sum of squares obtained by fitting this model. Such possibility was hinted in Shapiro and Wilk (1965) and Nelson and Hahn (1972). Let R= ~(X-Afl)tQ

I(X-Afl) (4.1)

: v/X'(Q -1 - Q-1A(A'Q-IA)-IA'Q-1)X ,

Estimation of scale parameter based on a fixed set of order statistics

167

where fl = (/~, 6)' and A are given in (2.1). Since R2/(n - 2 ) is an unbiased estimator of 62, an estimator based on R seems to be reasonable for estimating 6. Shapiro and Wilk (t965) used R 2, instead of the sample variance, to derive a test for normality in case of incomplete samples. However, they did not get into any details either in that paper or in any subsequent publications. The idea of using R as an alternative to estimate 6 was also mentioned by Nelson and Hahn (1972) without any details. Here, we attempt to study such an estimator in a little more details. We consider the estimator with minimum mean squared error among the class of estimators {kR, 0 < k < oc}. Let r = ~ / y ' (Q-1 - Q - ' A ( A ' Q - I A ) - I A ' Q - ' ) y

(4.2)

where Y = (X - / r e ) / 6 . The distribution of r, which is same as that of R/6, depends only on the form of the generating distribution function F but not on parameters. It is easy to see that ^'~

6 R ~-

nE(@2R

(4.3)

has the minimum mean squared error in {kR, 0 < k < oc}. The expectation of r can be simplified and given an explicit, closed-form representation for uniform and exponential distributions. These are derived later in this section. Such derivation for most other generating distribution functions is, however, difficult and needs to be worked out using Monte Carlo simulation. Values of E(r) have been calculated in Table 2 and 3 for some families of location-scale distributions and different sample sizes.

4.2. A combined estimator Now that we have two different estimators of 6, one is given in (2.3) and the other in (4.3), one might want to develop yet another estimator by having the "best" linear combination of them. Let zz

e'Q -1 (ea' - a e ' ) Q - l Y ( e t Q - l e ) ( a t Q - l a ) - (etQ-la) 2 '

(4.4)

and Er 2 -- ErE(zr) CI = Er2Ez2 _ E2(zr)

z

(n - 2) - ErE(zr) (n - 2)Ez 2 - E2(zr)

E z 2 E r - E(zr) EzaEr - E(zr) Cr = Er2Ez2 _ E2(zr) = (n - 2)Ez 2 - E2(zr)

(4.5)

(4.6)

168

S.K.

Sarkar and IV. Wang

Then the following combined estimator (4.7)

OC = C16 q- G R

has the minimum mean squared error among all such linear combinations of & and R, and hence, provides an improvement over both & and &~' in the sense of having smaller mean squared error. However, like &~', &c is also a biased estimator of the scale parameter, and the bias is given by (4.8)

Bias(&c) = [CI + C~E(r)- 1] 0- . The mean squared error of this combined estimator is given by MSE(&c) = [(1

-

C 1 -

GE(r)] 0-2

(4.9)

.

The combined estimator has negative bias since Bias(#c) = - M S E ( i r c ) / a < O. In order to see the extent of improvement of &c over ~, we have performed a Monte Carlo study for families of normal and logistic distributions on an IBM 3090 mainframe. The constants C1, G, and E(r) were first estimated through 100,000 pseudo-random samples and the ratio P of the mean squared error of the resulting combined estimator to the variance of the best unbiased L-estimator was then computed. The program was written in F O R T R A N language, calling some IMSL subroutines for pseudo-random numbers generation. These calculations are reported in Tables 1, 2 and 3 for certain sample sizes, complete as well as doubly censored. Although the constants C1 and Cr for doubly censored sample from exponential with n observed order statistics are exactly known, which are (n - 1)In and 0, respectively, with the ratio P being exactly (n - 1)/n, we carried out our simulation also for this distribution and reported the results in the tables only to point out that the closeness of these simulated constants to their known actual values gives credence to the accuracy of the simulation results for normal and logistic. The improvement of &c over 6- is encouraging, particularly for small samples. We have also made a comparison of the combined estimator with the maximum likelihood estimator for logistic distribution. The maximum likelihood estimator requires solving the following system of nonlinear equations: rle -rl r2 ~ 2e -r' l + e -r~ ~ l + e r, t - n - 2 _i=1, , -1- -+- 0e - Z --/,/

r l Yl e - r~ q - r2Yn 1 + e -r~

1 q- e -Y.

+ ~ i=1

Yi - ~'~ 12~-e-~ + e---------ff= 0 ' i=1

where Y~is the (rl + i)th standardized order statistic (i = 1 , . . . , n; rl is the number of observations censored from left and r2 is the number from right), and is computationally intensive like 6 o But, unlike 6c, the maximum likelihood esti-

Estimation of scale parameter based on a fixed set of order statistics

169

Table 1 Values of C1, Cr and P for the combined estimator for families of normal, logistic and exponential distributions N

n

rj

r2

Logistic

Normal

Exponential

C1

Cr

P

C~

C~

P

C1

C~

P

3

3

0

0

0.7337

0.0670

0.7795

0.7123

0.0511

0.7478

0.6652

0.00

0.6667

4

3 3 4

0 1 0

1 0 0

0.7271 0.7298 0.7768

0.0547 0.0510 0.0592

0.7653 0.7657 0.8411

0.7065 0.7092 0.7724

0.0453 0.0416 0.0373

0.7382 0.7385 0.8141

0.6648 0.6571 0.7494

0.00 0.01 0.00

0.6667 0.6666 0.7500

5

3 3 3 4 4 5

0 1 2 0 1 0

2 1 0 1 0 0

0.7240 0.7243 0.7235 0.7764 0.7769 0.8024

0.0465 0.0373 0.0472 0.0511 0.0507 0.0524

0.7569 0.7509 0.7568 0.8327 0.8327 0.8749

0.7021 0.7088 0.7037 0.7683 0.7657 0.8071

0.036t 0.0386 0.0339 0.0372 0.0394 0.0316

0.7276 0.7361 0.7277 0.8098 0.8095 0.8523

0.6653 0.6656 0.6708 0.7468 0.7512 0.8009

0.00 0.00 0.01 0.00 0.00 0.00

0.6667 0.6667 0.6667 0.7500 0.7500 0.8000

6

3 3 3 3 4

0 1 2 3 0

3 2 1 0 2

0.7183 0.7201 0.7217 0.7212 0.7781

0.0458 0.0308 0.0287 0.0419 0.0442

0.7506 0.7422 0.7423 0.7509 0.8272

0.7009 0.7064 0.7101 0.6983 0.7678

0.0256 0.0320 0.0268 0.0291 0.0319

0.7191 0.7291 0.7294 0.7190 0.8036

0.6669 0.6655 0.6685 0.6633 0.7489

0.00 0.00 0.00 0.00 0.00

0.6667 0.6667 0.6667 0.6667 0.7500

4 4 5 5 6

1 2 0 1 0

1 0 1 0 0

0.7826 0.7812 0.8060 0.8081 0.8250

0.0370 0.0416 0.0455 0.0441 0.0440

0.8243 0.8276 0.8698 0.8702 0.8972

0.7700 0.7662 0.8163 0.8087 0.8393

0.0349 0.0333 0.0240 0.0292 0.0226

0.8091 0.8034 0.8513 0.8507 0.8781

0.7541 0.7521 0.8027 0.7932 0.8349

0.00 0.00 0.00 0.00 0.00

0.7500 0.7500 0.8000 0.8000 0.8333

3 3 3 3 3

0 1 2 3 4

4 3 2 1 0

0.7184 0.7158 0.7155 0.7192 0.7196

0.0395 0.0285 0.0252 0.0238 0.0378

0.7464 0.7362 0.7337 0.7364 0.7465

0.6966 0.7027 0.7079 0.7034 0.6967

0.0221 0.0276 0.0241 0.0266 0.0219

0.7123 0.7223 0.7252 0.7224 0.7123

0.6681 0.6683 0.6712 0.6645 0.6662

0.00 0.00 0.01 0.00 0.00

0.6667 0.6667 0.6666 0.6667 0.6667

4 4 4 4 5

0 l 2 3 0

3 2 1 0 2

0.7790 0.7810 0.7855 0.7780 0.8102

0.0395 0.0329 0.0290 0.0403 0.0396

0.8232 0.8181 0.8184 0.8231 0.8663

0.7694 0.7710 0.7714 0.7640 0.8063

0.0253 0.0303 0.0299 0.0300 0.0281

0.7980 0.8050 0.8050 0.7976 0.8466

0.7449 0.7482 0.7480 0.7528 0.7984

0.00 0.00 0.00 0.00 0.00

0.7500 0.7500 0.7500 0.7500 0.8000

5 5 6 6 7

1 2 0 1 0

1 0 1 0 0

0.8198 0.8105 0.8316 0.8330 0.8410

0.0315 0.0394 0.0375 0.0367 0.0383

0.8652 0.8663 0.8941 0.8943 0.9125

0.8095 0.8086 0.8368 0.8393 0.8543

0.0285 0.0265 0.0235 0.0221 0.0212

0.8506 0.8468 0.877l 0.8773 0.8955

0.7996 0.8013 0.8305 0.8367 0.8534

0.00 0.00 0.00 0.00 0.00

0.8000 0.8000 0.8333 0.8333 0.8571

7

Note: N is the sample size, n is the number of observed order statistics, rl is the number of observations censored from left, and r2 is the number of observations censored from right.

mator

has some added disadvantages.

computational

As mentioned

in Harter

and Moore

difficulties may arise if the initial values are wrongfully

solving the maximum

likelihood

equations.

(1967),

guessed for

Also, the rate of convergence

for the

170

S. K. Sarkar and W. Wang

Table 2 Values of E(r), Cb C,,, Bias/a and P for the combined estimator for families of normal and exponential distributions with N= 10

rl r2 Normal

Exponential

E(r)

C1

Cr

2.6793 2.4823 2.2749 2.2727 2.0507

0.8942 0.8747 0.8650 0.8487 0.8511

Bias/~

P

E(r)

C1

Cr

Bias/or

P

0 0 0 1 0

0 1 2 1 3

0.0192 -0.0543 0.0249 , -0.0634 0.0264'-0.0749 0.0333 -0.0756 0.0289 -0.0896

0.9426 0.9319 0.9208 0.9173 0.9063

2.5963 2.4003 2.2003 2.2043 1.9763

0.9159 -0.0061 -0.1000 0.8998 0.8843 0.0019 -0.1111 0.8889 0.8773 -0.0011 -0.1250 0.8750 0.9056 -0,0139 -0.1249 0.8744 0.8502 0.0035 -0.1429 0.8571

1 0 1 2 0

2 2.0451 0.8555 0.0258 -0.0917 4 1.8029 0.8368 0.0297 -0.1098 3 1.7959 0.8453 0.0231 -0.1133 2 1.7929 0.8415 0.0247 -0.1142 5 1.5201 0.8126 0.0322 -0.1385

0.9050 0.8871 0.8849 0.8836 0.8588

1,9808 1.7310 1,7322 1.7323 1.4532

0.8633 -0.0031 -0.1429 0.8571 0.8172 0.0093 -0.1666 0.8332 0.8238 0.0055 -0.1667 0.8333 0.8377 -0.0025 -0.1667 0.8333 0.7946 0.0037 -0.2000 0.8000

1 2 0 1 2

4 3 6 5 4

1.5115 1.5121 1.1857 1.1747 1.1741

0.8128 0.8290 0.7784 0.7732 0.7843

0.0287 0.0164 0.0324 0.0302 0.0175

-0.1439 -0.1462 -0.1832 -0.1914 -0.1951

0.8540 0.8532 0.8150 0.8071 0.8044

1.4574 1.4506 1.1271 1.1340 1.1276

0.8270 -0.0184 0.7830 0.0117 0.7522 -0.0019 0.7666 -0.0146 0.7466 0.0030

-0.1999 -0.1999 -0.2500 -0.2499 -0.2500

0.7996 0.7998 0.7500 0.7498 0.7500

3 0 1 2 3

3 7 6 5 4

1.1712 0.7493 0.7371 0.7334 0.7358

0.7972 0.7146 0.7098 0.7069 0.7060

0.0056 0.0324 0.0222 0.0187 0.0169

-0.1963 -0.2611 -0.2738 -0.2793 -0.2816

0.8037 0.7378 0.7257 0.7204 0.7182

1.1256 0.7054 0.7072 0.7060 0.7055

0.7409 0.0081 0.6680 -0.0020 0.6591 0.0107 0.6604 0.0089 0.6683 -0.0024

-0.2500 -0.3333 -0.3333 -0.3333 -0.3333

0.7499 0.6667 0.6666 0.6666 0.6667

solution o f these e q u a t i o n s depends heavily on the a m o u n t o f censoring. O n the o t h e r hand, as o u r s i m u l a t i o n results (in T a b l e 3) indicate, 6c and the m a x i m u m likelihood e s t i m a t o r h a v e similar p e r f o r m a n c e in terms o f bias and m e a n sq u ar ed error. Hence, because o f the c o m p u t a t i o n a l p r o b l e m s associated with the maxim u m likelihood estimates, &c appears to be a g o o d alternative for estimating the scale p a r a m e t e r for logistic distribution. W e have chosen a sample o f size 10 an d considered different possible types o f d o u b l e censoring in o u r s i m u l a t i o n study. O n e t h o u s a n d samples were g e n e r a te d in each scenario. A modified Powell hybrid m e t h o d was e m p l o y e d by calling the I M S L s u b r o u t i n e D N E Q N J (user has to p r o v i d e the J o c o b i a n ) to solve the m a x i m u m likelihood equations. In all these cases, the initial values for the l o c a t i o n a n d scale p a r a m e t e r s are set to zero an d one, respectively, which are the values used to generate the r a n d o m numbers. It is i m p o r t a n t to p o i n t o u t that H a r t e r and M o o r e (1967) also c o n d u c t e d a M o n t e C a r l o study o f the m a x i m u m likelihood estimators for the p a r a m e t e r s in the family o f logistic distributions t h r o u g h 1,000 simulations o f sample sizes 10 and 20. They, however, used a different p r o c e d u r e to solve the m a x i m u m likelihood equations. W e like to e m p h a s i z e that their simulated biases an d the m e a n sq u ar ed errors for the m a x i m u m likelihood scale e s t i m a t o r are very close to w h a t we have observed in o u r study.

Estimation of scale parameter based on a fixed set of order statistics

171

Table 3 Values of E(r), Cb Cr, Bias/{r and P for the combined estimator, and of Bias/a and P for the maximum likelihood estimator for the logistic distribution with N = 10 n

rl

r2

Combined estimator

ML estimator

E(r)

C1

Cr

Bias/a

P

Bias/a

P

10 9 8 8 7

0 0 0 1 0

0 1 2 l 3

2.6465 2.4593 2.2529 2.2552 2.0299

0.9002 0.8966 0.8770 0.8819 0.8788

0.0108 0.0098 0.0143 0.0128 0.0069

-0.0713 -0.0793 -0.0908 -0.0892 -0.1072

0.9279 0.9202 0.9082 0.9100 0.8927

-0.0550 -0.0595 -0.0959 -0.0829 -0.0887

0.9325 0.8689 0.9303 0.8310 0.9132

7 6 6 6 5

1 0 1 2 0

2 4 3 2 5

2.0308 1.7753 1.7868 1.7875 1.4899

0.8547 0.8288 0.8411 0.8479 0.8061

0.0204 0.0233 0.0188 0.0157 0.0207

-0.1039 -0.1299 -0.1254 -0.1240 -0.1631

0.8944 0.8684 0.8736 0.8752 0.8359

-0.1012 -0.1249 -0.1279 -0.1301 -0.1453

0.8619 0.8615 0.9169 0.8525 0.8100

5 5 4 4 4

1 2 0 1 2

4 3 6 5 4

1.5002 1.5036 1.1606 1.1673 1.1680

0.8202 0.8076 0.7743 0.7762 0.7785

0.0150 0.0249 0.0099 0.0145 0.0156

-0.1574 -0.1550 -0.2143 -0.2069 -0.2032

0.8421 0.8436 0.7856 0.7928 0.7964

-0.1514 -0.1640 -0.2114 -0.2295 -0.2118

0.8594 0.8443 0.7963 0.7761 0.8124

4 3 3 3 3

3 0 1 2 3

3 7 6 5 4

1.1660 0.7240 0.7305 0.7309 0.7321

0.7797 0.6802 0.6903 0.6845 0.6917

0.0157 0.0269 0.0240 0.0370 0.0286

-0.2020 -0.3004 -0.2922 -0.2884 -0.2873

0.7976 0.6990 0.7073 0.7103 0.7119

-0.2202 -0.2969 -0.2824 -0.3091 -0.3295

0.7953 0.7258 0.7087 0.7254 0.6835

4.3. A n o t h e r possible nonlinear e s t i m a t o r W e p r o p o s e a n o t h e r n o n - l i n e a r e s t i m a t o r o f a in t h i s s u b s e c t i o n . L l o y d ' s b e s t unbiased L-estimator was obtained by minimizing the usual weighted leastsquares objective function, namely, (X

-/~e

-

~a)'Q -1 (x

-

~e

-

aa)

.

This objective function does not take into account the fact that Var(X), the m a t r i x u s e d i n w e i g h i n g t h e r e s i d u a l s , a l s o i n v o l v e s a 2, a n d it a p p e a r s t h a t a m o r e n a t u r a l o b j e c t i v e f u n c t i o n is h o t , a ) = ~-f2(X - # e - a a ) t Q

l(X -/~e - aa)

.

(4.10)

W e d e v e l o p a n e s t i m a t o r o f a b a s e d o n t h i s o b j e c t i v e f u n c t i o n . I n p a r t i c u l a r , we f i n d (~/, &t) t h a t m i n i m i z e h ( # , a ) , a n d p r o p o s e ~t, o r a m u l t i p l e o f it, as a n a l t e r n a t e e s t i m a t o r o f o-. T h e first p a r t i a l d e r i v a t i v e s o f h ( # , a ) w i t h r e s p e c t t o # a n d a, r e s p e c t i v e l y , a r e

172

S. K. Sarkar and W. Wang

0h(u, o) _

0g

2 e , Q _ 1 ( x - g e - o-a)

0-2

Oh(g, a)

_

00-

2 (X - ge - a a ) ' Q l(X - ge - o-a) 0-3

2 a,Q_ l (X - ge - o-a)

0-2

Equating these two derivatives to zero, we have e ' Q - ' (X - ge - o-a) = 0 , (x - ge)'Q-'

(X - ge -

(4.11)

o-a) = 0 .

