Exponential Distribution: Theory and Methods (Mathematics Research Developments)

MATHEMATICS RESEARCH DEVELOPMENTS

EXPONENTIAL DISTRIBUTION: THEORY AND METHODS

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MATHEMATICS RESEARCH DEVELOPMENTS

EXPONENTIAL DISTRIBUTION: THEORY AND METHODS

M. AHSANULLAH AND

G. G. HAMEDANI

Nova Science Publishers, Inc. New York

Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Ahsanullah, M. (Mohammad) Exponential distribution : theory and methods / Mohammad Ahsanullah, G.G. Hamedani. p. cm. Includes bibliographical references and index. ISBN 978-1-61324-566-8 (eBook) 1. Distribution (Probability theory) 2. Exponential families (Statistics) 3. Order statistics. I. Hamedani, G. G. (Gholamhossein G.) II. Title. QA273.6.A434 2009 519.2'4--dc22 2010016733

Published by Nova Science Publishers, Inc. New York

To Masuda, Nisar, Tabassum, Faruk, Angela, Sami, Amil and Julian MA To Azam , Azita , Hooman , Peter , Holly , Zadan and Azara GGH

Contents Preface 1.

ix

Introduction 1.1 Preliminaries

2. 2.1 2.2 2.3 2.4

Order Statistics Preliminaries and Definitions Minimum Variance Linear Unbiased Estimators Based on Order Statistics Minimum Variance Linear Unbiased Predictors (MVLUPs) Limiting Distributions

1 3 11 11 18 24 27

3.

Record Values 3.1 Definitions of Record Values and Record Times 3.2 The Exact Distribution of Record Values 3.3 Moments of Record Values 3.4 Estimation of Parameters 3.5 Prediction of Record Values 3.5 Limiting Distribution of Record Values

31 31 31 38 44 46 48

4.

Generalized Order Statistics 4.1 Definition 4.2 Generalized Order Statistics of Exponential Distribution

51 51 52

vi

Contents

5.

Characterizations of Exponential Distribution I 5.1 Introduction 5.2 Characterizations Based on Order Statistics 5.3 Characterizations Based on Generalized Order Statistics

65 65 66 86

6.

Characterizations of Exponential Distribution II 6.1 Characterizations Based on Record Values 6.2 Characterizations Based on Generalized Order Statistics

99 99 120

References

121

Index

143

Preface The univariate exponential distribution is the most commonly used distribution in modeling reliability and life testing analysis. The exponential distribution is often used to model the failure time of manufactured items in production. If X denotes the time to failure of a light bulb of a particular make, with exponential distribution, then P(X>x) represent the survival of the light bulb. The larger the average rate of failure, the bigger will be the failure time. One of the most important properties of the exponential distribution is the memoryless property; P(X>x+y|X>x) = P(X>y). Given that a light-bulb has survived x units of time, the chances that it survives a further y units of time is the same as that of a fresh light-bulb surviving y units of time. In other words past history has no effect on the light-bulb’s performance. The exponential distribution is used to model Poisson process, in situations in which an object actually in state A can change to state B with constant probability per unit. The aim of this book is to present various properties of the exponential distribution and inferences about them. The book is written on a lower technical level and requires elementary knowledge of algebra and statistics. This book will be a unique resource that brings together general as well as special results for the exponential family. Because of the central role that the exponential family of distributions plays in probability and statistics, this book will be a rich and useful resource for Probabilists, Statisticians and researchers in the related theoretical as well as applied fields. The book consists of six chapters. The first chapter describes some basic properties of exponential distribution. The second chapter describes order statistics and inferences based on order statistics. Chapter three deals with record values and chapter 4 presents generalized order statistics. Chapters 5 and 6 deal with the characterizations of exponential distribution based on order statistics, record values and generalized order statistics. Summer research grant and sabbatical leave from Rider University enabled the first author to complete part of the book. The first author expresses his sincere thanks to his wife Masuda for the longstanding support and

x

M.Ahsanullah and G.G.Hamedani

encouragement for the preparation of this manuscript. The second author thanks his family for their encouragement during the preparation of this work. He is grateful to Marquette University for partial support during preparation of part of this book. The authors wish to thank Nova Science Publishers for their willingness to publish this manuscript.