(4.12)

Multiplying (4.11) by g and adding to equation (4.12), we get X ' Q 1(X - ge - o-a) = 0 .

(4.13)

The solutions to equations (4.11) and (4.13) are given by a ' Q - l ( e X ' - X e ' ) Q 1x X'V1X /2t = - e,Q_l(ea, _ a e ' ) Q i x - e'V2X '

(4.14)

e'Q l ( e x ' - Xel)Q-1X X/V3 x &~ = e ' Q - l ( e a ' - a e ' ) Q 1x - a'V3X

(4.15) '

where Vl = Q 1

Q-lae, Q 1 etQ - l a (= Q-laelQ

v2 = Q - l

Q -el,eQa _' Ql a -1 ( = Q - l e a ' Q -1, i f e ' Q

V3 = Q - I

Q lee'Q -1 e'Q le

1, i f e i Q 1a = 0) , la = 0) ,

and (4.16)

Note that the best unbiased L-estimators of g and 0-, as given in (2.2) and (2.3) can be rewritten as

/2 = 6 =

a ' Q I(eEX' E X e ' ) Q 1X (EX)'V1X e'Q - l ( e a ' - a e ' ) Q IEX - e'V2(EX) ' e ' Q - I ( e E X ' - E X e ' ) Q 1X (EX)'V3X e ' Q - l ( e a ' - a e ' ) Q 1EX a'V3(EX) '

(4.17)

(4.18)

where EX is the expectation vector of X. It is of interest to observe that the only difference between/2 and/2t is that EX in formula (4.17) is replaced by X in (4.14). Similar difference exists between & and 6t in formulae (4.18) and (4.15).

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Estimation o f scale parameter based on a f i x e d set o f order statistics

Next, we need to show that ~t and #t as given in (4.14) and (4.15) are really the minima for the objective function h(#, o-). Since/~t and &t satisfy equations (4.11), (4.12) and (4.13), we have 0,22h(~ ' 2 , _~ o# o-),,=/,.... <=~2t2eO e ,

o-0# 02 02 O--a-h(#, o-) u=g,,~=<-- ,,----a-h(#, o-) {~=~,,a=6-,= °72 =--4e, Q 1a o#o,oand

O04h(#,o-) a=~,,,~=a-,: &72 a, O _1( X - ,hie) = ,5.22 ( X -

~te)'Q-l(X- ~te) a)

Thus, the Hessian matrix is that 2 (&te'Q-le n = = ~o.-3 &t e'Q - l a

&te'Q-la ) (X-flte)'Ql(x-~te)

•

dt

Noting from (4.11) that #t e'Q-la = e'Q-1X - ~t e'Q -le , we have e'Q-le (X - [,te)'Q l ( x -- hte ) - (&t e'Q la)2 =e'Q leX'Q 1X-(e'Q-1X) 2 , which is always greater than or equal to zero (Cauchy-Schwartz inequality). We also know that it equals zero if and only if X = c e for some constant c. Since X is the vector of order statistics, P(X = c e) = 0 for any c. Hence, the Hessian matrix H is positive definite with probability one. In other words, we prove that/)t and &t minimize the function h(#, o-) given in (4.10). If the generating density function is symmetric around the origin and the censoring is also symmetric,/~t and &t can be further simplified as e'Q ix ) t - - e,Q_le,

x ' ( Q-1 Qe'Q-'elee'QI'~x.] &t = a,Q_lX

Therefore, ~t = ~ but R2

&t-- a,Q-la ~- F& ,

(4.19)

S.K. Sarkar and W. Wang

174

where R is given in (4.1). In this case and under the conditions given in Theorem 1, #t is always greater than 6- and has a positive bias. 4.4. E x a m p l e s

The estimators given in the previous sections can be simplified for some special families of location-scale distributions. Some examples are given below. EXAMPLE 1. Let X be a complete sample of order statistics from the normal distribution with mean # and standard deviation o-. Then, we have Q le = e and a1Q-le = 0. Consequently, the best unbiased L-estimators of # and a given in (2.2) and (2.3) are now simplified as /2 = X ,

6 = a'Q-1X/a'Q

la .

The estimators given in (4.17) and (4.18) are now /~t = X ,

&t =

X'Q 1X - N X 2 a,Q_lX

As for the residual-based estimators, not much simplification can be obtained except that the R given in (4.1) and z in (4.4) can be rewritten as / R = VX'Q-1X

- NX 2 - a'Q-la

~-2

and

z

_ a'Q -IY atQ-l~

EXAMPLE2. Consider the family of two-parameter uniform distributions with the generating density function if -½_<x_<½ .

f(x)=l,

Let Xk+l < . . . < XN k be a doubly Type II censored symmetric sample from a two-parameter uniform distribution with k censored observations each side. If the suffixes of the a vector and the Q matrix start from k + 1, then 1 i ai=-~+N+l,

i=k+l,...,N-k

,

and for k + 1 < i < j < N - k , qij =

i(N - j + 1)

(N + 1)2(N + 2)

Let qij denote the (i,j)th element of Q-1. Then, for j > i, f-l, j=i+l, i=k+l,...,N-k, ---- ~ 2, j = i = k + l , . . . , N - k , (N+ 1)(X+2) 0, j > i + l qij

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Estimation o f scale parameter based on a fixed set of order statistics

(See Arnold, Balakrishnan, and Nagaraja 1992). So e'Q -1 = ( N + 1)(N+ 2)(1,0,0,...,0, 1) ,

a'Q-1 = ( N + I ) ( N + 2 )

1

k

1

N-k+

-~+~-@--~,0,0,...,0,-~-~

e ' Q - l e - 2 ( N + 1)(N+ 2),

N+I

1.)

e'Q-la = 0 ,

e'Q-1X = (N + 1)(N + 2)(X~+I + X u - k )

,

a'Q-IX = (N + 1 - 2k)(N + 2)(XN k --Xk+l)/2 , i/ N - k

2

N-k-1

XtQ-1X = 2(X + 1)(X + 2)~i=k~+lX/ -- i=k+l ~ X/Y/+l

/

Thus, =

+ XN-k 2 '

=-o+1

=

(N+ 1) (XN-k ~

Xk~ l ~

N-k-1

'

and (~-'~N-k ~i y2 4(N + 1 ) \z--,i=k+l

O't--

-- ~z-~i=k+l " ~ N - k - 1 Y i X i + l _ ~2~

( N + 1 - 2 k ) ( X N - k -- Xk)

By using the above expressions, we can also simplify the residuals-based estimators, &~ and/rc. EXAMPLE 3. Let Xr,+l <<_'" <--XN-r2 be a doubly Type II censored sample available from the two-parameter location-scale exponential distribution and Yr,+I <_ " " <<_Yu-r2 be the corresponding standardized order statistics. Let Wrt+l =bYrd+l,

Wii = ( N - i +

1)(Y/- Y/ 1),

i=rl

+2,.

where b=

1)2 2_,i=1 V/x-•r,+l 1/(N - i +

Then, W/'s are independent and r~+l E(Wr,+1) = c = b ~ 1 / ( N i=1

i + 1),

Var(W~,+l) = 1 ,

,U-r2

,

176

S. K. Sarkar and W. Wang

E(Wi)=I,

Var(Wi)=l,

i=rl+2,...,N-r2

.

Let D be the t r a n s f o r m a t i o n matrix such that W = DY. Thus, D Q D ' = I, D A = ( D e , D a ) =

0

e

"

V " N - ~+2 W ~ / ( n - 1). Then, the best unbiased L-esLet n = N - rl - r2 and W0 = z__.i:~, timator of the scale p a r a m e t e r given in (2.3) can be rewritten as

6 =

(De)'(DQD')

l[De(Da)' - D a ( D e ) ' ] ( D Q D ' ) - I D X

( ( D e ) t ( D Q D ' ) - I De) ( ( D a ) ' ( D Q D ' ) - I D a ) - ( ( D e ) ' ( D Q D ' ) - I D a ) 2 = (De)1[De(Da)'-

= Woa

Da(De)'IDX

b2(n-

1)

N-r2

=

,

Z (X-i+l)(Xi-Xi_i)/(n-1) i=rl +2

and r given in (4.2) simplifies to

r=

=

W'(I - DA(A'D'DA)-IA'D')W =

I

I,/° W

0

o' i _ @ i ee,

) W

\z~i=r, +2 i} w, 2 -

~-]

:

(n -

1)sv~,

i=rl +2

where N-r2 E Vi2 -

$2 =

Wi

1/(n -

1),with V/= v,N_r2

i=rl+2

,

m--d=rl+2 W/ i = rl + 2, . . . , N - r2 .

Since N ' s are independent standard exponential variables (i = rl + 2 , . . . , N - r2), ~ ' s have the joint density function l

f ( < +2,. • •, ~N-r2) -- -(n -- 2)! N-r 2

if Z

vi=

1, a n d 0 _ < vi_< 1, i = r l + 2 , . . . , N - r 2 ,

i=rl +2

and is independent of W0 (see, for example, R a o 1973, p. 215). It is easy to see that R = ra = (n-

1)SvW0a = ( n -

1)Sv& .

177

Estimation o f scale parameter based on a f i x e d set o f order statistics

For n : 3, we have

E(r)

V~EIW2- v/31

=T =T

Ix-yle X-Ydxdy

=

e x

( y - x)e Ydy] dx

(x - y ) e -y dy +

2

v~ 2

/o

e-X[x + e -x - 1 + e X]dx

2 So, 6~ = ~ - R

.

In this case, the mean squared error for 6~ is 1 o2 which is equal to the variance of 6. Similar to the derivation of 6, we have {7t :

~N-r2 mi 2 i=rl+2 G x_~N_r ~ -Z-ai=rl -+2 Wi

N r2

(n-1)Woo- ~

N r2

Vta=(n-1)& Z

i=rl +2

~2

i=rl +2

and (DX)' (I - D,(oe)"~DX /)'

(De)'(I

bc / =v,+~ -c~~,r ~ )DX ba'

De(Da)'

5. Extensionof the positivity results to censored scale regression model 5.1. Censored scale regression m o d e l

Let us first introduce a censored linear model which includes model (1.3) as a special case. Suppose that there are k different populations with distribution functions from the same location-scale family Cg(F), having a common scale parameter a(> 0) but having different location parameters. The ith population has the location parameter #i = Z~/~, where Zi's are known p x 1 vectors of covariates,/~ (p x 1) is a vector of unknown coefficient parameters. Both ~ and a are unknown constants across all the k populations. Suppose that a random sample of size Ni is drawn from the ith population and that only ni of the order statistics of the sample are observed and the remaining (N~ - ni) observations are

178

s. K. Sarkar and W. Wang

censored. Let Xi = (X/l,... ,X/hi)', where X/1 _< ..- _< X,.n, are the ni observed order statistics for the ith population. They may be any fixed set of ni order statistics of the sample and are not necessarily the first n,- order statistics. If we define Yi = (Xi - Z I ~ e i ) / a = (Y/I,..., Y/hi)', where ei = ( 1 , . . . , 1)': n i x 1, then Y~I < "'" < Y/n, become the corresponding n i order statistics based on a random sample of size Ni from the generating distribution function F(y). Suppose that E(Yi) = ai

and

Cov(Yi) = Qi -

Obviously, the ai's and Qi's are free of parameters, depending only on the form of F. Let us denote the n x 1 (n = ~-]~/k.l hi) vector of observed order statistics from the k samples by X = ( X 'l , . , ., Xk), ! t and the corresponding column vector of a~'s by a. Also, let Q = diag(Q1,... , Qk), an n x n block-diagonal matrix whose ith diagonal matrix is Qi. Then, we have E(X)=Rfl+aa

and

Cov(X)=a2Q,

(5.1)

where R = D Z with D = d i a g ( e l , . . . , ek), Z = ( Z 1 , . . . , Zk)'. This model often arises in accelerated life testings. The k populations can be k different testing conditions. Crawford (1970) gave an example of such a situation where a new Class B insulation for electric motors was evaluated using temperature accelerated life testing on 40 motorettes. Ten motorettes were put on test together at each of four temperatures (150°C, 170°C, 190°C, 220°C). At the time the analysis was performed, only some motorettes had failed. In this example, we have four populations corresponding to four different testing conditions (temperatures), and

zl/

=/30 + fl z, ,

where z~ is the ith testing temperature. Model (5.1) has been studied by many authors (Nelson and Hahn 1972, 1973; Hamouda 1988; and for the simple regression model, see Moussa 1972; Leone and Hamouda 1973; Hamouda and Leone 1974). When k = 1, model (5.1) can be reduced to model (1.3). General discussion on model (5.1) can be found in the book written by Balakrishnan and Cohen (1991), 5.2. The B L U E o f the scale p a r a m e t e r

Since a appears as a regression coefficient in this model, the same approach for obtaining the best unbiased L-estimator of cr in model (1.3) can be used (as derived by Nelson and Hahn 1972, 1973) to obtain the following best unbiased L-estimator of the scale parameter when the rank of Z is p _< k: 6 - a'[Q-1 - Q 1 R ( R ' Q - 1 R ) - I R ' Q - ~ ] X at[Q - 1 - Q - I R ( R ' Q - 1 R ) 1R' Q 1]a

(5.2)

It should be noted that the estimation of a is still possible even when the rank of Z is k _
Estimation of scale parameter based on a fixed set of order statistics

179

the same as (5.1) with Z and fl replaced, respectively, by Z * = (ZZ') ½ and fl* = (ZZ')-Izfl. Therefore, & in this case is given by (5.2) with Z replaced by Z*, a k x k matrix of rank k. The estimator given in (5.2) may be computationally laborious to obtain in practice, particularly for large sample sizes. Therefore, Nelson and Hahn (1972, 1973) introduced a two-step method and obtained simple linear unbiased estimators (SLUEs) of the parameters in model (5.1), which require computation of the inverses of smaller order (< n) matrices. However, as noted by Nelson and Hahn (1972, 1973), the SLUEs are generally not the BLUEs. A necessary and sufficient condition for the SLUEs to be the BLUEs were given by Escobar (1986). Another possible way to approximate the BLUEs is to replace the a vector and the Q matrix by their approximations as discussed in Section 2. Like the "nearly unbiased, nearly best" L-estimator of the scale parameter in model (1.3), the estimator obtained in this way should be very efficient if the sample sizes for all the k populations are large. 5.3. Positivity o f the B L U E

The following results concerning the positivity of the best unbiased L-estimator (5.2) were proved by Bai, Sarkar, and Wang (1996). Their proofs are omitted here. THEOREM 2. When the rank of Z is k _
THEOREM 3. When the rank of Z is p < k, the estimator (5.2) is positive with probability one if all the following conditions are satisfied: a) f ( y ) is symmetric about zero; b) the set of order statistics from each population is complete or symmetrically censored; and c) either 2 _< ni <_ 3 for all i or f ( y ) is log-concave in y.

6. Concluding remarks Discussion of some new results and ideas on estimation of the scale parameter in a location-scale family of distributions based on a fixed set of order statistics has been the main focus in this article. In addition to reporting a new result that provides an analytical proof of a conjecture on the positivity of Lloyd's best unbiased L-estimator, we put forward a number of new nonlinear estimators. The result supporting that one could possibly use the combined estimator in place of the maximum likelihood estimator for the purpose of estimating the scale parameter of a logistic distribution is an important finding relating to these new

180

S. K. Sarkar and W. Wang

estimators. With regard to the other nonlinear estimators, we have not attempted here to make a comprehensive study but intend to do so in future research.

References Aitken, A. C. (1935). On least squares and linear combinations of observations. Proc. Roy. Soc. Edinburgh 55, 4248. All, M. M. and L. K. Chan (1964). On Gupta's estimates of the parameters of the normal distribution. Biometrika 51,498 501. Arnold, B. C., N. Balakrishnan and H. N. Nagaraja (1992). A First Course in Order Statistics. John Wiley & Sons, Inc.. Bai, Z. D., S. K. Sarkar and W. Wang (1996). Positivity of the best unbiased L-estimator of the scale parameter with complete or selected order statistics from location-scale distribution. Statist. Prob. Lett. 32, 181 188. Balakrishnan, N. (1992). Handbook o f the Logistic Distribution. Marcel Dekker, Inc., New York. Balakrishnan, N. and A. C. Cohen (1991). Order Statistics and Inference." Estimation Methods. Academic, Boston. Barnett, V. and T. Lewis (1984). Outliers in Statistical Data. 2nd ed. New York, John Wiley & Sons. Blom, G. (1958). Statistical Estimates and Transformed Beta-Variables, Almqvist and Wiksell, Uppsala, Sweden. Blom, G. (1962). Nearly best linear estimates of location and scale parameters. In: A. E. Sarhan and B. G. Greenberg, eds., Contributions to Order Statistics 3446, Wiley, New York. Chernoff, H. and G. J. Lieberman (1954). Use of normal probability paper. J. Amer. Statist. Assoc. 49, 778-785. Crawford, D. E. (1970). Analysis of incomplete life test data on motorettes. Insulation/Circuits, 16, No. 11, 4348. Das Gupta, S. and S. K. Sarkar (1982). On TP2 and log-concavity, Inequalities in Statistics and Probability, IMS Lecture Notes-Monograph Series 5, 54-58. David, H. A. (1981). Order Statistics (2nd ed.). New York, Wiley. David, F. N. and N. L. Johnson (1954). Statistical treatment of censored data. I. Fundamental formulae. Biometrika 41,228-240. Escobar, L. A. (1986). The equivalence of regression-simple and best-linear-unbiased estimators with type II censored data from a location scale distribution. J. Amer. Statist. Assoc. 81, 210 214. Gan F. F. and K. J. Koehler (1990). Goodness-of-fit tests based on P-P probability plots. Technometrics 32, 289-303. Gupta, A. K. (1952). Estimation of the mean and standard deviation of a normal population from a censored sample. Biometrika 39, 260-273. Hamouda, E. M. (t988). Inference in regression problems based on order statistics, Comm. Statist. Part A-Theory and Methods 17, 2343-2367. Hamouda, E. M. and F. C. Leone (1974). The O-BLUE estimators for complete and censored samples in linear regression. Technometrics 16, 441 446. Harter, H. L. (1988). Weibull, log-weibull and gamma order statistics. Handbook o f Statistics 7, eds. Krishnaiah and Rao, New York, North Holland. Harter, H. L. and A. H. Moore (1967). Maximum-likelihood estimation, from censored samples, of the parameters of a logistic distribution. J. Amer. Statist. Assoc. 62, 675-684. Iglewicz, B. and D. C. Hoaglin (1993). How to Detect and Handle Outliers. The ASQS Basic References in Quality Control: Statistical Techniques, Vol. 16 Milwaukee, WI: American Society for Quality Control. Karlin, S. (1968). Total Positivity. Stanford University Press, Stanford: CA. Lawless, J. F. (1982). Statistical Models and MethodLfor Lifetime Data. New York, John Wiley.