M. Ahsanullah G. G.Hamedani

About the Authors Dr. M.Ahsanullah is a Professor of Statistics at Rider University. He earned his Ph.D. from North Carolina State University ,Raleigh, North Carolina. He is a Fellow of American Statistical Association and Royal Statistical Society. He is an elected member of the International Statistical Institute. He is editor-in-Chief of Journal of Applied Statistical Science and Co-editor of Journal of Statistical Theory and Applications. He has authored and co-authored more than twenty books and published more than 200 research articles in reputable journals. His research areas are Record Values, Order Statistics, Statistical Inferences, Characterizations of Distributions etc. Dr. Hamedani is a Professor of Mathematics and Statistics at Marquette University in Milwaukee Wisconsin. He received his doctoral degree from Michigan State University, East Lansing, Michigan in 1971. He is Co-Editor of Journal of Statistical Theory and Applications and Member of Editorial Board of Journal of Applied Statistical Science and Journal of Applied Mathematics, Statistics and Informatics. Dr. Hamedani has authored or co-authored over 110 research papers in mathematics and statistics journals. His main research areas are characterizations of continuous distributions and differential equations

Chapter 1

Introduction The exponential family of distributions is a very rich class of distributions with extensive domain of applicability. The structure of the exponential family allows for the development of important theory as it is shown via a body of work related to this family in the literature. We will be using some terminologies in the next few paragraphs which will formally be defined later in the chapter. To give the reader some ideas about the nature of the univariate exponential distribution, let us start with a basic random experiment, a corresponding sample space and a probability measure. We follow the usual notational convention: X, Y , Z, . . . stand for real-valued random variables; boldface X , Y , Z , . . . denote vector-valued random variables. Suppose that X is a real-valued continuous random variable for the basic experiment with cumulative distribution function F and the corresponding probability density function f . We perform n independent replications of the basic experiment to generate a random sample of size n from X: (X1 , X2, . . ., Xn). These are independent random variables, each with the same distribution as that of X. If Xi 0 s are exponential random variables with cumulative distribution function F (x) = 1 − e−λx , x ≥ 0, where λ > 0 is a parameter, then ∑ni=1 Xi is distributed as Gamma with parameters n and λ. The random variable 2λ ∑ni=1 Xi has a Chisquare distribution with 2 n degrees of freedom. Consider a series system (a system which works only if all the components work) with independent components with common cumulative distribution function F (x) = 1 − e−λx , x ≥ 0, and let T be the life of the system. Then P (T > t) = P (min1≤i≤n Xi > t) =

2

M. Ahsanullah and G.G. Hamedani

P (X1 > t, X2 > t, . . ., Xn > t) = ∏ni=1 P(Xi > t) = e−nλt , which is an exponential random variable with parameter nλ. Let N be a geometric random variable with probability mass function P (N = k) = p qk−1, k = 1, 2, . . . where p + q = 1. Now if Xi 0 s are independent and identically distributed with cumulative distribution function F (x) = 1 − e−λx , x ≥ 0 and if V = ∑Ni=1 Xi is the geometrically  compounded random d d variable, then pV = Xi = means equal in distribution . To see this, let L (t) be the Laplace transform of V , then 

−tV

L (t) = E E e

|N




∞

=

∑

k=1

 −1 t t −k k−1 pq = 1+ . 1+ λ λp



d

Thus, p V = Xi . Suppose the random variable X has cumulative distribution function F (x) = 1 − e−λx , x ≥ 0, and Y = [X], the integral part of X, then Y has the geometric distribution with probability mass function P (Y = k) = pqk , k = 0, 1, . . . and p = 1 − e−λ , P (Y = y) = P (y ≤ X < y + 1) = F (y + 1) − F (y)   = e−λy − e−λ(y+1) = 1 − e−λ e−λy . Let Xk,n denote the kth smallest of (X1 , X2, . . ., Xn). Note that Xk,n is a function of the sample variables, and hence is a statistic, called the kth order statistic. Our goal in Chapter 2 is to study the distribution of the order statistics, their properties and their applications. Note that the extreme order statistics are the minimum and maximum values: X1,n = min{X1, X2, . . ., Xn}, andXn,n = max{X1, X2, . . ., Xn}. If X has cumulative distribution function F (x) = 1 − e−λx , x ≥ 0,
then 1 − n F1,n (x) = P (X1,n ≥ x) = e−nλx and Fn,n (x) = P (Xn,n ≤ x) = 1 − e−λx . Record values arise naturally in many real life applications such as in sports, environment, economics, business, to name a few. Let X be a random variable. We keep drawing observations from X and, from time to time, an observation will be larger than all the previously drawn observations: this observation is

Introduction

3

then called a “record”, and its value a record value, or, more precisely, an upper record value. The first observation is obviously a record. We call it the first record. The second upper record is the first observation whose value is larger than that of the first one. We can define the lower records similarly by considering lower values. In Chapter 3 we will study record values, in particular when the underlying random variable X has an exponential distribution. Order statistics and record values are special cases of generalized order statistics. Many of their properties can be obtained from the generalized order statistics. In chapter 4, we have presented generalized order statistics of exponential distribution. The problem of characterizing a distribution is an important problem which has attracted the attention of many researchers in recent years. Consequently, various characterization results have been reported in the literature. These characterizations have been established in many different directions. The goal of Chapters 5 and 6 is to present characterizations of the exponential distribution based on order statistics and based on generalized order statistics (Chapter 5) as well as based on record values (Chapter 6). For the sake of self-containment, we mention here some elementary definitions, which most of the readers may very well be familiar with them. The readers with knowledge of introduction to probability theory may skip this chapter all together and go straight to the next chapter.

1.1. Preliminaries Definition 1.1.1. A random or chance experiment is an operation whose outcome cannot be predicted with certainty. We denote a random experiment with E. Throughout this book “experiment” means “random experiment”.

4

M. Ahsanullah and G.G. Hamedani Examples 1.1.2. (a) Flipping a coin once. (b) Rolling a die once.

Definition 1.1.3. The set of all possible outcomes of an experiment E is called the sample space for E and is denoted by S. Examples 1.1.4. Sample spaces corresponding to Examples (a) and (b) above are: Sa = {H, T }, H for heads and T for tails; Sb = {1, 2, . . ., 6}. Note that the set {even, odd} is also an acceptable sample space for E of Example 1.1.2 (b), so sample space is not unique. Event 1.1.5. An event is a collection of outcomes of an experiment. Hence every subset of sample space is an event. We denote events with capital letters A, B, C , . . .. We denote two events are called mutually exclusive if they have no common elements. Definition 1.1.6. A probability function is a real-valued set function defined on the power set of S (P (S)), denoted by P, whose range is a subset of [0, 1], i.e. P : P (S) → [0, 1], satisfying the following Axioms of probability (i) P (A) ≥ 0 for any A ∈ P (S). (ii) P (S) = 1. (iii) If A1, A2 , . . . is a sequence (finite or infinite) of mutually exclusive events (subsets) of S ( or elements of P (S) ), then P(A1 ∪ A2 ∪ · · · ) = P (A1 ) + P (A2 ) + · · · . Definition 1.1.7. A random variable (rv for short) is a real-valued function defined on S, a sample space for an experiment E. We denote rv 0 s with capital letters X,Y, Z, . . . (as mentioned before) and their values with lower case letters x, y, z, . . .. Range of a rv X is the set of all possible values of X and is denoted by R (X) .

Definition 1.1.8. A rv X is called (i) discrete if R (X) is countable; (ii) continuous if R (X) is an interval and P (X = x) = 0, for all x ∈ R (X) ; (iii) mixed if X is neither discrete nor continuous. Definition 1.1.9. Let X be a rv. The cumulative distribution function (cd f ) of X denoted by FX is a real-valued function defined on R whose range is a subset of [0, 1]. FX is defined by FX (t) = P (X ≤ t),

t ∈ R.

Properties of cd f FX : (i) limt→+∞ FX (t) =01 ; (ii) FX is non-decreasing on R; (iii) FX is right-continuous on R. Proposition 1.1.10. The set of discontinuity points of a distribution function is at most countable. Remark 1.1.11. A point x is said to belong to the support of the cd f F if and only if for every ε˙ > 0, F (x + ε) − F (x − ε) > 0. The set of all such points is called the support of F and is denoted by Supp F. We will restrict our attention, throughout this book, to continuous rv 0 s, in particular exponential rv. Definition 1.1.12. Let X be a continuous rv with cd f FX . Then the probability density function (pd f ) of X (or pd f corresponding to cd f FX ) is denoted by f X and is defined by ( d FX (t), if derivative exists, fX (t) = dt 0, otherwise. Remark 1.1.13. Since FX is continuous and non-decreasing, its derivative exists for all t, except possibly for at most a countable number of points in R. We define f X (t) = 0 at those points.

6

M. Ahsanullah and G.G. Hamedani Properties of pd f f X : (i) fX (t) ≥ 0 for all t ∈ R; R (ii) R f X (t)dt = 1.

Definition 1.1.14. The rv X has an exponential distribution with location parameter µ (−∞ < µ < ∞) and scale parameter σ (σ > 0) if its cd f is given by ( 0, t < µ, FX (t) = −λ(t−µ) , t ≥ µ, 1−e where λ = σ1 . Graph of FX for µ = 0 and different values of λ It is clear that dtd FX (t) exists everywhere except at t = µ, so the corresponding pd f of FX is given by ( λe−λ(t−µ) , t > µ, fX (t) = 0, otherwise,

Figure 1.1. Graph o f FX f or µ = 0 and di f f erent values o f λ.

Introduction

7

Graph of fX for µ = 0 and different values of λ We use the notation X ∼ E (µ, λ) for such a rv. The rv X ∼ E (0, λ) will be denoted by X ∼ E (λ). We use the notation X ∼ E (1) for the standard exponential random variable.