Estimation o f scale parameter based on a fixed set of order statistics

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Leone, F. C. and E. M. Hamouda (1973). Relative efficiencies of O-BLUE estimators in simple linear regression. J. Amer. Statist. Assoc. 68, 953-959. Leslie, J. R., M. A. Stephens and S. Fotopoulos (1986). Asymptotic distribution of the Shapiro-Wilk W for testing for normality. Ann. Statist. 14, 1497-1506. Lloyd, E. H. (1952). Least-squares estimation of location and scale parameters using order statistics, Biometrika 39, 88-95. Moussa, E. A. (1972). Estimation and robustness of efficiency of linear models for small complete and censored samples. Ph.D. Thesis, University of Iowa. Nelson, W. and G. J. Hahn (1972). Linear estimation of a regression relationship from censored data: Part I - Simple methods and their applications. Technometries 14, 247-269. Nelson, W. and G. J. Hahn (1973). Linear estimation of a regression relationship from censored data: Part II - Best linear unbiased estimation and theory. Technometrics 15, 133-150. Parrish, R. S. (1992). New tables of coefficients and percentage points for the W test for normality. J. Statist. Camp. Simul. 41, 169 185. Rao, C. R. (1973). Linear Statistical Inference and Its Applications. John Wiley & Sons, Inc., New York. Sarhan, A. E. and B. G. Greenberg (Eds.) (1962). Contributions to Order Statistics. Wiley, New York. Sarkadi, K. (1981). On the consistency of some goodness of fit tests. Proc. Sixth Conf. Probab. Theory, Brasov, 1979, Bucuresti 195-204. Ed. Acad. R. S. Romania. Shapiro, S. S. and M. B. Wilk (1965). An analysis of variance test for normality (complete samples). Biometrika 52, 591-611. Shapiro, S. S. and M. B. Wilk (1968). Approximations for the null distribution of the W statistic. Technometries 10, 861 866. Shapiro, S. S. and M. B. Wilk (1972). An analysis of variance test for the exponential distribution (complete samples). Technometries 14, 355-370. Shapiro, S. S., M. B. Wilk and H. J. Chen (1968). A comparative study of various tests for normality. J. Amer. Statist. Assoc. 63, 1343-1372.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 17 © 1998 Elsevier Science B.V. All rights reserved.

"7 [

Optimal Linear Inference Using Selected Order Statistics in Location-Scale Models

M. Masoom Ali and Dale Umbach

1. Introduction

One of the major earliest works in statistical inference using selected order statistics was done by Mosteller (1946). In this pioneering work he advocated estimation of location/scale parameters using a few optimally selected order statistics, particularly for large sample size. These procedures were developed as a compromise between lack of efficiency and quickness and ease of computation. In general, it has been observed that for most distributions eMciencies of 90% or more are achieved with seven or even fewer optimally chosen observations. These estimates are based on linear combinations of the selected order statistics, which are Best Linear Unbiased Estimates. Lloyd (1952) introduced BLUE's to construct linear estimates. Since the coefficients of these linear combinations are functions of means and covariances of order statistics, the estimates can be numerically computed for small sample size. Ogawa (1951) considered the problem of estimating location/scale parameters for large samples and introduced the Asymptotically Best Linear Unbiased Estimates. Sarhan and Greenberg (1962) gave a comprehensive account of the estimation problem using a few selected order statistics which was addressed up until that point in time. Mosteller's (1946) paper, along with Ogawa's (1951) paper and Sarhan and Greenberg's (1962) book, laid the foundation for this area of research. Some of the other works on estimation of parameters are found in Yamanouchi (1949), Kulldorff (1961a,b 1963a,b 1964, 1973), Harter (1961, 1971), Siddiqui (1963), Sfirndal (1962, 1964), Eisenberger (1968), Eisenberger and Posner (1965), Chart (1969, 1970), Chan and Chan (1973), Chan and Cheng (1973, 1988), Chan and Mead (1971a), Chan, Chan and Mead (1971, 1973), Gupta and Gnanadesikan (1966), Hassanein (1968, 1969a,b, 1971, 1972, 1974, 1977), Kaminsky (1972, 1973, 1974), Ogawa (1960, 1976), Saleh (1966, 1967), Saleh and Ali (1966), and Sarhan and Greenberg (1967). The articles by Kubat and Epstein (1980) and Koutrouvelis (1981) are among the first to address the problem of estimating the quantiles of a distribution using

183

M. M. Ali and D. Umbaeh

184

a few optimally selected order statistics. Since then Ali, Umbach and Hassanien (1981a,b), Umbach, Ali and Hassanein (1981a,b), Ali, Umbach and Saleh (1982, 1992), Ali Umbach, Saleh and Hassanein (1983), Saleh (1981), Saleh, Ali and Umbach (1983a, 1985) Saleh, Hassanein and Ali (1992), Umbach, Ali and Saleh (1986), and Umbach and Ali (1996b) have also considered the problem of estimating quantiles. Some works on testing hypotheses and constructing confidence intervals using optimal spacings were also published. Sarhan and Greenberg (1962) address some problems in their book, Saleh, Ali and Umbach (1984), Ali, Umbach and Saleh (1985), Eisenberger (1968), Hassanein, Saleh and Brown (1986a), Saleh, Ali and Umbach (1985), Saleh, Hassanein and Ali (1988), Saleh, Hassanein and Brown (1985), Saleh and Sen (1985) and Umbach, Ali and Saleh (1986) also considered the testing problem using optimal spacing. Finally, Ali and Umbach (1989a) and Umbach and Ali (1990, 1991, 1996a) developed goodness-of-fit tests using a few selected order statistics. Chan and Cheng (1988) mention in their survey paper some earlier works in this field along with examples of applications of optimal spacings. For example, Benson (1949) used two out of seventy observations to estimate the standard deviation of the thread strength in textiles. As another example of application, Eisenberger and Posner (1965) used selected sample quantiles for data compression in space telemetry. These estimators, based on a few selected order statistics, gained popularity in the statistical literature because of their computational simplicity, high efficiency, and robust behavior under departures from distributional assumptions.

2. Preliminaries

Let X I , 352, . . . , X n be a random sample of size n from a population with distribution function G and density function 9. Let XI:~ < X2:n < -.. < X,:, be the corresponding order statistics. Let X,,:,, X,2:n, . . . , X,,:, be k order statistics of arbitrary ranks subject only to the restriction 1 _< nt < n2 < -.. < nk _< n. The joint density of these k order statistics is given by h(Xnl:n,xn2:n,..., Xn+:n) = n! rTk+',

k+l

k

II[G(xni:n)

11i< kni - ni-1 - 1)! i=~

- G(x<

,:,)]

ni--ni_l

--1

I-Ig(x,i:n)

(2.1)

i=1

where no = 0, nk+l = n + l, G(x,0:~) = 0 and G(x,k+t:,) = 1. Now, let Ui = GO(/) for i = l, 2 , . . . , n . Then U1, U 2 , . . . , U, is a random sample of size n from the uniform distribution over (0,1). Let UI:, < U2:, < ... < U,:n be the corresponding order statistics. Then the joint density of U , , : , , U,2:, , . . . , U,+:, for 1 < nl < n2 < . . ' < nk < n is given by I~k+l,

n!

k+l

I'I[ttni:n

i=1 V'v/i - - h i - 1 - - 1 ) ! i=1

--

.~i_t:.]/~i . . . . 1-1

(2.2)

Optimal linear inference using selected order statistics in location-scale models

185

for 0 < un~:n < u,2:n < "'" < Unk:n < 1 with un0:, = 0 and u.k+~:n= 1. With Vi = - ln(1 - U/), for i = 1, 2, . . . , n, V/has an exponential distribution with mean 1. Then the corresponding order statistics, Vl:, < V2:~ < ... < V~:n, have the joint density n!exp -

vi:,

=n!exp

-

i=1

(n-i+l)(vi:,-vi

(2.3)

1:~)

"=

for 0 < vnl:n < Vn2:n < "'" < 1Ank:n < 00, with v0:n = 0. Hence, with Z i = ( n - i + 1)(V/:,- V/-l:n), for i = 1, 2 , . . . , n , Z1, Z2, . ,Z, are independently and identically distributed as exponential with mean 1. Thus V/:n can be expressed as V~:,

Z1 + Z2 n n- 1 +

Zi + n-i÷l

(2.4)

This produces the Renyi (1953) representation of X/:, as Xi:n = a - l ( g i : n ) ~- G-l(1 - exp(-V~:,)) .

(2.5)

This indicates why the exponential distribution plays a central role in the study of order statistics. In particular, this representation has been used by Chernoff, Gastwirth and Johns (1967) to study the asymptotic distributional properties of linear combinations of order statistics. Order statistics arise naturally in the estimation of the quantile function, (2.6)

Q(p) = inf{x]G(x) >_ p } .

Given a random sample of size n from some population with distribution function G, an estimate of G is the empirical distribution function, G~(x), given by Gn(x) =

{o orx .l i-~

1

for X/-I:~ _< x < X/:n for X,:,_<x .

(2.7)

Since G,(x) -+ G(x) in probability as n ~ oc, a natural estimate of Q(p) is given by the sample quantile function, Q, (p), given by Q~(p) = G21(p) := inf{xlGn(x ) >_p} = X~.,.

for -i - < p1_ < /'/

i .

(2.8)

n

Parzen (1979) modified this estimator slightly through interpolation. He suggests using 0,(p)=n (/-p)Xi

l:n+n(p

i-I.)X~.~. /2

for i - 1 < p < _ -i. T/

(2.9)

n

Under the assumption that the form of G is known up to some unknown parameters, efficiencies of the preceding estimators can be greatly improved upon by using a few selected order statistics. In particular, when sampling from a

186

M.

M. Ali and D. Umbach

location-scale model, estimation by the use of selected order statistics is often used. This is typically accomplished by first estimating the location and scale parameters and then using these estimators to form an estimate of Q(p). Thus, estimation of the parameters will be considered at this point. Let X1, X2, . . . , X ~ be a random sample of size n from a distribution with distribution function Fx,~ and density function f~,6, where F~.,~(x) = F ( ( x - 2)/3) and f ~ . , ~ ( x ) = f ( ( x - 2 ) / 3 ) / 6 with location 2 E I R and scale 6 E I R +. Let XI:, <X2:n < ... <X,:, be the corresponding order statistics and let Y,.:, = (X/:n - 2 ) / 6 for i = 1, 2, . . . , n. Then YI:~ < Y2:~ < " " < Y~:n are order statistics corresponding to a random sample from F. Hence, with X/:, = 2 + bY/:,, for i = 1, 2, . . . , n, and E(Y/:,) = ei and Cov(Y/:,, Yj:n) = Yij, E(~:~) = 2 + 6e;

(2.10)

Cov(X/:n,Xj:~) = 62vii .

(2.11)

and

Using these results, one can find the Best Linear Unbiased Estimators (BLUE) of )o and/or 3 based on selected order statistics as follows. Denote the ranks of the k selected order statistics by nl < n2 < " " < nk. Express X~,:~ as X , ~ : , = ) c + 6 c ~ n i + e n ~ for i = 1, 2, . . . , n ,

(2.12)

where E(e~i) = 0 and Cov(e,,, % ) = 32v~,j. In matrix notation, this is X=AO+e=21+6e+e

,

(2.13)

where

x --(x., :., xn2:.,..., xok:.)' A =(1, x)

o

6/ =(c~n,,

~n2, - • •, c%) ~

1 = ( 1 , 1 , . . . , 1) t

e =(ent,em_,

• • • ,enk) t

(2.14)

•

By the Gauss-Markov theorem, the BLUE for O based on Xn,:,, X,2:~,... , Xn~:~ is 6 = (A'V-1A)-IA'V i x ,

(2.15)

where V is the matrix whose elements are vninj. The variance-covariance matrix for is 62(A'V-IA) 1. If 6 is known, then the B L U E of 2 is

= (ltV-1l) l l t v - l ( x -

6~)

(2.16)

Optimal linear inference using selected order statistics in location-scale models

187

with Var()~) = 6 2 ( 1 W 11) 1

(2.17)

If 2 is known, then the BLUE of 6 is

; -~- (o~tv-lo~)-l~tv-i(x - 21)

(2.18)

Vat(3) = 62(0(V-1~¢) i

(2.19)

with

These estimates can all be computed once the first two moments of the order statistics are known. This demonstrates the importance of tables of moments of order statistics. Depending on one's focus and which parameters may be known, one may use a variety of criteria to choose the optimal set of k order statistics at this point. This will be the focus of the next section. Suffice it to say here that analytic results on optimal selection of order statistics are quite rare. The result is the generation of lengthy tables for each possible sample size. To simplify things, asymptotic results are often used. For the asymptotic case, (Pl,P2,... ,p/r) is called a spacing of the k selected order statistics if 0 < pl < P2 < ' < pk < 1. Here, the ranks of the order statistics are obtained from ni = Inpi] + 1, for i = 1, 2, . . . , n, where [.] represents the greatest integer function. For a fixed spacing, the asymptotic distribution of selected order statistics was obtained by Mosteller (1946). THEOREM (Mosteller). Let 0 < Pl < P2 < " " < Pk < 1 be fixed. Suppose that X = (Xn~:~, Xn2:n, . . . , X,~:,) ~ is a vector of k order statistics from a random sample of size n from distribution function G with associated density g, where ni = [npi] + 1, for i = 1, 2, . . . , k. Suppose that g(G-l(pi)) > 0 and that g and g' are continuous in a neighborhood of G-l(pi) for i = 1, 2, . . . , n. Then X h a s a kvariate normal distribution with

E(Xn,:,)=G 1(pi)

for i = l ,

2,..., k

and

(2.20)

Cov(X,,,,X,j,) •

= 1.

"

n

p,(1 - pj)

g(G-l(pi))g(G l(pj))

for i _ < j = l ,

2, . . . , k .

Under the location-scale model, with G = F~.6, note that E(X) = 21 + 6u, where U = (Ul,U2,.--,Uk)

' = (F I(pl),F-1Co2),... , F l(pk))' .

(2.21)

Also, the variance-covariance matrix of X is given (62/n)W, where the (i,j)th element of W is given by ( p i / j ~ ) ( ( 1 - p j ) / f j ) for i<_j, where f,. = f ( u i ) for i = 1, 2, . . . , k. Let B = (1, u). Then the Gauss-Markov theorem yields the Asymptotically Best Linear Unbiased Estimator (ABLUE) of O as

188

M. M. Ali and D. Umbach

= (BtW I B ) - I B t w - I / .

(2.22)

The asymptotic variance-covariance matrix for ~) is (62/n)(B'W-1B) 1. The coefficients of ~) and the elements of W 1 have been expressed in many different ways by different authors. In particular, W -1 is a tri-diagonal matrix. The main diagonal elements can be expressed as + Pi-----1)pi1

fi2~o~

f o r / = 1, 2, . . . , k ,

(2.23)

where p0 = 0 and pk+l = 1. The diagonals above and below the main diagonal have non-zero values where the (i, i - 1) th and the ( i - 1, i)th elements are --J~f-I

for i = 2, 3, . . . , k .

(2.24)

Pi -- Pi 1

All other elements of W -1 are zero. Quantities K1, K2, and K3 are introduced in Sarhan and Greenberg (1962) as

k+l

K1 = I'W 11 = Z

~

-- J~-l) 2

(2.25)

i=1 p i - - P i - I k+l --ft lUi 1)2 K2 = u'W-~u = Z (fiui i--1

(2.26)

Pi -- Pi 1

k+l K 3 = ltW 1/t = Z (J~ui - j } I Ui-1 )(ft" - j } 1) , i=i Pi - Pi 1

(2.27)

where additionally fo = fouo = f~+luk+l = fk+l = 0. With A = KIK2 - K~ this yields

k k (~ = (~2,3)' = (~-~. aiX,,:n, ~ biXni:n) t i=1 i-1

(2.28)

where

diiK2 ~ - d ~ ai=~

1 pi 1

fi+l ~ ) pi+l (2.29)

A

\

Pi - pi-1

bi= ~- \- pi- pi~l

Pi+1 - Pi

]

pi+-~l--p~i I

f i-KJ ~3 ~- I A - Pi- 1 Pi+ft'+1- ~ . ) 1

for i = 1, 2, ...

(2.30) k.

Also, the variance-covariance matrix for (2,3)' can be expressed as (b2/n)M, where

Optimal linear inference using selected order statistics in location-scale models

-K3

M = S

189

(2,31)

K1

If 6 is known, then the A B L U E of 2 is k

a*Xn,:~- a~7

(2.32)

i = 1, 2, . . . , k

(2.33)

: ( I ' W - 1 1 ) - l l ' w - I ( x - au) : Z i=1

1

with

a; =fi@i-fi-1Pi-1 Pi+lfi+'--~) for and Var()0 = (a2/n)(l'W 11)-'

=

¢]2/(nK1)

(2.34)

.

If 2 is known, then the A B L U E of ~ is

= (u'W-lu):lu'w-l(x

k - )~l) = ~_b~Xn,:n - .~ K3

i=1

(2.35)

/£2

with = f i ( ~ u ~ i 7 J~_IHi_I b*

\

fi+lUi+l_ _ - ~ u i ~

Pi-Pil

Pi+l-Pi

for i :

1, 2, ...

k

(2.36)

I

and

Var(6) =

(a2/n)(utW-1/4)

1 = (52/(nK2) .