Figure 1.2. Graph o f f X f or µ = 0 and di f f erent values o f λ. We observe that the condition P(X > s + t|X > s) = P (X > t) is equivalent to 1 − F (s + t) = (1 − F (s)) (1 − F (t)). Now, if X is a non-negative and nondegenerate rv satisfying this condition, then cd f of X will be F (x) = 1 − e−λx , x ≥ 0. To see this, note that condition 1 − F (s + t) = (1 − F (s)) (1 − F (t)) will lead to the condition 1 − F (nx) = (1 − F (x))n , for all n ≥ 1 and all x ≥ 0,
n that is, 1 − F (x) = 1 − F( nx ) . The solution of this last equation with boundary conditions F (0) = 0 and F (∞) = 1 is F (x) = 1 − e−λx . The hazard rate ( f (x) / (1 − F (x))) is constant for E (µ, λ). In fact E (µ, λ) is the only family of continuous distributions with constant hazard rate. It can easily be shown that the constant (λ) hazard rate of a continuous cd f F together with boundary conditions F (0) = 0 and F (∞) = 1 imply that F (x) = 1 − e−λx .

8

M. Ahsanullah and G.G. Hamedani

The linear exponential distribution with increasing hazard rate has pd f of the form 2 f (x) = (λ + θx) e−(λx+θx /2), λ, θ > 0, x ≥ 0, 2 and the corresponding cd f is F (x) = 1 − e−(λx+θx /2) , λ, θ > 0, x ≥ 0. The hazard rate is λ + θx. If θ = 0, then it is the exponential with cd f F (x) = 1 − e−λx .

If X ∼ E (λ), then P (X > s + t|X > s) = P (X > t) for all s,t ≥ 0. This property is known as memoryless property of the standard exponential random variable (or distribution). The pth quantile of a rv X is defined by F −1 (p). For X ∼ E (λ), we have
ln(1−p) F −1 (p) = − λ . The first, second and fourth quartiles are 1λ ln 43 , ln2 λ and ln 4 respectively. λ Definition 1.1.15. Let X be a continuous rv with pd f f X , then the rth moment of X about the origin is defined by 0

r

µr = E[X ] =

Z R

xr f X (x) dx,

r = 0, 1, . . . ,

provided the integral is absolutely convergent. Note that throughout this book we will use the notation E [h (X)] = R h (x) dFX (x) for the expected value of the rv h (X).

R

Remarks 1.1.16. 0 0 0 0 (a) µ0 = 1, µ1 = E [X] is expected value or mean of X . σ2X = µ2 − µ12 is variance of X and σX is standard deviation of X. 0 (b) The rth moment of X about µX = µ1 is defined by µr = E [(X − µX )r ] =

Z R

(x − µX )r f X (x) dx,

r = 1, 2, . . . ,

provided the right hand side (RHS) exists. Note that µ2 = σ2X . (c) It is easy to show that from µr 0 s one can calculate µ0r 0 s and vice versa. In fact if the moments about any real number a are known, then moments about

Introduction

9

any other real number b can be calculated from those about a. Moments about zero, µ0r 0 s , are the most common moments used. Example 1.1.17. Let X ∼ E (λ). Find all the moments of X which exist. Solution: 0

µr =

Z ∞ 0

xr λe−λx dx =

Γ (r + 1) , λr

r = 1, 2, . . . .

Definition 1.1.18. Let X be a continuous rv with pd f f X . The MGF (Moment Generating Function) of X denoted by MX (t) is defined by   Z MX (t) = E etX = etx f X (x) dx, R

for those t 0 s for which the RHS exists. Properties of MGF : (i) MX (0) = 1; 0 (r) (r) (ii) MX (0) = µr , r = 1, 2, . . ., where MX (0) is the rth derivative of the MGF evaluated at 0. Example 1.1.19. For X ∼ E (µ, λ), the MGF is MX (t) =

Z ∞ µ

e λe tx

−λ(x−µ)

dx = e

µt

Z ∞ 0

λe−(λ−t)x dx = λeµt (λ − t)−1 , if t < λ,

from which we obtain 0 1 (1) µ1 = MX (0) = µ + , λ 0 2 2µ (2) 2 µ2 = MX (0) = µ + + 2 . λ λ

2 1 2 So, µX = µ + λ1 , σ2X = µ2 + 2µ λ + λ2 − (µ + λ ) =

1 λ2

and σX = λ1 .