(2.37)

3. Optimality criteria for estimation

The question now arises as to which set of order statistics to select. For the small sample case, this typically results in checking all (~) subsets of k order statistics to determine the optimum collection once the optimality criterion has been established. This is typically straightforward on a computer. For the asymptotic case, the programming is a bit more complex because optimizing over 0 < Pl < / ) 2 < " " < Pk < 1, is required. One cannot simply check all possible collections of order statistics. For the case where one of the parameters is known, it is clear that selecting the order statistics to minimize the variance of the estimator is appropriate. This leads to minimization of 62(l'V 11)-1 or maximization of l ' V - l l when 6 is known for the small sample case and to maximization of K1 in the asymptotic case. If 2 is

190

M. M. Ali and D. Umbach

known, one minimizes 62(e'V-le) ~ or maximizes ~tv-l~ for the small sample case and maximizes/£2 for the asymptotic case. When both parameters are unknown, the situation is a bit more interesting. One approach is to select the order statistics to minimize the generalized variance, Var(2)Var(6) - Cov2@ 6). This is, of course, the determinant of the variancecovariance matrix of ®. This leads to maximization of det(A'V-1A) for the small sample case and to minimization of det(M) = 1/A in the asymptotic case. This leads to maximization of A = K1K2 - K 2. In many early papers in this area, the optimal spacings for the parameters of some specific distributions were obtained by maximizing K1, K2, or A over 0 < pl < P2 < "'" < Pk < 1. However, in many cases this approach proved to be mathematically intractable or computationally tedious. Hence alternative criteria and approaches were developed to attack the problem. In their survey paper, Chan and Cheng (1988) mention the use of dynamic programming and other numerical optimization techniques to overcome some of the computational difficulties in this maximization process. Hassanein (1969a) takes a different approach. He minimizes the total variance, i.e. Var(2) + Var(6). This is the trace of the variance-covariance matrix. In the asymptotic case, the total variance is (a2/(nk))(K2 + K~). I f f is symmetric and a symmetric spacing is used, (Pi = 1 -pk-i+l), then K3 = 0. This special case leads to minimization of K21 + Ki-1. One might also take a minimax approach to the problem. This entails minimizing max(Var(),), Var(6)). For the asymptotic case, this reduces to minimizing max(K1/A, K2/k). For the symmetric case mentioned above, this entails maximization of min(K1,/(2). An approximate solution to the optimal spacing has been proposed by Sfirndal (1962) and Eubank (1981a, 1986). Let I represent the Fisher information matrix for &,a. Now, since 2 and 6 are location and scale parameters, it can be shown that 1 i= j

1 (J1 J 3 ) J3 J2

'

(3.1)

where J1, J2, and J3 do not depend on 2 and 6. Define 0(w) = (~q(w), O2(w))', where

02

I//l(W) =n-Z-~.af(F I(W)) O2(w)

=

Ow 02 __F Ow2

-1

( w ) f ( F -l(w)) .

(3.2) (3.3)

Then, under suitable regularity conditions, Eubank (1981 a) shows that as k --+ oc, optimal spacings are given by pi = H - l ( i / ( k + 1)), where H -I is the quantile function for the density h, given by

Optimal linear inference using selected order statistics in location-scale models

1

1~2(w)12/3/ f2 1O2(w)lZ/3dw

for 6 known, for )o known,

[0(w),j_lqt(w) ]1/3/f~[qt(w),j_lqt(w)l,/3dw

for both unknown.

IlPl(W)l

h(w) =

2/3

191

/fo

ItPl(W)l 2/3dw

(3.4) Not much attention was focused on the problem of estimating the quantiles using a few selected order statistics until the early 1980's. Kubat and Epstein (1980), Koutrouvelis (1981), Eubank (1981b), Ali, Umbach and Hassanein (19818), and Umbach, Ali and Hassanein (1981b) are some of the papers which created more interest in the problem of optimal selection of order statistics in general and in the problem of estimating quantiles using optimal spacings in particular. The general approach to estimating quantiles using a few selected order statistics is outlined below. The ~th quantile ofF~,a is given by Q(~) = 2 + 6F -1 (~). The BLUE or A B L U E of Q(~) is given by )~+ 6F -1 (~), whose variance is Var(2) + (F l(~))2Var(6)+ 2F 1(~)Cov(2, 6). For the asymptotic case, this means one must minimize ( K 2 q- ( f

l ( ~ ) ) 2 K 1 - 2F l(~)K3)/a .

(3.5)

The optimal spacing in this case almost always depends on the value of {. Thus, if one is going to estimate Q(~) for more than one value of 4, another criteria for choosing the spacing is required. This is the focus of the remainder of this section. Let q(2, 6) be a sufficiently smooth function of the location and scale parameters that one may wish to estimate. Examples of such include the quantile function Q({) = 2 + 6 F 1(~) ,

(3.6)

the hazard function f ( ( t - 2)/6) h(t) = 6(1 - F ( ( t - ),)/6))

'

(3.7)

and the survival function s(t) = 1 - F ( ( t - 2)/6) .

(3.8)

N

Consider q(2, 6) as an estimator of q(2, 6). The local linearization theorem as in Rao (1965) can be used to obtain the asymptotic distribution of q(2, 6). Oq Oq Oq Oq • Thus, if ~ , 37, 2 ag' 2 and 0;C37 2 all exist bS:' for - v o < ,~ < ec and 0 < 6 < oo, then q(2, 6) is asymptotically unbiased with an expression of the asymptotic variance of q(2, 6) given by / \ [, 2 - ~ Oq Oq Var(q(2, 5)) = Var(2)/0q|2+\~j V a r ( 5 ) ( ~ ) +2Cov(2, 6 ) ~ 62 = --a~'M¢o , n

(3.9)

192

M. M. Ali and D. Umbach

where a~, = (Oq Oq) ' ~'Oa A similar expression for the variance of the maximum likelihood estimator of q(2, 6) can be developed. Let )o and ~ represent the maximum likelihood estimators. Then the asymptotic variance of the maximum likelihood estimator can be expressed as q(2, c5) as

(+)2

Var(q()~, 6)) = Var(2) ~-~

+Var(~) \ ~ j

^ OqOq

+ 2Coy()., c5)~-~ 0a 62

= --ofJ-l(~o n

(3.10)

.

Thus, the asymptotic efficiency of q(2, ~) can be expressed as AE(q(i, 3)) - Var(q(2, 3)) Var(q(2, a))

~o'J la~ ogM~o

(3.11)

Finally, by the Courant-Fisher Theorem, see Rao (1965, p. 48), AE(q(2, 6)) lies between vl and re, where vl _< v2 are the characteristic roots of J - I M ~. Now, the smaller characteristic root, Vl can be expressed as 1

1)1 -- 2(J1J2 - j 2 )

{J2K1 q- J1K2 - 2J3K3.

- [ ( J 2 K 1 - J I K 2 ) 2 -}- 4(J2K3 - J3K2)(J1K3 - J3K,)] 1/2 }

(3.12)

while the larger characteristic root, v2 can be expressed as 1 V2-

2(JlJ2 _j2)

{J2K1 + J I K 2 - 2J3K3.

@ [ ( J 2 g l - J 1 g 2 ) 2 -]- 4(J2K3 - J3K2)(J1K3 - J 3 g l ) ] 1/2 } •

(3.13)

I f f is symmetric then J3 = 0. If also the spacing is symmetric then K3 = 0. In this case vl is min(K1/J1,K2/J2) and v2 is max(K1/Ji,K2/J2). One approach is to choose the spacing to maximize trace(J-1M 1) = vl + V2

J2KI + J1K2 - 2J3K3

(3.14)

J, J2 -

For the symmetric case, this reduces to K~/J1 + K2/J2. This is motivated by the total variance criterion. Consideration of the generalized variance concept leads to m i n i m i z i n g

Optimal linear inference using selected order statistics in location-scale models

193

d e t ( j - l M -1) = vlv2 (3.15)

K1K2 - K~ g

g2 -

Since J1, J2, and J3 are fixed, this criterion results in the same spacing as minimizing the generalized variance. Since Vl serves as a lower bound for the asymptotic efficiency of q()o, 3), a conservative spacing is obtained by choosing the spacing 0 < pl < p2 < " < pk < 1 to maximize vl. The beauty of any of these last three methods of choosing a spacing is that they yield a spacing which is independent of the function or functions of 2 and 6 being estimated. Thus, the spacing so generated may be considered robust for the estimation of 2 and 6 as well. It should be noted that a similar approach could be taken to compare O with other estimators. One could simply replace (3.1) with (1/n) times the inverse of the variance-covariance matrix of the estimator of ® for which a comparison is to be made. Such estimators might include the full sample BLUE as an example.

4. Specific distributions In this section these results are applied to specific distributions. The so-called life distributions play a central role because observations from these distributions are often naturally ordered.

4.1. Exponential distribution

The distribution function of the exponential distribution is given by

F~.,~(x)

l" 0 1 - e x p ( - ( x - 2)/c5))

for x < 2 for 2 _< x .

(4.1)

The lack-of-memory property makes it a natural distribution as a model for lifetime analysis. Estimation of the parameters of this distribution has been carried out by many authors. Harter (1961) estimates 2 with 6 known and c5 with 2 known in small samples with k = 1 or 2. Kulldorff (1963a) considers the same problem with more order statistics, Both show that the optimal selection of order statistics is the same whether estimating 2 or 6 alone. For the two parameter problem, minimization of the generalized variance was carried out for small samples by Sarhan and Greenberg (1963) for k --- 2 and by Saleh (1967) for larger values of k. Saleh also considers censoring. Saleh, Ali and Umbach (1983b) extend this to samples censored in the middle. Siddiqui (1963) uses a slightly different approach to the problem with k = 2. For the asymptotic case, note that ui = - l n ( 1 - p i ) and j) = 1 - p i . These can be substituted in K1, K2, and K3, and then K1, /(2, and A maximized over

194

M . M . A l i a n d D. U m b a c h

0 < Pl
< ' • ' < Pk < 1

directly. However, f o r

this distribution

it is c o n v e n i e n t

to carry out the maximization over the ui's, i.e. over 0 < ul < u2 < -. • < uk < ec. After substitution and simplification, one finds K1 -

1

(u,

k

K2=~

- - u1' 2 2

eui - - eui t

i-I

K3

(4.2)

e u~ - 1 u0=0

(4.3)

_ e ulul- 1

(4.4)

O g a w a (1960) maximizes (4.3) for estimation o f 6 with 2 k n o w n for k = 1 ( 1) 15. He presents the spacings and coefficients o f the A B L U E for 6. This table was reproduced in Sarhan and Greenberg (1962) and has since been used extensively in m a n y applications. A s y m p t o t i c results are quite interesting when 2 is u n k n o w n . F o r the estimation o f 2 when 6 is known, one maximizes (4.2), which occurs for ul = 0 or pl = 0. Thus, the first order statistic Xl:n should be selected. W h e n both parameters are u n k n o w n , the generalized variance criterion leads to maximization o f A-

/£2 e u~ - 1

{ ul "/2 \e"~ - 11

After the reparameterization ti = ui+l following expressions are obtained I./2

/(2 -- e u~ _ 1

(4.5)

-

Ul

for i --- l, 2 , . . . , k - 1, (to = 0) the

+ e - U , X-"k-1 (ti . .- ti-1) . . 2 i=1

et~ -- eti 1

u2 e ul -- 1

e "'K~

(4,6)

After simplification, A has the following f o r m e-Ul

A - ~

eut _

K~

(4.7)

which is to be maximized over 0

0

<ec

.

(4.8)

Notice that Kj depends only on q, t2, . . . , tk-1 and has the same f o r m as (4.3). Thus the results o f O g a w a (1960) can be used to optimize K~. N o w e - " ' / ( e "~ - I) is maximized at Ul = 0 or pl = 0. But pl -- 0 is not covered by Mosteller's Theorem (1946). Sarhan, Greenberg and O g a w a (1963) consider the asymptotic case for k = 2. Saleh and Ali (1966) try to patch things up by justifying the choice pl = 1 / ( n + 1) for arbitrary k. Saleh (1966) extends these results to the censored case. Estimation o f the quantile function Q(~) = 2 - 6In(1 - ~)

(4.9)

Optimal linear inference using selected order statistics in location-scale models

195

has been considered by U m b a c h , Ali and Hassanein (198 l a). They minimize the variance of the BLUE, 2 - aln(1 - ~), for k = 2 with n = 2(1)30. This work was extended to k = 3(1)6 for n = k(1)100 in Ali, Umbach and Saleh (1982). For asymptotic results, note that (3.5) can be rewritten as K ; q (F-1 ({))2 eu,-1

2~u,l (~)Ul) 1 e'-I J e-U,K~

.((F41(~)-)2 2F4'(~)<-) e"' =eU'+\ e,-1 e,-1 /K;

(4.,0)

Minimization of (4.10) was carried out by Ali, Umbach and Hassanein (1981a) for k = 2 and by Saleh, Ali and Umbach (1983b) for k = 4(2)14. Again, this was done in two stages. First Ogawa (1960) was used to maximize K~. Then the resulting function of Ul was minimized. For certain values of { the first order statistic is selected. In these cases pl = 1/(n + 1) was again used. A more careful approach to maximization of zXwas taken in Ali, Umbach and Saleh (1985). Here the first order statistic was included in the selected group and then k was maximized over 0 < p2 < p3 < " < Pk < 1.

4.2. Logistic distribution Another distribution that has had a lot of attention in the area of optimal spacing is the logistic distribution. Even though it is not itself a life distribution it is a simple transformation of one, the so-called log-logistic distribution. The form of the distribution function to be considered here is 1 b3~,a(x) = 1 + e x p ( - ( x - 2)/6)

(4.11)

Small sample work for this distribution has been somewhat sparse. Chan, Cheng and Mead (1971) consider the estimation of the parameters in small samples using the generalized variance criterion. Beyer, Moore and Hatter (1976) extended these results by including censoring. In considering the asymptotic case, compact expressions u, = - ln((1 - p i ) / p i ) and f,. = (1 - p i ) / p i are available. However, these do not lead to easily manipulated expressions for KI, K2, and K3 as they did in the exponential case. Nonetheless, asymptotic results are rather plentiful for this distribution. Gupta and Gnanadesikan (1966) consider estimation of 2 and a alone for k = 2 or 3. They note the complexity involved in the problem of minimizing the generalized variance. Hassanein (1969a) attacked the joint estimation problem by consideration of the total variance. Chan, Cheng and Mead (1972) minimize the generalized variance for k = 2, 3, and 4. For larger values of k, Chan and Cheng (1972) find the optimal estimator of 2 with 6 known. Later, Chan and Cheng (1974) extend these results to the censored case. Hassanien (1974) minimizes the generalized variance for k = 2(1)10.

196

M. M. Ali and D. Umbach

Estimation of the quantile function 0(4) = 2 - 61n((1 - ~)/~)

(4.12)

was considered by Eubank (1981 b). Using Eubank's approach, Saleh, Hassanein, and Ali (1992) consider the problem of finding ABLUE's of the quantile function for k = 2(1)10. Ali and Umbach (1989b) present exact results for k = 2, 3, and 4. Saleh0 Ali and Umbach (1992) provide the full asymptotic results. Ali and Umbach (1994) also find the spacings that maximize vl in (3.12) subject to the spacing being symmetric, the so-called conservative spacing. To carry this out, note the Fisher information matrix (3.1) for the logistic distribution has J1 = 1/3, J2 = (g2 + 3)/9, and J3 = 0. This yields v,

=

729 . f 722 -~- 3 . i mln - - h l , - (~22+3)2 ]. 9

/£2 ]

3 J~

(4.13)

to be maximized. The results of the maximization process are reproduced in Table 4.1 for illustrative purposes. 4.3. Extreme value distribution

The extreme value distribution provides a natural model for lifetimes. In this work, the Type 1 (smallest) extreme value distribution, whose distribution function is given by F~,6(x) = 1 - e x p ( - e x p ( ( x - 2)/6)) ,

(4.14)

is considered. All results are easily transformed to cover the "largest" extreme value distribution, since the "largest" is simply the negative of the "smallest." Hassanein (1969b) finds estimates of the parameters based on 2 or 3 order statistics for small samples, n _< 20. He also gives asymptotic results for k -- 2, 3, and 4 based on the total variance criterion. Hassanien (1972) extends these results to k = 2(1)10 in the small sample case. He also considers censoring. For this distribution ui = i n ( - l n ( 1 - p i ) ) and J} = - ( 1 - p i ) l n ( 1 - p i ) . Even after substituting these into K1, K2, and /£3, the expressions are still very untractable. Numerical results, however, are plentiful for this distribution. This can be attributed to the importance of the distribution in life testing. Hassanein (1968) provides asymptotic results for estimation of the parameters separately. He considers joint estimation of 2 and ~ by minimizing the generalized variance for k = 2. Chan and Mead (1971a) and Chan and Kabir (1969) extend the results to consider more order statistics. Hassanien (1972) minimizes the total variance for the joint estimation of 2 and 6. Kulldorff (1973) shows that the optimal spacing for location in a Type 1 extreme value distribution is essentially the same as the optimal spacing for scale in a Type 2 distribution. Estimation of the quantile function Q(~) = 2 + 61n(-ln(1 - ~))

(4.15)

Optimal linear inference using selected order statistics in location-scale models

197

Table 4.1

Optimal spacing, coefficients, and characteristic roots for the conservative spacing for the logistic distribution. k=2

k=3

k=4

k=5

k=6

k=7

k=8

pl a~ b~ P2 a2 b2 P3 a3 b3 p4 a4 b4 p5 a5 b5 P6 a6 b6 /~ a7 b7 P8 a8 b8

0.1503 0.5000 -0.2887 0.8497 0.5000 0.2887

0.1029 0.1587 -0.2309 0.5000 0.6826 0.0000 0.8971 0.1587 0.2309

0.0552 0.0431 -0.1011 0.2282 0.4569 -0.1745 0.7718 0.4569 0.1745 0.9448 0.0431 0.1011

0.0404 0.0213 -0.0724 0.1690 0.2098 -0.1699 0.5000 0.5379 0.0000 0.8310 0.2098 0.1699 0.9596 0.0213 0.0724

0.0259 0.0093 -0.0430 0.1098 0.0813 -0.1077 0.2749 0.4095 -0.1227 0.7251 0.4095 0.1227 0.8902 0.0813 0.1077 0.9741 0.0093 0.0430

0.0197 0.0052 -0.0321 0.0850 0.0474 -0.0842 0.2124 0.2197 -0.1331 0.5000 0.4552 0.0000 0.7876 0.2197 0.1331 0.9150 0.0474 0.0842 0.9803 0.0052 0.0321

0.0141 0.0028 -0.0220 0.0617 0.0258 -0.0592 0.1527 0.1019 -0.0987 0.3066 0.3696 -0.0930 0.6934 0.3696 0.0930 0.8473 0.1019 0.0987 0.9383 0.0258 0.0592 0.9859 0.0028 0.0220

K1 K2 /£3

0.2170 0.9311 0.0000

0.2909 0.9779 0.0000

0.2762 1.1849 0.0000

0.3077 1.2072 0.0000

0.2995 1.2849 0.0000

0.3159 1.2971 0.0000

0.3110 1.3343 0.0000

vl v2

0.6511 0.6511

0.6838 0.8726

0.8286 0.8286

0.8442 0.9231

0.8986 0.8986

0.9071 0.9477

0.9331 0.9331

was considered by Hassanein, Saleh and Brown (1984) and (1986). They also make some applications to testing. Umbach and Ali (1996b) find the spacings that maximize Vl in (3.12), the conservative spacing. To carry this out, note the Fisher information matrix (3.1) for the extreme value distribution has J1 = 1, "]2 = (1 -- 7) 2 -F- 2 ' and J3 = 1 - 7, where 7 is Euler's constant. Using these values vl can be expressed as

vl = 7

3

{((1 - 7) 2 -}- 7z2/6)Xl + K 2 - 2(1 - 7)/£3

- I{((1 - 7) 2 + Tc2/6)K1 -- K2} 2 q- 4{((1 - 7) 2 -t- rr2/6)K3 - (1 - 7)K2}{K3 -- (1 -- 7)K1}] 1/2 }

(4.16)

198

M . M . A# and D. Umbach

A steepest d e s c e n t a l g o r i t h m was u s e d to m a x i m i z e this e x p r e s s i o n for k = 2(1)8. T h e results o f this p r o c e s s a r e r e p o r t e d in T a b l e 4.2. F'or a c o n c r e t e e x a m p l e o f the use o f s u c h a table, c o n s i d e r the d a t a o n p a g e 257 o f A r n o l d , B a l a k r i s h n a n , a n d N a g a r a j a (1992). T h e r e t h e y r e p o r t the a n n u a l rainfall at the L o s A n g e l e s C i v i c C e n t e r f o r the y e a r s 1890 t h r o u g h 1989. E v e n t h o u g h t h e d a t a is n o t a r a n d o m s a m p l e , it f o l l o w s a n e x t r e m e v a l u e d i s t r i b u t i o n q u i t e well. S u p p o s e t h a t e s t i m a t e s o f 2 a n d 6 b a s e d o n k = 5 o r d e r statistics are desired. F r o m T a b l e 4.2, t h e f o l l o w i n g a r e f o u n d , pl = 0.0927, p2 = 0.3626, P3 = 0.9029, 1o4 = 0.9762, p5 = 0.9963 . (4.17) T h u s , since n = 100, t h e r a n k s o f the selected o r d e r statistics are 10, 37, 91, 98, a n d 100. T h e d a t a yields Table 4.2 Optimal spacing, coefficients, and characteristic roots for the conservative spacing for the extreme value distribution k=2

k=3

k=4

k=5

k=6

k-7

k-8

0.1705 0.3759 -0.3722 0.9358 0.6241 0.3722

0.2700 0.4996 -0.4132 0.9499 0.4462 0.2827 0.9939 0.0542 0.1305

0.0614 0.0538 -0.1007 0.2707 0.3374 -0.2700 0.8845 0.4984 0.1698 0.9824 0.1103 0.2009

0.0927 0.0824 -0.1415 0.3626 0.3962 -0.2534 0.9029 0.4014 0.1712 0.9762 0.0954 0.1544 0.9963 0.0247 0.0693

0.0284 0.0195 -0.0411 0.1280 0.0742 -0.1172 0.3320 0.3048 -0.2144 0.8480 0.4208 0.0926 0.9575 0.1386 0.1781 0.9929 0.0421 0.1020

0.0401 0.0279 -0.0565 0.1707 0.1011 -0.1441 0.4085 0.3362 -0.1899 0.8669 0.3598 0.1101 0.9548 0.1148 0.1428 0.9870 0.0467 0.0949 0.9978 0.0135 0.0427

0.0152 0.0094 -0.0207 0.0703 0.0321 -0.0599 0.1819 0.0831 -0.1155 0.3735 0.2780 -0.1787 0.8211 0.3670 0.0536 0.9335 0.1444 0.1461 0.9798 0.0655 0.1162 0.9965 0.0205 0.0590

KI /(2 /(3

0.6257 1.1411 0.2645

0.6847 1.2487 0.2895

0.8092 1.4756 0.3421

0.8429 1.5371 0.3564

0.8850 1.6140 0.3742

0.9046 1.6497 0.3825

0.9234 1.6840 0.3904

vl v2

0.6257 0.6257

0.6847 0.6847

0.8092 0.8092

0.8429 0.8429

0.8850 0.8850

0.9046 0.9046

0.9234 0.9234

pl al bl P2 a2 b2 P3 a3 b3 P4 a4 b4

P5 a5 b5 P6 a6

b6 p7 a7

b7 P8 a8 b8

Optimal linear inference using selected order statistics in location-scale models

199

n~0 = 7.38, n37 = 11.80, n91 = 23.65, n98 = 30.57, n~00 = 34.04 . (4.18) The table also yields al = 0.0824, a2 = 0.3962, a3 = 0.4014, a4 = 0.0954, a5 = 0.0247 . (4.19) Thus, ~. --- 0.0824 • 7.38 + 0.3962 - 11.80 + 0.4014.23.65 + 0.0954 • 30.57 + 0.0247 • 34.04 -= 18.53 .

(4.20)

The table also yields bl = - 0 . 1 4 1 5 ,

b2 = - 0 . 2 5 3 4 ,

b 3 = 0.1712, b 4 = 0.1544, a5 = 0.0693 .

(4.21) Thus, = - 0.1415 • 7.38 - 0.2534- 11.80 + 0.1712.23.65 + 0.1544- 30.57 + 0.0693 • 34.04 = 7.09 .

(4.22)

Table 4.2 also yields KI = 0 . 8 4 2 9 , K 2 = 1.5371, and / £ 3 = 0 . 3 5 6 4 . Thus A = 0.8429,1.5371 - 0 . 3 5 6 4 2 = 1.1686. Substituting these in (2.31), the asymptotic variance o f ~ is (62/100)(1.5371/1.1686). Using 7.09 for 6, this is approximately 0.6612. Again, using (2.31) the asymptotic variance o f 6 is (62/100) (0.8429/1.1686). Using 7.09 for 6, this is approximately 0.3626.

4.4. Weibull distribution The Weibull distribution is second only to the exponential distribution in importance as a life distribution. Specifically, consider the Weibull distribution given by

F~,~(x)

f 0 1 - e x p ( - ( x - 2)/6) ~)

tbr x < )~ for fl < x .

(4.23)

In some applications, authors consider e k n o w n and get optimal spacings for the estimation o f 2 and 6 for various values o f c~. Using this approach, ui = ( - l n ( 1 _pi))O/~) and J} = e(1 -Pi)(-ln(1 -pi)) "@'.As before, these do not lead to simple forms for g l , K 2 , and /£3. Nonetheless, numerical results are available for the joint A B L U E o f fl and 6 for k = 2, 4, and 6 based on the generalized variance criterion in Hassanein (1971). A n o t h e r fruitful a p p r o a c h comes from considering the shape-scale Weibull distribution, whose distribution function is given by

F~,B(x) =

{ 0 1 - exp(-(x//~) ~)

for x < 0 for 0 <_ x .

(4.24)

200

M. M. All and D. Umbach

The natural logarithm of a random variable with this distribution has the extreme value distribution (4.14) with 2 = In fi and 6 = 1/co Thus, the estimators k

lnAfl =

~ailnX~,:~ i=l

=

=- ~ bilnXn~:,

=

(4.25)

k

1/~

3

i=1

are obtained. Thus, for an arbitrary spacing, estimators of the form = e i and ~ = =1

(4.26)

can be considered. Using this approach Mann (1970) obtains the optimal spacing for the scale parameter. Umbach and All (1996b) used the same approach to find a conservative spacing for the Weibull distribution. Using the estimators (4.20), they find that j) and ~ are asymptotically unbiased and obtain the asymptotic variance-covariance matrix for (fl, ~) as (fl2/n)M*, where M * = A -1

7

-9

"~

(4.27)

Here, f i , g2, K3, and A are calculated using the extreme value distribution (4.14). They also note that for the Weibull distribution (4.18), the information matrix is given by l*

1j,

1 (

~2

-(1-7)fi

B2)

(4.28)

Let q(~, fl) represent a sufficiently smooth function of ~ and ft. The asymptotic efficiency of q(& fl) can be expressed as ~ t J *-~ ¢o

aE(q(& fl)) - re'M'co

(4.29)

Thus, AE(q(~ fl)) lies between v~ and v~, where vl < v~ are the characteristic roots ofJ*-~M *-~. Interestingly, it turns out that v~ and v~ do not depend on e and fi, even though J* and M* do. In fact, v~ can be expressed as 3 v~ =~-5 {((1 - y) 2 + ~2/6)K1 +I£2 + 2(1 - y)K3 - [{((I - 7) 2 + zc2/6)Xl - K2} 2 + 4{((1

- 7) 2 + a:2/6)K3 +

(1

(4.30)

- y ) K 2 } { K 3 + (1 - y ) K l } ] 1/2 } .

They present the results of numerically maximizing v~ for k = 2(1)8.

Optimal linear inference using selected order statistics in location-scale models

201

4.5. N o r m a l distribution

The normal distribution has received a fair amount of attention in this area also. Here the mean is 2 and the standard deviation is 6. One of the earliest works in selecting order statistics for the location scale model was an application to the normal distribution by Yamanouchi (1949). He justifies the use of the spacing Pi = (i - ½)/k. Kulldorff (1963b) finds the optimal spacing using the generalized variance criterion under symmetric spacing. Kulldorff (1964) takes an interesting approach. He uses simple estimates based on the selected order statistics, such as their average, instead of the B L U E or ABLUE. He finds optimal selections of order statistics within these simplified classes. Chan and Chan (1973) consider optimal spacings for the joint estimation of 2 and 6. Hassanein, Saleh and Brown (1986b) and Saleh, Hassanein and Ali (1988) consider estimation of the quantiles of the normal distribution. Ali and Umbach (1994) find the spacings that maximize v~ in (3.12) under symmetric spacings. For this problem J1 -- 1, J2 = 2, and J3 ~ 0. Symmetric spacings reduce (3.12) to vl = min{Kl,/£2/2} .

(4.31)

They present results of maximizing (4.25) for k = 2(1)8. 4.6. C a u c h y distribution

Consider the Cauchy distribution whose distribution function is given by " F~,~(x) -- ~ + - arctan rc

.

(4.32)

This distribution has a knack for producing interesting results. It happens with selected order statistics as well. For this distribution ui = tan(Tz(pi - ½)) and 1

Y~ = 7t(l + tan2(Tr(pi - ½)))

(4.33)

As usual, these do not produce expressions for Kl, K2, and K3 that can be easily manipulated. Nonetheless, some activity is reviewed below. Bloch (1966) considers estimation of 2 with known 6. He shows that the spacing pl = .13, p2 = .40, P3 --- .50, p4 = .60, p5 = .87 is 95% efficient. Chan (1970) finds the optimal spacings for the estimation of 2 alone and 6 alone. Chan, Chan and Mead (1973) show that the generalized variance is minimized when the uniform spacing pi = i / ( k + 1) is used. Cane (1974) restricts attention to symmetric spacings. She confirms that the symmetric spacing is optimal and tables the coefficients of the estimators for k = 2(1)10. Balmer, Boulton and Sack (1974) show that for estimating 2 with known 6 that an asymmetric spacing may be optimal. They show that this is indeed the case when k -- 4m - 1 for some positive integer m.

M. M. All and D. Umbach

202

For the Cauchy distribution, note that J1 = 1/2, J2 = 2, and J3 = 0. By restricting to a symmetric spacing, vl in (3.12) is given by v~ = min{2K1,/£2/2} .

(4.34)

Maximization of this expression was carried out by Umbach (1994) for k = 2, 3, 4, and 5. He finds that a uniform spacing is optimal under this criterion only for k = 3.

4.7. Pareto distribution Another distribution that has received quite a bit of attention in this area is the Pareto distribution. There are a number of forms of the distribution that can be considered. First consider the form

F;,6(x) = { 0

for x < 2 + 6 for ;+6_<x.

(4.35)

For the small sample problem, Vgnnman (1976) uses the generalized variance criterion to find the BLUE of 2 and 6. Umbach, Ali and Hassanein (1981b) find the optimal spacing for the BLUE of the quantile function Q(~) = 2 + 6(1 - ~)'/~

(4.36)

based on two order statistics. Note that ui = ( 1 - p i ) ~ and J} = e ( 1 - P i ) (~+~)/~. Kulldorff and Vfinnman (1973) use these values to discuss the optimal spacing of the order statistics for estimation of 2 and 6 for known c~in the asymptotic case. Chan and Cheng (1973) discuss the optimal spacing for estimation of 6 with c~and 2 known. Umbach, Ali, and Hassanein (1981b) find the optimal spacing for the estimation of Q(~) for k=2. A few authors have considered the form of the distribution function given by {0

F,3(x)

for x < f l for f l < x

l-(x/fl)-~

(4.37) .

The natural logarithm of a random variable with this distribution has the exponential distribution (4.1) with 2 = in fi and 6 = 1/c~. Thus, estimators k

lnfl

~-

~

ai

lnX~,:.

=

)o

i=1

(4.38)

k

17a

=

~bilnX,,:,

=

i=l

can be formed. Hence, for an arbitrary spacing, estimators of the form = e i and ~ ==1 can be considered.

(4.39)

Optimal linear inference using selected order statistics in location-scale models

203

Koutrouvelis (1981) used this technique to estimate the parameters using the generalized variance criterion. Saleh, Ali and Umbach (1985) use this idea to form a nonlinear estimator of the quantile function fl(1 - ~)1/~ = e~(1 _ ~)a.

4.8. Other distributions A number of other distributions have received some attention in this area. For the gamma distribution these include SS.rndal (1964) and Hassanein (1977). Some results for the double exponential distribution are available in Cheng (1978), Ali, Umbach and Hassanein (1981a) and Umbach, Ali and Saleh (1986). Dyer and Whisenand (1973) consider optimal spacing for the Rayleigh distribution. See the bibliography for others. 5. Tests of significance In this section, tests of significance are considered. Specifically four null hypotheses are considered: HI : ,~ = 20, 6 unspecified, H2:6 = 60, 2 unspecified, H3:2 = 20,6 = 6o, H4: Q(~) = Qo, 2 and 6 otherwise unspecified. Define the quadratic form/2 as /2 = ( X - 21 - 8u)'W l ( X - 21 - 8u) .

(5.1)

Mosteller's Theorem implies that nf2/62 has a chi-square distribution with k degrees of freedom. Now,/2 can be orthogonally partitioned as/2 =/2o +/21, where (20= ( X - i l - 3 u ) t W

l(X-~l-3u)

,

/21 = K l ( ~ - 2 ) 2 + 2 K 3 ( ) . - , ~ ) ( 3 - 8 ) ÷ K 2 ( 3 - 8 )

(5.2) 2 .

(5.3)

See Saleh, Ali, and Umbach (1984). Thus, n/20/82 has a chi-square distribution with k - 2 degrees of freedom and n/21/a 2 has a chi-square distribution with 2 degrees of freedom The minimum of/21 under the restriction 2 = 20 is (~, - 2o)2A/K2. Following Anderson (1958, pp. 189-190), Saleh, Ali and Umbach (1984) propose the test statistic tl =

a g X T 2 7 ( i - 2) V/Oo/(k - 2)

(5.4)

for testing HI. Under H1, tl has a (central) t-distribution with k - 2 degrees of freedom. Under the alternative hypothesis, tl has a noncentral t-distribution with noncentrality parameter

M. M. Ali and D. Umbach

204

(5.5) The spacing that maximizes the power of the test is obtained by maximizing A/K2. This is, of course, the spacing that minimizes the variance of the ABLUE of 2 with 3 unknown. The minimum of (21 under the restriction 3 = 6o is (~ - 3o)2A/K1. Again, they propose the test statistic

t2 --

(3 - 3)

/f2o/(k-

(5.6)

2)

for testing//2, which under H2 has a t-distribution with k - 2 degrees of freedom. Under the alternative hypothesis, t2 has a noncentral t-distribution with noncentrality parameter

x/SX

l (3 - 30)

(5.7)

The spacing that maximizes the power of the test is obtained by maximizing A/K1. This is, of course, the spacing that minimizes the variance of the ABLUE of 3 with 2 unknown. For symmetric distributions and symmetric spacings, these reduce to maximization of K1 and K2, respectively. For testing H3, Saleh and Sen (1985) propose the test statistic L ° = nf2~/62o. Under //3, L° has a chi-square distribution with 2 degrees of freedom. They consider the Pitman-A.R.E. of the test, which turns out to be exactly (3.11). Thus, one could consider any of the three optimality criteria at the end of Section 3 to choose the spacing. In a similar manner to the development of tl and t2, Saleh, Umbach and Ali (1984) propose the test statistic

t4= )~+6F ~ ~ 21(~)-QO )--

~K 2

+(F

A

l(~))2K~-2F 1(~)K3

(5.8)

for testing H4. Again, t4 has a t-distribution with k - 2 degrees of freedom under H4. Under the alternative hypothesis t4 has a noncentral t-distribution with noncentrality parameter (2 + 6F-1 ( ~ ) - Q O ) ~ K 2 nA 6 + (F -l(~))2K 1

--

2F

(5.9) I(~)K

3

The power of the test is maximized by choosing the spacing that maximizes this noncentrality parameter. This spacing also minimizes (3.5). Here again, the duality with the estimation problem is evident.

Optimal linear inference using selected order statistics in location-scale models

205

6. T e s t i n g g o o d n e s s - o f - f i t

In this section, tests concerning the suitability of the assumed underlying distribution, F, of the location scale model, F~,~(x) = F((x - 2)/3) are considered. The test statistics will vary depending on which parameters are assumed known. First, consider the goodness-of-fit hypothesis //5: ~,~(x) = Fo((~ - 4o)13o) ,

(6.1)

where Fo is a completely specified distribution function, with density fo, and 2o and 6o are specified. For a fixed spacing, let Wo and uo be the values of W and u, respectively, calculated using F = Fo and f = fo. See (2.21) and following. Now, let 05 = (X - 2ol - 3oUo)'W~-l (X - 4ol - 3oUo) •

(6.2)

U n d e r / / 5 , nf25/32o has a limiting chi-square distribution with k degrees of freedom. Umbach and Ali (1990) propose testing Hs with a critical region given by 2 n05/32o > z~(k), where zZ(k) represents the ( 1 - c~)100%-ile of the chi-square distribution with k degrees of freedom. For a few different spacings, they show that the power of the test is comparable to Pearson's test, but that it is somewhat less powerful than the Kolmogorov-Smirnov test. Modifications are necessary if one or both of the parameters is unknown. For example, consider 06: F2,f(x) ~-- Fo((X - 4)/3o) ,

(6.3)

where Fo is a completely specified distribution function and 3o is specified. A value for the location parameter 4 is not specified. To carry out the test, 4 must be estimated. For this use the estimator 4 of (2.32) calculated using F = Fo and f = fo and using 3 = 3o. Similar to I2s, let 06 = ( X - 21 - 3oUo)'Wo 1 (X - 21 -

3oUo)

.