For X ∼ E (λ), MX (t) = λ (λ − t)−1 , i f t < λ and (r)

MX (t) = λ (r!) (λ − t)−(r+1) , for r = 1, 2, . . . ,

10

M. Ahsanullah and G.G. Hamedani 0

Then µr =

r! λr

0

= λr µr−1, which is a recurrence relation for the moments of E (λ). n

The nth cumulant of a rv X is defined by Kn = dtd n ln MX (t) | t=0 . Here 0 ln 0 is used for natural logarithm. For X ∼ E (λ), MX (t) = λ (λ − t)−1 , t < λ and Kn = Γ (n) /λn. Remarks 1.1.20. If X1, X2, . . ., Xn form an independent sample from an exponential distribution with parameter λ, then ˆ = 1 , where X = 1 ∑n Xi; (i) method of moments estimator of λ is λ n i=1 X (ii) maximum likelihood estimator of λ is also λ∗ = X1 ; (iii) entropy of λ is 1 − ln λ. For E (µ, λ), the maximum likelihood estimators of
µ and respecλ are given by µ∗ = X1,n and λ∗ = 1/ X − X1,n tively, where, as mentioned before, X1,n = min{X1 , X2, . . ., Xn}. The entropy of E (µ, λ) denoted by E X is EX =

Z ∞ µ

(− ln f (x)) f (x) dx =

Z ∞

h i − ln λe−λ(x−µ) λe−λ(x−µ)dx = 1 − ln λ,

µ

which does not depend on location parameter µ. It is the same as the entropy of the exponential distribution E (λ).

Chapter 2

Order Statistics 2.1. Preliminaries and Definitions Let X1, X2, . . ., Xn be n independent and identically distributed (i.i.d.) rv 0 s with common cd f F and pd f f . Let X1,n ≤ X2,n ≤ · · · ≤ Xn,n denote the order statistics corresponding to 1 , X2, . . ., Xn. We call Xk,n, 1 ≤ k ≤ n, the kth order statistic based on a sample X1 , X2, . . ., Xn. The joint pd f of order statistics X1,n, X2,n , . . ., Xn,n has the form f 1,2,...,n:n (x1 , x2, . . ., xn) ( n! ∏nk=1 f (xk ) , −∞ < x1 < x2 < · · · < xn < ∞, = 0, otherwise. Let f

k:n

denote the pd f of Xk,n. From (2.1.1) we have

fk:n (x) =

Z Z

(2.1.1)

...

= n! f (x)

Z

f 1,2,...,n:n (x1 , . . ., xk−1 , x, xk+1, . . ., xn ) dx1 · · ·dxk−1 dxk+1 · · ·dxn

Z Z

...

Z k−1

n

∏ f (x j ) ∏ j=1

f (x j )dx1 · · ·dxk−1 dxk+1 · · ·dxn , (2.1.2)

j=k+1

where the integration is over the domain −∞ < x1 < · · · < xk−1 < xk+1 < · · · < xn < ∞.

12

M. Ahsanullah and G.G. Hamedani

n The symmetry of ∏k−1 j=1 f (x j ) with respect to x1 , . . ., xk−1 and that of ∏ j=k+1 f (x j ) with respect to xk+1 , . . ., xn help us to evaluate the integral on the RHS of (2.1.2) as follows:

Z Z

...

Z k−1

∏

j=1

=

n

f (x j )

∏

f (x j )dx1 · · ·dxk−1 dxk+1 · · ·dxn

j=k+1

k−1 1 (k − 1)! ∏ j=1

Z x −∞

f (x j ) dx j

1 (n − k)!

n

∏

Z ∞

j=k+1 x

f (x j ) dx j

= (F (x))k−1 (1 − F (x))n−k / (k − 1)! (n − k)!. (2.1.3) Combining (2.1.2) and (2.1.3), we arrive at fk:n (x) =

n! (F (x))k−1 (1 − F (x))n−k f (x) . (k − 1)! (n − k)!

(2.1.4)

Clearly, equality (2.1.3) immediately follows from the corresponding formula for cd f 0 s of single order statistics, but the technique, which we used to arrive at (2.1.3), is applicable for more complicated situations. The following exercise can illustrate this statement. The joint pd f f k(1),k(2),...,k(r):n (x1 , x2, . . ., xr ) of order statistics Xk(1),n , Xk(2),n , . . ., Xk(r),n , where 1 ≤ k (1) < k (2) < · · · < k (r) ≤ n, is given by fk(1),k(2),...,k(r):n (x1 , x2, . . ., xr ) =

r+1 r n! k( j)−k( j−1)−1 (F (x ) − F (x )) j j−1 ∏ ∏ f (x j ) , ∏r+1 j=1 j=1 (k ( j) − k ( j − 1) − 1)! j=1

if x1 < x2 < · · · < xr , and = 0, otherwise. In particular, if r = 2, 1 ≤ i < j ≤ n, and x1 < x2 , then f i, j:n (x1 , x2 ) =

n! (i − 1)! ( j − i − 1)! (n − j)!

× (F (x1 ))i−1 [F (x2) − F (x1 )] j−i−1 [1 − F (x2)]n− j f (x1 ) f (x2 ) .

Order Statistics

13

The conditional pd f of X j,n given Xi,n = x1 is

=

(n − i)! ( j − i − 1)! (n − j)!



f j|i,n (x2 |x1 )  j−i−1  n− j f (x2 ) F (x2 ) − F (x1 ) 1 − F (x2) . 1 − F (x1 ) 1 − F (x1) 1 − F (x1 )

Thus, X j,n given Xi,n = x1 is the ( j − i)th order statistic in a sample of n − i from 2 )−F(x1 ) . For F (x) = 1 − e−x , truncated distribution with cd f Fc (x2 |x1) = F(x1−F(x 1) x ≥ 0, we will have Fc (x2 |x1) = 1 − e−(x2−x1) , x2 ≥ x1 . If X ∼ E (1) and Z1,n ≤ Z2,n ≤ · · · ≤ Zn,n are the n order statistics corresponding to a sample of size n from X, then it can be shown that the joint pd f of Z1,n , Z2,n, . . ., Zn,n is ( n n!e−(∑i=1 zi ) , 0 ≤ z1 ≤ z2 ≤ · · · ≤ zn < ∞, f 1,2,...,n (z1 , z2, . . ., zn ) = 0, otherwise. Using the transformation Wi = (n − i + 1) (Zi,n − Zi−1,n ), i = 1, 2, . . ., n with Z0,n = 0, we obtain the joint pd f of W1, W2, . . .,Wn as ( n e−(∑i=1 wi ) , 0 ≤ wi < ∞, i = 1, 2, . . ., n, f1,2,...,n (w1 , w2, . . ., wn) = 0, otherwise. Thus, W1, W2, . . .,Wn are i.i.d. exponential with cd f F (w) = 1 − e−w , w ≥ 0. Hence we can write d

Zk,n =

W2 Wk W1 + +···+ , n n−1 n−k+1

k = 1, 2, . . ., n,

(2.1.5)

where Wi0 s are i.i.d. with cd f F (x) = 1 − e−x , x ≥ 0. d

Clearly, nX1,n = X ∼ E (1). Since E [Wi ] = 1, Var [Wi] = 1, from (2.1.5) it follows that   k k Wi 1 =∑ , E [Zk,n] = ∑ E n−i+1 i=1 i=1 n − i + 1   k k 1 Wi Var [Zk,n] = ∑ Var =∑ , 1 ≤ k ≤ n, 2 n−i+1 i=1 i=1 (n − i + 1)

14

M. Ahsanullah and G.G. Hamedani

and 

k

Wi Cov (Zk,n , Zs,n ) = ∑ Var n−i+1 i=1



k

=∑

i=1

1 (n − i + 1)2

,

k ≤ s.

h i k , k ≥ 1, n ≥ 1, then we have the following Furthermore, letting αki,n = E Xi,n theorems (see, Joshi, (1978)). Theorem 2.1.1. αk1,n = nk αk−1 1,n , k ≥ 1, n ≥ 1. Proof. αk1,n

=

Z ∞

k

x ne

−nx

dx =

0

−xk e−nx |∞ 0 +

Z ∞ 0

k kxk−1 e−nx dx = αk−1 . n 1,n

Theorem 2.1.2. αki,n = αki−1,n−1 + nk αk−1 i,n , k ≥ 1, 2 ≤ i ≤ n. Proof. For k ≥ 1 and 2 ≤ i ≤ n, αk−1 i,n =

n! (i − 1)! (n − i)!

Z ∞

xk−1 1 − e−x

0


i−1

e−(n−i+1)x dx.

Integrating by parts, we obtain αk−1 i,n

n! = [ (i − 1)! (n − i)!k −

Z ∞

Z ∞ 0

(i − 1) xk 1 − e−x

0

=

(n − i + 1) xk 1 − e−x

n! [n (i − 1)! (n − i)!k − (i − 1)

Z ∞ 0

k

Z ∞


i−2

x 1 − e−x


i−2

e−(n−i+1)xdx

e−(n−i+2)x dx]

xk 1 − e−x

0


i−1


i−1

e−(n−i+1)xdx

e−(n−i+1)x dx]

n n = αki,n − αki−1,n−1. k k Thus, k . αki,n = αki−1,n−1 + αk−1 n i,n

Order Statistics

15

Theorem 2.1.3. Let αi, j,n = E [Xi,n X j,n], 1 ≤ i < j ≤ n, then αi,i+1,n = α2i,n +

1 αi,n , n−i

1 ≤ i ≤ n − 1,

and αi, j,n = αi−1, j,n +

1 αi, j−1,n, n− j+1

1 ≤ i < j ≤ n,

j − i ≥ 2.

Proof.

=

  0 αi,n = E Xi,n Xi+1,n Z ∞Z ∞
i−1 −x −(n−i)x i+1 xi 1 − e−xi e ie dxi+1 dxi

n! (i − 1)! (n − i + 1)! 0 xi Z ∞
i−1 −x n! = xi 1 − e−xi e i Ixi dxi , (i − 1)! (n − i + 1)! 0

where Ixi =

Z ∞

−(n−i)xi+1

e

xi

dxi+1 =

xi+1 e−(n−i)xi+1 |∞ xi + (n − i)

Z ∞ xi

xi+1 e−(n−i)xi+1 dxi+1 .