(6.4)

Under H6, n06/32o has a limiting chi-square distribution with k - 1 degrees of freedom. Umbach and Ali (1990) propose testing//6 with a critical region given by n06/32o > z 2~ ( k - 1). They also present results of a Monte Carlo power study. Things are not so straightforward for H7: FL~(x ) = Fo( (X - 40)/3) ,

(6.5)

where Fo is a completely specified distribution function and 4o is specified. A value for the scale parameter 3 is not specified. To carry out the test, estimate 3 with of (2.35) calculated using F = Fo and f = fo and using 4 = 4o. Now, let

M. M. Ali and D. Umbach

206

07

:

(X -

2ol

-

6Uo)'Wol(X

-

2ol

-

6Uo)

•

(6.6)

Under 117, n~7/82 has a limiting chi-square distribution with k - 1 degrees of freedom. However, a test statistic cannot be built directly on this quantity, since 8 is an unknown parameter. Umbach and Ali (1990) propose the following fix for distributions Fo with finite variance ao2. Under//7, the sample variance S 2 converges in law to 82o2o. This yields (6.7)

n(27 . ~ 2 cro2 _ n O 7 O-2

82

S2

S2

converging in law to a chi-square distribution with k - 1 degrees of freedom. They propose the use of (nO7a2o)/S 2 as a test statistic for//7. Since alternatives t o / / 7 m a y contain distributions whose second moments are not finite, which would tend to make (6.7) small, they propose a two tailed test in this case. The case where both parameters are unknown is handled in a similar manner in Ali and Umbach (1989a). Consider

Ha: F),a(x) = Fo( (X - 2)/8) ,

(6.8)

where Fo is a completely specified distribution function. To carry out the test, estimate 2 with 2 and 8 with 6 calculated using F = Fo and f = )Co. Then calculate the test statistic (nOoa2o)/S 2 in a manner similar to the previous case. The limiting distribution is chi-square with k - 2 degrees of freedom in this case. They propose a two-tailed test in this case also. The results of an extensive Monte Carlo simulation of power are included in their work. LaRiccia (1991) presents a test based on all of the order statistics for Hs. Umbach and Ali (1996a) propose a variant which is based on a few order statistics. It is outlined below. Choose appropriate functions hi, h2, • •., hr and define

hj = (hj(uol), hj(uo2), ...,hj(uog))'

for = 1, 2, . . . , r ,

(6.9)

where Uoi is the :helement of uo. F o r m the linear model F

X = 21 + 6uo +

~-~/~:hj +e

,

(6.10)

j=l

where e has mean 0 and variance-covariance matrix (82/n)Wo. Now, instead of testing//8 directly, they test H9:/3

= 0

,

(6.11)

where/3 = (/31, / 3 2 , . . . , fir)'The X-matrix for a regression analysis based on the previous linear model can be partitioned as

Optimal linear inference using selected order statistics in location-scale modeb

Xt = (1, u0) X2 = (hi, h2, . . . , hr) .

207

(6.12)

Now, for i and j = 1, 2, let Cii = Xi'WolXi, forming C :

ICtt

C121

L C21

C22 J "

Let C ij represent the corresponding elements of C -l. Then the regression estimator of p can be expressed as j~ = (C21XI t -Jr C 2 2 X 2 t ) W o l X

.

(6.13)

Thus, for this model,/] is asymptotically normal with mean/~ and variance covariance matrix given by (62/n)C 22. Under/49, the quadratic form 32

(6,14)

is asymptotically chi-square with r degrees of freedom. To form a test statistic with a chi-square distribution under//9, they replace 6 in the denominator of (6.14) with 3. Thus, they propose the test statistic 32

(6.15)

for testing the goodness-of-fit hypothesis H9. They include results of a fairly extensive Monte Carlo simulation that indicates the test statistic (6.15) converges to its limiting chi-square distribution faster than does LaRiccia's statistic. Umbach and Ali (1991) have also developed a test of exponentiality with both location and scale unspecified based on a few selected order statistics. It is based on the non-central t-distribution.

References [1] Adatia, A. and L. K. Chan (1981). Some relations between stratified, grouped and selected order statistics samples. Scand. Actuar. J. 193-202. [2] Ali, M. Masoom (1994). Inference base on optimal inference: Some recent results. Proc. ICCS IV, Lahore, Pakistan 7, 145-153. [3] Ali, M. Masoom and D. Umbach (1984). Testing Pareto quantiles using selected order statistics. J. Statist. Stud. 4, 107 123. [4] Ali, M. Masoom and D. Umbach (1989a). A Shapiro-Wilk type goodness-of-fit test using a few order statistics. J. Statist. Plann. Infer. 22, 251-261. [5] Ali, M. Masoom and D. Umbach (1989b). Estimation of quantiles of symmetrically truncated logistic distribution using a few optimally selected order statistics. J. In[or. Opt. Sci. 10(2), 303-307. [6] Ali, M. Masoom and D. Umbach (1990). Nonlinear estimation for the log-logistic distribution using selected order statistics. J. Statist. Stud. 10, 1-8.

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[7] Ali, M. Masoom and D. Umbach (1994). Estimating functions of the parameters of normal and logistic distributions using conservative spacings. Pak. J. Statist. 10(1)A, 179-188. [8] Ali, M. Masoom, D. Umbach and K. M. Hassanein (1981a). Estimation of quantiles of exponential and double exponential distributions based on two order statistics. Commun. Statist.Theory. Meth. 10, 1921-1932. [9] Ali, M. Masoom, D. Umbach and K. M. Hassanein (1981b). Small sample quantile estimation of Pareto populations using two order statistics. Aligarh J. Statist. 1,139-164. [10] All, M. Masoom0 D. Umbach and A. K. Md. E. Saleh (1982). Small sample quantile estimation of the exponential distribution using optimal spacings. Sankhy8 B 44, 135-142. [11] Ali, M. Masoom, D. Umbach and A. K. Md. E. Saleh (1985). Tests of significance for the exponential distribution based on selected order statistics. Sankhy~ B 47, 310-318. [12] All; M. Masoom, D. Umbach and A. K. Md. E. Saleh (1992). Estimating life functions of chi distribution using selected order statistics, l i E Trans. 24(5), 88-98. [13] Ali, M. Masoom, D. Umbach, A. K. Md. E. Saleh and K. M. Hassanein (1983). Estimating quantiles using optimally selected order statistics. Commun. Statist.- Theory. Meth. 12(19), 2261-2271. [14] Anderson, T. W. (1958). Introduction to Multivariate Statistical Analysis. John Wiley & Sons. [15] Arnold, B. C., N. Balakrishnan and H. N. Nagaraja (1992). A First Course in Order Statistics. John Wiley & Sons, Inc., New York. [16] Balakrishnan, N. and A. Clifford Cohen (1991). Order Statistics and I n f e r e n c e - Estimation Methods. Academic Press, Inc. [17] Balmer, D. W., M. Boulton and R. A. Sack (1974). Optimal solutions in parameter estimation problems for the Cauchy distribution. J. Amer. Statist. Assoc. 69, 238-242. [18] Benson, F. (1948). A note on the estimation of mean and standard deviation from quartiles. J. Roy. Statist. Soc. B 11, 21-100. [19] Beyer, J. N., A. H. Moore and H. L. Harter (1976). Estimation of the standard deviation of the logistic distribution by the use of selected order statistics. Technometrics 18, 313 332. [20] Bloch, D. (1966). A note on the estimation of the location parameter of the Cauchy distribution. J. Amer. Statist. Assoc. 61, 852 855. [21] Bofinger, E. (1975). Optimal condensation of distributions and optimal spacing of order statistics. J. Amer. Statist. Assoc. 70, 151-154. [22] Cane, G. J. (1974). Linear estimation of parameters of the Cauchy distribution based on sample quantiles. J. Amer. Statist. Assoc. 69, 243 245. [23] Chart, L. K. (1969). Linear quantile estimates of the location and scale parameters of the Cauchy distribution. Statist. Hefte 10, 277-282. [24] Chan, L. K. (1970). Linear estimation of the location and scale parameters of the Cauchy distribution based on sample quantiles. J. Amer. Statist. Assoc. 65, 851-859. [25] Chan, L. K. and N. N. Chan (1973). On the optimum best linear unbiased estimates of the parameters of the normal distribution based on selected order statistics. Skand. Aktuarietidskr. 1973, 120-128. [26] Chan, L. K., N. N. Chan and E. R. Mead (1971). Best linear unbiased estimates of the parameters of the logistic distribution based on selected order statistics. J. Amer. Statist. Assoc. 66, 889 892. [27] Chan, L. K., N. N. Chan and E. R. Mead (1973). Linear estimation of the parameters of the Cauchy distribution using selected order statistics. Utilitas Math. 3, 311-3 l 8. [28] Chan, L. K. and S. W. H. Cheng (1971). On the student's test based on sample percentiles from the normal, logistic, and Cauchy distributions. Technometrics 13, 12%137. [29] Chan, L. K. and S. W. Cheng (1972). Optimum spacing for the asymptotically best linear estimate of the location parameter of the logistic distribution when samples are complete or censored. Statist. Hefte 13, 41- 57. [30] Chan, L. K. and S. W. Cheng (1973). On the optimum spacing for the asymptotically best linear estimate of the scale parameter of the Pareto distribution. Tamkang J. Math. 4, 1 21.

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[31] Chan, L. K. and S. W. Cheng (1974). An algorithm for determining the asymptotically best linear estimate of the mean from multiply censored logistic data. J. Amer. Statist. Assoc. 69, 1027-1030. [32] Chan, L. K. and S. W. Cheng (1982). The best linear unbiased estimates of parameters using order statistics. Soochow J. Math. 8, 1-13. [33] Chan, L. K. and S. W. Cheng (1988). Linear estimation of the location and scale parameters based on selected order statistics. Commun. Statist. - Theory. Meth. 17(7), 2259-2278. [34] Chan, L. K., S. W. H. Cheng and E. R. Mead (1972). An optimum t-test for the scale parameter of an extreme-value distribution. Naval Res. Logist. Quart. 19, 715-723. [35] Chan, L. K. and A. B. M. L. Kabir (1969). Optimum quantiles for the linear estimation of the parameters of the extreme-value distribution in complete and censored samples. Naval Res. Logist. Quart. 16, 381 404. [36] Chan, L. K. and E. R. Mead (197la). Linear estimation of the parameters of the extreme-value distribution based on suitably chosen order statistics. IEEE Trans. on Reliab. R-20, 74-83. [37] Chan, L. K. and E. R. Mead (1971b). Tables to facilitate calculation of an asymptotically optimal t-test for equality of location parameters of a certain extreme-value distribution. IEEE Trans. on Reliab. R-20, 235 243. [38] Cheng, S. W. (1975). A unified approach to choosing optimum quantiles for the ABLE's. J. Amer. Statist. Assoc. 70, 155-159. [39] Cheng, S. W. (1978). Linear quantile estimation of parameters of double-exponential distribution. Soochow J. Math. 4, 39-49. [40] Cheng, S. W. (1983). On the most powerful quantile test of the scale parameter. Ann. Inst. Statist. Math. 35, 407-414. [41] Chernofl, H. (1971). A note on optimal spacings for systematic statistics, Mimeographed Report. Department of Statistics, Stanford University. [42] Chernoff, H., J. L. Gastwirth and M. V. Johns Jr. (1967). Asymptotic distribution of linear combinations of functions of order statistics with applications to estimation. Ann. Moth. Statist. 38, 52-72. [43] Cs6rg6, M. (1983). Quantile Processes with Statistical Applieations. Society for Industrial and Applied Mathematics, Philadelphia. [44] Dalenius, T. (1950). The problem of optimum stratification. Scand. Aktuarietidskr. 33, 203- 213. [45] David, H. A. (1981). Order Statistics. 2nd edn. John Wiley and Sons, Inc., New York. [46] Dyer, D. D. (1973). Estimation of the scale parameter of the chi distribution based on sample quantiles. Technometrics 15, 489-496. [47] Dyer, D. D. and C. W. Whisenand (1973). Best linear unbiased estimator of the parameter of the Rayleigh distribution-Part II: Optimum theory for selected order statistics. IEEE Trans on Reliab. R-22, 229 231. [48] Eisenberger, I. (1968). Testing the mean and standard deviation of a normal distribution using quantiles. Technometrics 10, 781-792. [49] Eisenberger, I. and E. C. Posner (1965). Systematic statistics used for data compression in space telemetry. J. Amer. Statist. Assoc. 60, 97-133. [50] Epstein, B. (1956). Simple estimators of parameters of exponential distributions when samples are censored. Ann. Inst. Statist. Math. 8, 15-26. [51] Epstein, B. (1960). Estimation from life test data. Technometrics 2, 447-454. [52] Eubank, R. L. (1981a). A density-quantile function approach to optimal spacing selection. Ann. Statist. 9, 494-500. [53] Eubank, R. L. (1981b). Estimation of the parameters and quantiles of the logistic distribution by linear functions of sample quantiles. Scand. Actuar. J. 1981, 229 236. [54] Eubank, R. L. (1986). Optimal Spacing Problems. In: Encyclopedia in Statistical Science (N. L. Johnson and S. Kotz, eds.), Vol. 6, pp. 452-458, John Wiley & Sons, New York. [55] Gupta, S. S. and M. Gnanadesikan (1966). Estimation of the parameters of the logistic distribution. Biometrika 53, 565-570.

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[56] Gupta, S. S., A. S. Qureshi and B. K. Shah (1967). Best linear unbiased estimators of the parameters of the logistic distribution using order statistics. Technometrics 9, 43-56. [57] Hammersley, J. M. and K. W. Morton (1954). The estimation of location and scale parameters from grouped data. Biometrika 41, 296-301. [58] Harter, H. L. (1961). Estimating parameters of negative exponential populations from one or two order statistics. Ann. Math. Statist 32, 1078-1090. [59] Harter, H. L. (1971). Some optimization problems in parameter estimation. In: Optimizing Methods in Statistics (J. S. Rustagi, ed.), pp. 33-62, Academic Press, New York. [60] Hassanien, K. M. (1968). Analysis of extreme-value data by sample quantiles for very large samples. J. Amer. Statist. Assoc. 63, 877-888. [61] Hassanien, K. M. (1969a). Estimation of the parameters of the logistic distribution of sample quantiles. Biometrika 56, 684-687. [62] Hassanien, K. M. (1969b). Estimation of the parameters of the extreme value distribution by use of two or three order statistics. Biometrika 56, 429436. [63] Hassanien, K. M. (1971). Percentile estimators for the parameters of the Weibull distribution. Biometrika 58, 673-676. [64] Hassanien, K. M. (1972). Simultaneous estimation of the parameters of the extreme value distribution by sample quantiles. Technometrics 14, 63-70. [65] Hassanein, K. M. (1974). Linear estimation of the parameters of the logistic distribution by selected order statistics for very large samples. Statist. Hefte 15, 65-70. [66] Hassanien K. M. (1977). Simultaneous estimation of the location and scale parameters of the gamma distribution by linear functions of order statistics. Seand. Actuar. J. 88-93. [67] Hassanein, K. M., A. K. Md. E. Saleh and E. F. Brown (1984). Quantile estimates in complete and censored samples from extreme-value and Weibull distributions. IEEE Trans. on Reliab. R-33, 370-373. [68] Hassanein, K. M., A. K. Md. E. Saleh and E. F. Brown (1985). Best linear unbiased quantile estimators of the logistic distribution using order statistics. J. Statist. Comput. Simul. 23, 123-131. [69] Hassanein, K. M., A. K. Md. E. Saleh and E. F. Brown (1986a). Estimation and testing of quantiles of the extreme-value distribution. J. Statist. Plann. Inf. 14, 389-400. [70] Hassanein, K. M., A. K. Md. E. Saleh and E. F. Brown (1986b). Best linear unbiased estimators for normal distribution quantiles for sample sizes up to 20. 1EEE Trans. on Reliab. R-35, 327-329. [71] Hosking, J. R. M. and J. R. Wallis (1987). Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29, 339 349. [72] Kaigh, W. D. and P. A. Lachenbruch (1982). A generalized quantile estimator. Commun. Statist.-Theor. Meth. 11, 2217-2238. [73] Kaminsky, K. S. (1972). Confidence intervals for the scale parameter using optimally selected order statistics. Teehnometries 14, 371 383. [74] Kaminsky, K. S. (1973). Comparison of approximate confidence intervals for the exponential scale parameter from sample quantiles. Teehnometries 15, 483~487. [75] Kaminsky, K. S. (1974). Confidence intervals and tests for two exponential scale parameters based on order statistics in compressed samples. Teehnometrics 16, 251-254. [76] Kaminski, K. S. and P. I. Nelson (1975). Best linear unbiased prediction of order statistics in location and scale families. J. Amer. Statist. Assoc. "70, 145-150. [77] Kappenman, R. F. (1987). Improved distribution quantile estimation. Commun. Statist.-Simul. Comput. 16, 30~320. [78] Koutrouvelis, I. A. (1981). Large sample quantile estimation of Pareto laws. Commun. Statist. Theor. Meth. A(10)2, 189-201. [79] Kubat, P. and B. Epstein (1980). Estimation of quantiles of location-scale distributions based on two or three order statistics. Technometric~ 22, 575-581. [80] Kulldorff, G. (1961a). Contributions to the Theory of Estimation .from Grouped and Partially Grouped Samples. Almqvist and Wiksell, Stockholm.