Thus, αi,n =

Z ∞Z ∞
i−1 −x −(n−i)x n! (n − i) i+1 xi xi+1 1 − e−xi e ie dxi+1 dxi (i − 1)! (n − i + 1)! 0 xi Z ∞
i−1 −(n−i+1)x n! i − x2i 1 − e−xi e dxi (i − 1)! (n − i + 1)! 0

= (n − i) αi,i+1,n − (n − i) α2i,n . Upon simplification, we obtain αi,i+1,n = α2i,n +

1 αi,n , n−i

1 ≤ i ≤ n − 1.

For j > i + 1,   αi, j−1,n = E Xi,nX 0j−1,n

n! (i − 1)! ( j − i − 1)! (n − i + 1)!

16

M. Ahsanullah and G.G. Hamedani ×

Z ∞Z ∞ 0

xi

xi 1 − e−xi

=


i−1

(e−xi − e−x j ) j−i−1e−xi e−(n− j+1)x j dx j dxi

n! (i − 1)! (n − i − 1)! (n − j)!

Z ∞ 0

xi 1 − e−xi


i−1

e−xi Jxi dxi ,

where Jxi =

Z ∞ xi

(e−xi − e−x j ) j−i−1 e−(n− j+1)x j dx j Z ∞

= (n − j + 1)

xi

x j (e−xi − e−x j ) j−i−1 e−(n− j+1)x j dx j

− ( j − i − 1)

Z ∞ xi

(e−xi − e−x j ) j−i−2 e−(n− j+2)x j dx j .

Thus, n! αi, j−1,n = (i − 1)! ( j − i − 1)! (n − j)! ·[(n − j + 1) − ( j − i − 1)

Z ∞

Z

xi ∞ xi

−xi

x j (e

Z ∞ 0

xi 1 − e−xi

−x j j−i−1 −(n− j+1)x j

−e

)

e


i−1

e−xi

dx j

(e−xi − e−x j ) j−i−2e−(n− j+2)x j dx j ]dxi

= (n − j + 1) αi, j,n − (n − j + 1) αi−1, j,n . Upon simplification, we arrive at αi, j,n = αi−1, j,n +

1 αi, j−1,n, n− j+1

1 ≤ i < j ≤ n,

j − i ≥ 2.

The relation, in this case, to uniform rv is interesting. If we let U be a uniformly distributed rv on (0, 1) and Ui,n is the ith order statistic from U, then it can be shown that d

d

Xi,n = − lnUn−i+1,n or equivalently Xi,n = − ln (1 −Ui,n ). Let f 1,...,r−1,r+1,...,n|r (x1 , . . ., xr−1, xr+1 , . . ., xn|v) denote the joint conditional pd f of order statistics X1,n , . . ., Xr−1,n , Xr+1,n, . . ., Xn,n given that Xr,n = v. We suppose that fr:n (v) > 0 for this value of

Order Statistics

17

v, where fr:n ,as usual, denotes the pd f of Xr,n. The standard procedure gives us the required pd f : f 1,...,r−1,r+1,...,n|r (x1 , . . ., xr−1, xr+1, . . ., xn |v) = f1,2,...,n:n (x1 , . . ., xr−1, v, xr+1, . . ., xn ) / fr:n (v) . (2.1.6) Upon substituting (2.1.1) and (2.1.4) in (2.1.6), we obtain f1,...,r−1,r+1,...,n|r (x1 , . . ., xr−1, xr+1, . . ., xn |v) r−1

= (r − 1)! ∏

j=1

n f (x j ) f (x j ) (n − j)! ∏ , F (v) j=r+1 1 − F (v)

x1 < · · · < xr−1 < xr+1 < · · · < xn , (2.1.7) and equal zero otherwise. Finally, we would like to present Fisher’s Information, I, for the order statistics from E (λ). Fisher’s Information for a continuous random variable X with pd f f (x, λ) and parameter λ, under certain regularity conditions, is given by  2  ∂ I = −E ln ( f (X, λ)) . ∂λ2 The exponential distribution E (λ) satisfies the regularity conditions and the Fisher’s Information for order statistics from this distribution are as follows: For X1,n ,



 ∂2  −λnX  1 I1 = −E ln nλe = 2. 2 ∂λ λ

For X2,n , 



∂2 ln{n(n − 1)λ(1 − e−λX )e−λ(n−1)X } I2 = −E ∂λ2 # " X 2 e−λX 1 =E 2+ λ (1 − e−λX )2

18 =

Z ∞ 0

"