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[81] Kulldorff, G. (1961b). Estimation from Grouped and Partially Grouped Samples. John Wiley & Sons, New York. [82] Kulldorff, G. (1963a). Estimation of one or two parameters of the exponential distribution on the basis of suitably chosen order statistics. Ann. Math. Statist. 34, 1419 143l. [83] Kulldorff, G. (1963b). On the optimum spacing of sample quantiles from a normal distribution. Part I. Scand. Aktuarietidskr. 46, 143-156. [84] Kulldorff, G. (1964). On the optimum spacing of sample quantiles from a normal distribution. Part II. Scand. Aktuarietidskr. 47, 71-87. [85] Kulldorff, G, (1973). A note on the optimum spacing of sample quantiles from the six extreme value distributions. Ann. Statist. 1, 562-567. [86] Kulldorff, G. and K. V~.nnman (1973). Estimating location and scale parameters of a Pareto distribution by linear functions of order statistics. J. Amer. Statist. Assoc. 68, 218-227. [87] La Riccia, V. N. (1991). Smooth goodness-of- fit tests: A quantile function approach. J. Amer. Statist. Assoc. 86, 427-431. [88] LikeL J. (1985). Estimation of the quantiles of Pareto distribution. Math. Operationsfors. und Statist., Ser. Statist. 16, 541-547. [89] Lloyd, E. H. (1952). Least-squares estimation of location and scale parameters using order statistics. Biometrika 39, 88-95. [90] Mann, N. R. (1970). Estimators and exact confidence bounds for Weibull parameters based on a few ordered observations. Technometrics 12, 345-361. [91] Mik6, V. (1971). Efficiency-robust systematic linear estimates of location. J. Amer. Statist. Assoc. 66, 594-601. [92] Miyamoto, Y. (1972). Three-parameter estimation for a generalized gamma distribution by sample quantiles in large samples. J. Japan Statist. Soc. 3, 9-18. [93] Mosteller, F. (1946). On some useful "inefficient" statistics. Ann. Math. Statist. 17, 377-408. [94] Ogawa, J. (1951). Contributions to the theory of systematic statistics. I. Osaka Math. J. 3(2), 175-213, [95] Ogawa, J. (1960). Determination of optimal spacings for the estimation of the side parameters of an exponential distribution based on sample quantiles. Ann. Inst. Statist. Math. 12, 135 141. [96] Ogawa, J. (1976). A note on the optimal spacing of the systematic statistics - normal distribution. In: Essays in Probability and Statistics (Ikeda et al., eds.), pp. 467-474, Shinku Tsusho, Tokyo. [97] Parzen, E. (1978). A density-quantile function perspective on robust estimation. In: Robustness in Statistics (R. U Laudner and G. N. Wilkinson, eds.), pp.237-258, Academic Press, New York. [98] Rao, C. R. (1965). Linear Statistical Inference and Its Applications. John Wiley & Sons, Inc., New York. [99] Reiss, R.-D. (1980). Estimation of quantiles in certain nonparametric models. Ann. Statist. 8, 87-105. [100] Renyi, A. (1953). On the theory of order statistics, Acta. Math. Acad. Sci. Hung. 4, 191-231. [101] Rukhin, A. L. and W. E. Strawderman (1982), Estimating a quantile of an exponential distribution. J. Amer. Statist. Assoc'. 77, 159-162. [102] Saleh, A. K. Md. E. (1966). Estimation of the parameters of the exponential distribution based on order statistics in censored samples. Ann. Math. Statist. 37, 1717-1735. [103] Saleh, A. K. Md. E. (1967). Determination of the exact optimum order statistics for estimating the parameters of the exponential distribution from censored samples. Technometries 9, 279-292. [104] Saleh, A. K. Md, E. (1981). Estimating quantiles of exponential distribution. In: Statistics and Related Topics, North Holland Publishing. Co-editors: C~org6, M., Dawson, D., Rao, J. N. K., and Saleh, A. K. Md. E., 279-283. [105] Saleh, A. K. Md. E. and M. M. Ali (1966). Asymptotic optimum quantiles for the estimation of the parameters of the negative exponential distribution. Ann. Math. Statist. 37, 143 151. [106] Saleh, A. K. Md. E., M. Masoom Ali and D. Umbach (1983a). Estimating the quantile function of location- scale family of distributions based on a few selected order statistics. J. Statist. Plann. Infer. 8, 75-86.

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[107] Saleh, A. K. Md. E., M. Masoom Ali and D. Umbach (1983b). Simplified estimation of the parameters of exponential distribution from samples censored in the middle. Commun. Statist. Simula. Computa. 12(5), 609-627. [108] Saleh, A. K. Md. E., M. Masoom Ali and D. Umbach (1984). Tests of significance using selected sample quantiles. Statist. Prob. Lett. 2(5), 295-298. [109] Saleh, A. K. Md. E., M. Masoom All and D. Urnbach (1985). Large sample estimation of Pareto quantiles using selected order statistics. Metrika 32, 49-56. [110] Saleh, A. K. Md. E. and K. M. Hassanein (1986). Testing equality of location parameters and quantiles of s(> 2) location-scale distributions. Bull. lnst. Math. Acad. Sinica 14, 3949. [111] Saleh, A. K. Md. E. K. M. Hassanein and M. Masoom Ali (1988). Estimation and testing of hypotheses about the quantile function of the normal distribution. J. Infor. Opt. Sei. 9(1 ), 85-98. [112] Saleh, A. K. Md. E., K. M. Hassanein and M. Masoom Ali (1992). Estimation of quantiles using selected order statistics. Handbook o f the Logistic Distribution, ed: N. Balakrishnan, pp. 98-113, Marcel and Dekker, New York. [113] Saleh, A. K. Md. E., K. M. Hassanein and E. F. Brown (1985). Optimum spacings for the joint estimation and tests of hypothesis of location and scale parameters of the Cauchy distribution. Commun. Statist. - Theory. Meth. 14, 247-254. [114] Saleh, A. K. Md. E. and P. K. Sen (1985). Asymptotic relative efficiency of some joint-tests for location and scale parameters based on selected order statistics. Commun. Statist. - Theory. Meth. 14, 621 633. [115] Sarhan, A. E. and B. G. Greenberg (eds.) (1962). Contributions to Order Statistics. John Wiley and Sons, Inc., New York. [116] Sarhan, A. E. and B. G. Greenberg (1967). Linear estimates for doubly censored samples from exponential distribution with observations also missing from the middle. Bull. 1SI, 36 th Session 42, Book 2, 1195- 1204. [117] Sarhan, A. E., B. G. Greenberg and J. Ogawa (1963). Simplified estimates for the exponential distribution. Ann. Math. Statist. 34, 102-116. [118] S/irndal, C. E. (1962). Information from Censored Samples. Almqvist & Wiksells, Uppsala. [119] S~irndal, C. E. (1964). Estimation of the parameters of the gamma distribution by sample quantiles. Technometrics 6, 405-414. [120] Shelnutt, J. W., III (1966). Conditional Linear Estimation of the Scale Parameters of the Extreme Value Distribution by the Use of Selected Order Statistics. Unpublished M.S. Thesis, Air Force Institute of Technology. [121] Shelnutt, J. W., III, A. H. Moore and H. L. Harter (1973). Linear estimation of the scale parameter of the first asymptotic distribution of extreme values. IEEE Trans. on Relaib. R-22, 259-264. [122] Siddiqui, M. M. (1963). Optimal estimators of the parameters of negative binomial distributions from one or two order statistics. Ann. Math. Statist. 34, 117-121. [123] Siddiqui, M. M. and K. Raghunandanan (1967). Asymptotically robust estimators of location. J. Amer. Statist. Assoc. 62, 950-953. [124] Tischendorf, J. A. (1955). Linear Estimation Techniques Using Order Statistics. Ph.D. Thesis, Purdue University. [125] Ukita, Y. (1955). On the efficiency of order statistics. J. Hokkaido College o f Sci. and Art. 6, 54-65. [126] Umbach, D. (1994). Estimating functions of location and scale for the t distribution. Nonpar. Statist. 3, 369-377. [127] Umbach, D. and M. Masoom All (1990). Three goodness-of-fit tests based on selected order statistics. Soochow J. Statist. 16, 37-52. [128] Umbach, D. and M. Masoorn Ali (1991). A goodness-of-fit test for the exponential distribution. Pak. J. Statist. 7(3), 39-52. [129] Urnbach, D. and M. Masoorn Ali (1993). Conservative spacings for the estimation of functions of location and scale. J. InJb. Opt. Sci. 14, 309-319.

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[130] Umbach, D. and M. Masoom Ali (1996a). A smooth goodness-of-fit test using selected sample quantiles. J. Korean Statist. Soc. 25(3), 347-358. [131] Umbach, D. and M. Masoom Ali (1996b). Conservative spacings for the Weibull and extreme value distributions. Calcutta Statist. Assoc. Bull. 46(183 184), 169-180. [132] Umbach, D., M. Masoom Ali and K. M. Hassanein (1981a). Small sample estimation of the exponential quantiles with two order statistics. Aligarh J. Statist. 1, 113 120. [133] Umbach, D., M. Masoom Ali and K. M. Hassanein (1981b). Estimating Pareto quantiles using two order statistics. Commun. Statist. Theory. Meth. 10(19), 1933-1941. [134] Umbach, D., M. Masoom Ali and A. K. Md. E. Saleh (1984). Hypothesis testing for the double exponential distribution based on optimal spacing. Soochow J. Math. 10, 133-143. [135] Umbach, D., M. Masoom Ali and A. K. Md. E. Saleh (1986). Estimation of the parameters of the double exponential distribution using symmetric spacings. Soochow J. Math. 12, 99-108. [136] V~innman, K. (1976). Estimators based on order statistics from a Pareto distribution. J. Amer. Statist. Assoc. 71,704-708. [137] Weiss, L. (1963). On the asymptotic distribution of an estimate of a scale parameter. Naval Res. Logist. Quart. 10, 1-11. [138] Weissman, I. (1978). Estimation of parameters and large quantiles based on the K largest observations. J. Amer. Statist. Assoc. 73, 812-815. [139] Yamanouchi, Z. (1949). Estimates of the mean and standard deviation of a normal distribution from linear combinations of some chosen order statistics. Bull. Math. Statist. 3, 52-57.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 17 © 1998 Elsevier Science B.V. All rights reserved.

0

O

L-Estimation

J. R. M . H o s k i n g

1. Introduction We denote by Xk:n the k th smallest observation from a sample of size n drawn from the distribution of a random variable X. The ordered sample is XI:, _< X2:, _< ... _< Xn:,; the Xk:, are the order statistics. When basing inference on the order statistics, it is often convenient to work with linear combinations of them, of the form n

r = Zwi,.x,:n

.

(1)

i=1

These linear combinations are known as L-statistics or L-estimates. L-statistics have several advantages for use in statistical inference. If the rand o m variable Y is a linear transformation of X, Y = c~+ fiX, then the order statistics of X and Y satisfy Y,:n = e + flXi:n, and L-statistics computed from them satisfy (in an obvious notation) T (Y) = ct ~ wi,n ~- fiT (x). Thus T (r) = ct + fiT (x) if wi,n = 1 and T (y) = fiT (x) if ~ wi,, = 0; these are the required transformations for estimators of location and scale of the distribution of X, respectively. Because L-statistics are linear in the X~:,, they can often be more easily computed, and have their sampling properties evaluated, than more general estimators. By appropriate choice of the weights wi,~ it is possible to derive robust estimators, whose properties are not excessively dependent on correct specification of a statistical model. L-statistics are also a natural choice when the sample is censored, with only a subset of the order statistics, say X~:,, i = r, r + 1,... ,s (r > 1 or s < n) having been observed. In this survey of L-estimation we first describe some typical uses of L-estimates, illustrating the ways that L-statistics are commonly constructed. We then discuss in a little more detail some of the principal applications of L-statistics: summarizing data samples and estimating parameters and quantiles of univariate distributions. Finally we briefly discuss some applications of L-estimation to multivariate statistics; these tend to be much more complicated because it is not straightforward to extend the definition of order statistics to multivariate observations. 215

216

J.R.M. Hosking

Many estimators used in Statistics can be regarded as L-estimates, but not all are most conveniently treated as such, and there is considerable overlap between L-estimation and other approaches to parameter estimation and robust statistics. It should be borne in mind, therefore, that any account of L-estimation must necessarily overlap with several other topics, notably parameter estimation, kernel estimation, robust statistics and nonparametric regression.

2. Introductory examples 2.1 Efficient methods using the entire sample For a distribution with cumulative distribution function F((x - 2)/o-), completely specified except for a location parameter 2 and a scale parameter a, optimal L-estimates of the parameters can be derived from least-squares theory. The estimators were first obtained by Lloyd (1952). The order statistics of a random sample of size n have means and covariances of the form E&.

= =

+ 2 oej:

, ,

(2) (3)

where ~i:, and ~o~,j:,, are constants depending only the base distribution F. We write these expressions in vector form. Let 0 = [2 a] v be a vector containing the parameters; let X = [2(i. . . . . X,:~]"r be a vector containing the order statistics; let 1 be an n-vector each of whose elements is 1; let ¢ = [~l:n..- ~,:,]T; let ~ be the matrix with (i,j) element coij:,. Then we can write (2)-(3) as EX=21+a~=A0

,

varX = a2~ , where A is the n x 2 matrix A = [1

CJ.

We seek L-estimates )~ = y~,.W i .(~)v and ~ = y~.7i.n (~)v a/:n of 2 and a respectively; n Aim The best linear unbiased estimator ( " B L U E " ) 0 = [2 #IT of 0 is given by the Gauss-Markov theorem (see, e.g., Odell, 1982). It has the form 0 = (AT~"UIA)-IAT~[~ IX .

(4)

It is optimal in the sense that it has minimum variance among unbiased estimators, i.e. it satisfies E0 = 0 and, if 0 is any other L-statistic that is an unbiased estimator of 0, then the matrix vat 0 - var O is positive definite. Disadvantages of Lloyd's approach are that it requires the tabulation of means and variances of order statistics for every distribution and sample size for which it is used, and the inversion of the n x n matrix ~2. Gupta's (1952) "simplified linear estimates" overcome the second disadvantage by replacing the matrix ~ with the identity matrix. This gives surprisingly good results, at least for the Normal

L- Estima tion

217

distribution. Blom's (1958, 1962) "unbiased nearly best linear estimates" overcome both disadvantages. The covariance coi,j:n is first replaced by an asymptotic approximation (David, 1981, Eq. (4.6.5)), O')i'j:n ~

pi(1 - p j ) (n + 2)f(F-l(pi))f(F l(pj)) '

(5)

where p~ = i/(n + 1) and f(x) = dF(x)/dx is the probability density function of the standardized variate (X - 2)/a. This yields an ~ matrix whose inverse can be explicitly written down (e.g., Balakrishnan and Cohen, 1991, p. 217), and after some algebraic manipulations minimum-variance estimators are derived that require inversion of only a 2 x 2 matrix. The BLUEs and their derivatives are optimal only among the restricted class of L-estimates. There is no assurance that other estimators, such as maximumlikelihood estimators, may not be more efficient. However, it is possible to derive L-estimates that are asymptotically fully efficient in large samples (as n --+ oc). For location-scale models, the estimators were obtained by Bennett (1952) and more thoroughly studied by Chernoff et al. (1967). An L-statistic can be defined for arbitrary sample size n by permitting the weights wi,~ to approximate a smooth weight function, with nwi,, ~ J(i/(n + 1)). Chernoff et al. (1967) showed that by appropriate choice of the function J(.), estimates of location and scale parameters could be obtained that attained the Cramer-Rao lower bound asymptotically and were therefore maximally efficient. Consider the L-statistic

\n + l J

i=1

"

By Chernoff et al. (1967, Theorem 3), T is asymptotically Normally distributed, with asymptotic mean and variance given by

ET ~

I'

nvarT~2/

J(u)O(u) du , f

J(u)J(v)Q'(u)(~(v)u(1-v)dudv

,

0
LI(y) = - f ( y ) / f ( y ) ,

L2(y) = -{1 ÷ y f ( y ) / f ( y ) }

,

and define weight functions J(X)(.) and J(~)(-) by the matrix equation

[J()')(u)

J(Cr)(u)]=Ittl(Q(u))

L~z(Q(u))lI i v

where [ is the Fisher information matrix of a random sample drawn from the distribution of X. Then the statistics T()~) and T (~), defined as in (6) but with

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J . R . M . Hosking

weight functions J(~) and J(°) respectively, are asymptotically unbiased and maximally efficient estimators of 2 and ~. The foregoing discussion has concentrated on the case in which a complete sample is available. In doubly censored samples only a subset of the order statistics, say X~:~, i = r, r + 1 , . . . , s (r > 1 or s < n), is observed. Both the BLUEs and the asymptotically efficient L-estimates may also be derived for such samples, by essentially the same approach as for complete samples; however, the asymptotically efficient L-estimates take the slightly modified form

Wr,nXr:" ~- Ws,nXs:n ~- n -I i~=rJ

Xi:, ,

(7)

in which additional weight is given to the most extreme of the observed order statistics.

2.2. Quick methods using a few order statistics L-statistics that involve only a small number of order statistics are quicker to compute than statistics that involve the entire sample, yet can yield useful and accurate estimates of distribution properties. An example is the sample range, Rn = Xn:n - Xl:n, which has a long history. As well as giving an informal measure of the spread of the sample values, it can be used in formal inference procedures. For the N o r m a l distribution, for example, constants dn have been tabulated (e.g., David, 1981, p. 185) such that Rn/dn is an unbiased estimator of the standard deviation of the distribution. Relative to the minimum-variance unbiased estimator,

the estimator R,/d, has efficiency that decreases as the sample size increases, being 85% when n = l0 and 70% when n = 20. Formal hypothesis tests can also be based on the sample range. To test the hypothesis that the mean of a Normal distribution is equal to a specified value /t 0, for example, an analogue of the Student's t statistic is t* =_ (X - #o)/R,. Relative to the (optimal) t test, the test based on t* loses a fairly small amount of power (Lord, 1950). However, it can be sensitive to departures of the underlying distribution from exact Normality; Prescott (1975) suggested alternatives that are more robust. With the availability of modern computing power, the choice of statistics on the basis of computational simplicity has become less compelling. However, it may still be that data are difficult or expensive to obtain, and methods based on a few order statistics can still be worthwhile. Balakrishnan and Cohen (1991, p. 240) describe an experiment of Benson (1949) in which observations of textile strength made by an automatic testing machine were recorded as points on a chart, and note that "as accurate conversion of the points to numerical values would be time-consuming, and also an estimate based on inaccurate conversion would

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219

usually cause gross accumulated error, sample quantiles were used for estimation since they could easily be picked up from the chart". A useful approach in such cases is to determine an optimal choice of which order statistics to use for estimation. For location-scale models, this is straightforward in principle using the BLUE approach. For any subset of k order statistics, the B L U E and its accuracy based on these k order statistics can be derived; the optimal choice is then the subset that yield the L-estimate with the smallest variance. In practice an exhaustive search over all (~) subsets of k order statistics is prohibitively expensive, but becomes feasible for moderate values of n if the approximation (5) for coij:n and the analogous approximation {i:n ~ F-1 (Pi) for in are used to simplify the matrices A and £~ in the definition of the BLUE. Chan and Cheng (1988) investigated this approach and found its asymptotic efficiency to be quite high, even when only a small number of order statistics are used.

2.3. Robust estimation using L-statistics

The previous sections have described L-estimates that are in some sense optimal when the underlying distribution of the sample is known apart from location and scale parameters. In practice one may not have complete confidence that the underlying distribution can be exactly specified. In particular, the distribution may have longer tails than expected, the sample may be contaminated by values drawn from some other distribution, or some sample values may be subject to measurement error. These situations can give rise to outliers, observations that are discordant with the main body of the sample. Considerable effort has been directed towards devising statistics that are robust to deviations from assumptions of randomness and distributional form. Because outlying values are the most extreme order statistics, L-statistics that give zero weight to these order statistics are particularly good candidates to be robust estimators. A widely studied robust estimate of location is the trimmed mean, n

r

m~ = (n - 2r) -1 Z

X/:, ,

(8)

i=r+l

where r --- [n~]. Here the largest and smallest r observations, each representing a fraction ~ of the entire sample, are ignored when calculating the mean. Extreme cases of trimmed means are m0, the usual sample mean, and ma/2, interpreted as lim~71/2 m~, the sample median. These are the maximum-likelihood estimators of the centers of the Normal and double exponential distributions respectively. For estimating the center 2 of a symmetric distribution, the trimmed mean is unbiased, approximately Normally distributed in large samples (Bickel, 1965; Stigler, 1973) and can be used as the basis of a t-like statistic for hypothesis tests on the value of 2 (Tukey and McLaughlin, 1963). Appropriate choice of ~, the amount of trimming, depends on the degree of robustness required: larger amounts of trimming (larger values of ~) give protection against heavier-tailed distributions. For example, Crow and Siddiqui (1967) recommend a value of

220

J.R.M. Hosking

= 1/5 when possible distributions range from the Normal to the double-exponential, and values of ~ between 1/4 and 1/3, depending on n, when possible distributions range from the Normal to the Cauchy (a heavier-tailed distribution than the double exponential).

2.4 Adaptive L-statistics The appropriate degree of trimming of the mean depends on the distribution from which the sample is drawn, and in particular of the tail weight of the distribution. It is therefore reasonable to estimate the distribution's tail weight from the observed sample and to choose the degree of trimming on the basis of this estimate. This is an example of an adaptive L-statistics, in which the weights wi,, in (1) depend on the actual observed sample. Hogg (1967) proposed to choose the trimming parameter c~ based on the sample kurtosis, but later (Hogg, 1974) found that a better indicator of tail weight was

U(.2) - L(.2) °l

- vU

s)

'

where ~(fl) and L(fi) denote the averages of the [nil] largest and the [nil] smallest order statistics respectively. The population analogue of QI takes the value 1.75 for the Normal distribution and 1.93 for the double exponential distribution. Hogg (1974) therefore suggested the adaptive L-estimate

ml/8 ml/4

if Q1 < 1.81 , if 1.81 _< Q1 < 1.87 ,

m3/8 if QI_> 1.87 . De Wet and van Wyk (1979) proposed a modification of Hogg's scheme in which is a continuous function of Q1. This estimator compares favorably in performance with a number of other robust estimates of location (Barnett and Lewis, 1994, p. 146).

3. Single-sample problems 3.1. Estimation of location Estimation of a location measure of a probability distribution, given a random sample drawn from the distribution, can be divided into two cases. In parametric estimation, the distribution is regarded as being completely specified apart from a finite number of parameters. L-estimates in this case can be based on the BLUE or asymptotically efficient approaches described previously. In nonparametric estimation, no firm assumptions are made about the form of the distribution. In

221

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this case, choice of an appropriate estimator combines two considerations: efficiency or accuracy, when the distribution has some particularly plausible form; and robustness against the possibility that the distribution has some unexpected form or against contamination of the sample by aberrant observations drawn from a completely different distribution. Many robust L-estimates of location have been proposed, but robust estimation can be based on other approaches. Robust estimation in general has been described by Andrews et al. (1972), Huber (1981) and Hampel et al. (1986). Here we merely describe and compare a few of the more c o m m o n L-estimators. The sample mean is unbiased and an efficient estimator of location for the N o r m a l distribution, but is not particularly robust. The trimmed mean is defined in (8). The Winsorized mean is similar to the trimmed mean, but extreme observations are replaced by less extreme order statistics rather than being ignored altogether:

nr l n 1 (r+l)Xr+l:n+

)

Z Xi:n+(r+l)Xn-r:n i=r+2

,

(9)

where r = [n~]. The sample median is ~"x(,,+l)/2:n, 1 "r = ].1(X,/z:n +x,/2+l:,),

n odd, n even

Two other simple estimators, each based on three sample quantiles, are Gastwirth's (1966) estimator,

0.3X[n/3]+l:n + 0.42 + 0.3Xn [n/3]:~ ,

(10)

and the trimean, 0.25X[n/4]+l: n + 0.52 + 0.25X n [n/4]:n •

(11)

An example of the performance of some L-estimators of location is given in Table 1. The tabulated values are the variances, obtained by simulation of samples of size 20, of estimates of the center of symmetry of various symmetric distributions. The performance of the mean is seriously degraded when the parent distribution is non-Normal. The median is somewhat inefficient for the N o r m a l distribution and small deviations from it, but is the most robust of these statistics; it performs relatively well for the Cauchy distribution. The other location measures have intermediate performance characteristics. The foregoing discussion has concentrated on symmetric distributions - for which it is appropriate to use symmetric L-estimators, with wi,n = wn+l i,n in (1) because for such distributions the center of symmetry is a particularly obvious and natural measure of location. Robustness to asymmetry is also important in many applications, but is less easy to assess in general, because choice of a statistic to measure location in samples from asymmetric distributions is also a choice of a particular feature of the distribution as a location measure. Use of the sample median as a location measure, for example, implies that the population

222

J. R. M. Hosking

Table 1 Empirical variances of some L-estimators of location. Results reproduced from Exhibit 5 of Andrews et al. (1972). CN(~,/?) denotes a contaminated Normal distribution: X ~ N(#, er2) with probability 1 - ~, x ~ N O , , (/~o)2) with probability e. Sample size 20. Estimator

Distribution

Mean 10% trimmed mean 25% trimmed mean Median Gastwirth, (10) Trimean, (11)

Normal

CN(10%,3)

CN(10%,10)

Laplace

Cauchy

1.00 1.06 1.20 1.50 1.23 1.15

1.88 1.31 1.41 1.70 1.45 1.37

11.54 1.46 1.47 1.80 1.51 1.48

2.10 1.60 1.33 1.37 1.35 1.43

12548.0 7.3 3.1 2.9 3.1 3.9

median is regarded as the appropriate measure of location for an arbitrary asymmetric probability distribution. This may or may not be the case in practice, depending on the application for which the data are being used.

3.2. Estimation of scale Nonparametric estimation of scale is analogous to estimation of location for asymmetric distributions, in that choice of a particular scale measure for a data sample implies belief that a particular feature of a probability distribution is an appropriate measure of scale. Again, considerations of robustness apply - the effect of outliers and long-tailed distributions can be more serious for scale estimators than for location estimators (Davidian and Carroll, 1987) - and L-estimates are only one kind of estimate that may be considered. However, some reasonable scale measures for a probability distribution with quantile function Q(u) have the form of a linear functional of Q(.), flj(u)Q(u)du, with fd J(u) du = 0. In this case the smooth L-estimator (6) is a natural estimator of the scale measure. Some L-estimators of scale have a long history. The sample range, Xn:n --Xl:n, has been widely used as a quick estimator of scale, but it is not robust to outliers and its properties are dependent on the sample size. Gini's mean difference statistic

1
can be written as an L-estimator, G

,=,

2(2i - n

(n-Z

1)x':" ;

its history was traced back to Von Andrae (1872) by David (1968),

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223

Estimators initially derived for use with specific distributions may, of course, be suitable for more general use if they are suitably robust. An example is the asymptotically efficient L-estimator for the Normal standard deviation,

i=~l~D 1( i "~Yi-n \ n + l /

"

'

where ¢ - l ( u ) is the quantile function of the standard Normal distribution. A trimmed version of this estimator,

In(1 c~)]

E

i=[,~1+1

k,n + l J

' '

was recommended by Welsh and Morrison (1990) for making inferences in small contaminated samples, and was used by them in an astronomical application involving comparison of distributions of stellar velocities. Scale estimators that are analogs of trimmed means can alternatively be based on the ordered values Zi:, of the deviations Zi = IX/- m I of the sample values from the estimated location measure m. Such estimators include the trimmed standard deviation,

{ 1 [n,~] 2 }1/2 with m taken to be the mean or the trimmed mean m~, and the median deviation Z[,/2]:~, with m taken to be the median. These are not L-estimators as defined in (1), though they are clearly related to L-estimators. Related statistics have been used to test homogeneity of scale in several samples (Conover et al., 1981; Tiku and Balakrishnan, 1984; Carroll and Schneider, 1985).

3.3. Estimation of quantiles The quantile of nonexceedance probability p of a random variable X is denoted by Q(p) and is the value such that a proportion p of the realizations of X are less than or equal to Q(p). Quantiles are an important topic in statistical inference, being often of more direct interest than the parameters of the probability distribution that is fitted to the data. Nonparametric inference for quantiles, in which no assumptions are made about the distributions of the underlying random variable X, gives robust and often reasonably efficient procedures. Nonparametric confidence intervals for quantiles based on individual order statistics are described by David (1981, pp. 15-16). Here we are concerned with some recent developments that involve more complicated L-statistics. The sample quantile is the value analogous to Q(p) but computed from the observed data; for a random sample of size n the sample quantile of nonexceedance probability p is

224

J.R.M.

Q(p)=X/:n

Hosking

for --i-1 < p _ < - i . n

(12)

///

As a function of p, Q(p) has discontinuities at the points p = i / n , i = 1 , . . . , n - 1, and a continuous estimator may be preferred. It is reasonable to estimate Q(p) by X~:n when p = (i - 1 / 2 ) / n and by a weighted average of the order statistics whose indices are adjacent to the fractional value n p + 1/2 otherwise. Thus, following Parzen (1979), we are led to the estimator Q(p)=n

---p

Xi:n-}-n

p--

+l:n

n

for

i! n

2
n

with 0(P) left undefined f o r p < 1/(2n) o r p > 1 - 1/(2n). Further extensions of this approach involve all of the order statistics in the estimation of a single quantile, greatest weight being given to the order statistics X/:, for which i / ( n + 1) is close to p. Harrell and Davis (1982) defined a quantile estimator that is the L-statistic (1) with weights HD

/ ' ( n -~- l)

wi,n = wi, ~ =-- F ( p ( n + 1))F((1 - p ) ( n

+ 1))

tp(n+l) 1(1 - t ) (l-p)(n+l)-I dt .

× fi/n a(i 1)/~

Heuristically, this is justified by arguing that Q(p) .~ EX(n+l)p:~ for large n, and estimating the mean of the order statistic X(n+l)p:~, given by

EX/:.

=

/7(I"/ -l- 1) F(i)-~(n~

fl

-1 - - i) ~.

ui-l(1 - u)" iO(u) du

by the corresponding integral with Q ( u ) replaced by the sample quantile function Q(u) the sample quantile function is Q(p) as in (12), regarded as a function of p, 0 < p < 1. An alternative interpretation of the Harrell-Davis estimator, noted by Sheather and Marron (1990), is that it is the bootstrap estimator of EX(,+I)p:,. Kaigh and Lachenbruch (1982) proposed an estimator of Q(p) that is a U-statistic, the average, over all size-k subsamples drawn without replacement from the full size-n sample, of the sample quantile Q(p) for the subsample. This can also be written as an L-statistic, the weights in (1) being given by Wi'n =

KL

n -- i

n

Wi'n ~

k - r

k

'

where r = [p(k + 1)1 and (~) is defined to be zero if a < b. The parameter k must be chosen by the user. When k = n, the estimator is just the sample quantile 0(P) for the full sample. Smaller values o f k cause more order statistics to contribute to the estimator, and tend to reduce its variability, but may cause bias because EXp(k+l):k can be a poor approximation to Q(p) when k is small. Figure 1 shows

,

L-Estimation

03t

225

o3t

i,n ]

l 1

~,n n

1

n

Fig. t. Relative contributions of each observation to the Harrell-Davis and Kaigh Lachenbruch estimators of the upper quartile Q(0.75), for sample size n = 19; Kaigh-Lachenbruch estimator uses k=7.

examples of the weights of the Harrell-Davis and Kaigh-Lachenbruch estimators of the upper quartile Q(0.75). Sheather and Marron (1990) considered kernel estimators of Q(p). The idea of kernel estimation is to take a rough estimator of a function and a smooth "kernel function" and combine them via convolution to yield a smooth estimator. In the case of quantile estimation, the rough estimator is the sample quantile function Q(u), which has discontinuities at the points u = i/n. The kernel is a function K(.) that integrates to 1 and is generally taken to be positive and symmetric about 0, i.e., it is a symmetric probability density function. The kernel estimator is QK(p) = f./n 1 h - I K ( ( .

du

----~-~I~J(i-1)/nh 1g((bt-p)/h) du}Xi:n ; the latter form exhibits the kernel estimator as an L-estimator. The parameter h is a "bandwidth" that controls the amount of smoothing of the sample quantile function; the contribution of Q(u) to QK(p) is determined by the size of the distance ]u - p [ as a multiple of h. The alternative form

x ( i - ½ ) / n - p X:n i=l

h

K '-½)/"-P h

is often more convenient for computation and includes a normalization of the weights so that they sum to 1. Sheather and Marron (1990) pointed out that both the Harrell-Davis and K a i g ~ L a c h e n b r u c h quantile estimators can be regarded as kernel estimators. Their weight functions are essentially the density function of the beta distribution and the probability function of the hypergeometric distribution, respectively, and both of these can be approximated by the Normal density function for large n; the asymptotic bandwidths are h = { p ( 1 - p ) / ( n + 1)}1/2 for the Harrell-Davis estimator and h = { p ( 1 - p ) / k } ] / 2 for the Kaigh-Lachenbruch estimator.

226

J . R . M . Hosking

The mean square error of QK(p) as an estimator of Q(p) contains terms that are of asymptotic order n -1, n - l h and h 4 and signs that are +, - and + respectively (Sheather and Marron, 1990, Theorem 1). The O(n -1) term is the asymptotic variance of the sample quantile Q(p). The optimal choice of the bandwidth parameter is therefore h = O(n i/3), and the mean square error of the optimal kernel estimator differs only by a term of order n -4/3 from that of the sample quantile. This suggests that kernel estimators are not likely to give great improvements in accuracy over the sample quantile. In simulation experiments using samples of size 50 and 100 from the Normal, lognormal, exponential and double exponential distributions, Sheather and Marron (1990) found that this was generally the case, improvements of 15% in mean square error being the best that could be obtained overall. Nonetheless kernel estimates have been found to be useful in practice; recent examples of their use include Alam and Kulasekera (1993) and Moon and Lall (1994).

3.4. Estimation o f distribution parameters

For the nonparametric estimation problems discussed in the previous sections, it is sometimes unclear which features of a distribution or an estimator are most appropriate as the basis for inference, and L-estimates often have appeal on account of their simplicity and demonstrable robustness to outliers. In parametric problems of fitting a distribution to data, the asymptotic optimality of likelihoodbased methods is well established and alternative methods need to possess comparable desirable properties. Optimality properties of L-estimates can be based on the B L U E or asymptotic efficiency properties discussed above, and when these can be achieved simultaneously with a convenient explicit form for the estimators the L-estimates become candidates for practical use. The BLUE and related approaches have been outlined above, and are further described by David (1981, Sections 6.2-6.3) and Balakrishnan and Cohen (1991, Chapters 4 and 7). The asymptotically efficient estimators of Chernoff et al. (1967), also mentioned above, are useful in that for some distributions they have a simple explicit form. For example for the Normal distribution the estimators of the mean and standard deviation are t/

i=1

i=1

~

x/:n ;

for the Cauchy distribution, with probability density function f ( x ) = o- 17c 1{ 1 + ((x - 2)/o-)} -2, - o c < x < oc, the estimates of the parameters are )~

1+ = n~

sin(47rpi) tan(~rpi) '

8i~1 tan(~pi) cos 4(~pi)X/:n a = n

where pi = i / ( n + 1) - 1/2. The weights wi,n of these estimators are illustrated in Figure 2, and illustrate how for long-tailed distributions such as the Cauchy

227

L-Estimation

-:I[TTIII[ITTIIIIITITT -:] i

Win

_.,

2

Normal scale

it TTI[IFT 3t I

Wi, n

0

2

Cauchy location

-

-

3

TTT

TTT

0 1

,TTTT, ~e.

i

Cauchy scale

Fig 2. Relative contributions of each observation to Chernoff et at.'s estimates of parameters of the Normal and Cauchy distributions, for sample size n - 19.

efficient estimators give less weight to the extreme observations, which are often outliers. The methods discussed thus far have been concerned with location-scale families of distributions, but can be extended to more general distributions. An approach worthy of attention was used by Maritz and Munro (1967) for the generalized extreme-value distribution and by Munro and Wixley (1970) for the three-parameter lognormal distribution. It can be applied to any distribution with location and scale parameters and one or more shape parameters. It can be regarded as an iterative procedure based on Blom's modification of the BLUE. When a shape parameter fl is present the BLUE cannot be computed directly because the means and covariances of the order statistics in (2)-(3) depend on the unknown ft. To overcome this an iterative procedure is used. 1. Guess an initial value for ft. 2. Compute means and variances of order statistics, and the A and D matrices in (4), using the assumed value of ft. 3. Estimate the parameters, including fl, by the B L U E (4). 4. Repeat steps 2. and 3. until converged. Rather than use the full BLUE, to minimize computations it is convenient to use the asymptotic approximations (5) for O,)i,j: n and ~i:n F-1 (Pj) in the definition of the A and ~ matrices. Computation of the estimates is then very straightforward, comprising iterations of least-squares calculations. For many distributions this will be considerably simpler than full maximum-likelihood estimation. LaRiccia and Kindermann (1983) proved that the Munro Wixley parameter estimators for the three-parameter lognormal distribution are asymptotically as efficient as the maximum-likelihood estim