M. Ahsanullah and G.G. Hamedani # x2 e−λx 1 + n(n − 1)λ(1 − e−λx )e−λ(n−1)xdx λ2 (1 − e−λx )2 Z

∞ x2 1 + n(n − 1)λ e−λnx dx λ2 1 − e−λx 0 1 2n(n − 1) ∞ 1 = 2+ ∑ (n + k)3 . λ λ2 k=0

=

For Xr,n , r > 2,   2  ∂ n! −λX r−1 −λ(n−r+1)X Ir = −E λ(1 − e ln ) e ∂λ2 (r − 1)!(n − r)! # " (r − 1)X 2e−λX 1 =E 2+ λ (1 − e−λX )2 " # Z ∞ 1 (r − 1)X 2e−λX n! + = λ(1 − e−λx )r−1e−λ(n−r+1)xdx 2 −λX 2 λ (r − 1)!(n − r)! (1 − e ) 0 Z

∞ n! 1 + x2 λ(1 − e−λx )r−3e−λ(n−r+2)xdx λ2 (r − 2)!(n − r)! 0 Z ∞ (n − 1)! 1 n(n − r + 1) · = 2+ x2 λ(1−e−λx )r−3e−λ(n−r+2)xdx λ (r − 2) (r − 3)!(n − r + 1)! 0 1 n(n − r + 1) 2 = 2+ E[Xr−2,n ] λ (r − 2)λ2 ( #) " r−3 n(n − r + 1) r−3 1 1 2 1 . +(∑ ) = 2 1+ ∑ 2 λ (r − 2) k=0 (n − k) k=0 n − k

=

2.2. Minimum Variance Linear Unbiased Estimators Based on Order Statistics We will use MVLUEs for minimum variance linear unbiased estimators. Let us begin from MVLUEs of location and scale parameters. Suppose that X has an absolutely continuous cd f F of the form   (x − µ) , −∞ < µ < ∞, σ > 0. F σ

Order Statistics

19

Further, assume that E [Xr,n ] = µ + αr σ,

r = 1, 2, . . ., n,

Var [Xr,n] = vrr σ ,

r = 1, 2, . . ., n,

2

Cov (Xr,n , Xs,n ) = Cov (Xs,n , Xr,n ) = vrs σ2 ,

1 ≤ r < s ≤ n.

Let X 0 = (X1,n, X2,n , . . ., Xn,n) . We can write

X ] = µ11 + σα α, E [X

(2.2.1)

where 1 = (1, 1, . . ., 1)0 , α = (α1, α2, . . ., αn)0 , and

X ) = σ2 ∑, Var (X

where ∑ is a matrix with elements vrs , 1 ≤ r ≤ s ≤ n. Then the MVLUEs of the location and scale parameters µ and σ are µˆ =

o 1 n 0 −1 −1 α10 − 1α 0 ) ∑ α ∑ (α X, ∆

(2.2.2)

σˆ =

o 1 n 0 −1 −1 1 ∑ (11α 0 − α 10 ) ∑ X, ∆

(2.2.3)

and

where

 2 
 −1 −1 ∆ = α 0 ∑ α 1 0 ∑ 1 − α0 ∑ 1 .

The variances and covariance of these estimators are given by
σ2 α0 ∑−1 α , Var (ˆµ) = ∆
σ2 1 0 ∑−1 1 ˆ = , Var (σ) ∆

(2.2.4)

(2.2.5)

20

M. Ahsanullah and G.G. Hamedani

and


σ2 α0 ∑−1 1 ˆ =− Cov (ˆµ,σ) . (2.2.6) ∆ The following lemma (see Garybill, 1983, p. 198) will be useful in finding the inverse of the covariance matrix. Lemma 2.2.1. Let ∑ = (σrs ) be an n × n matrix with elements, which satisfy the relation σrs = σsr = cr ds , 1 ≤ r, s ≤ n,

for some positive numbers c 1 , c2 , . . ., cn and d1 , d2, . . ., dn. Then its inverse ∑−1 = (σrs ) has elements given as follows: σ11 = c2 /c1 (c2 d1 − c1 d2) , σnn = dn−1/dn (cn dn−1 − cn−1 dn) , σk+1k = σkk+1 = −1/ (ck+1 dk − ck dk+1) , σkk = (ck+1 dk−1 − ck−1 dk+1) / (ck dk−1 − ck−1 dk ) (ck+1 dk − ck dk+1 ), k = 2, 3, . . ., n − 1, and σi j = 0, Let f (x) =

(

if |i − j| > 1.




exp (− (x − µ) /σ), −∞ < µ < x < ∞, 0 < σ < ∞, 0, otherwise. 1 σ

We have seen that r

1 j=1 n − j + 1

E [Xr,n ] = µ + σ ∑ r

Var [Xr,n ] = σ2 ∑

1

j=1

and

(n − j + 1)2 r

Cov (Xr,n , Xs,n ) = σ2 ∑

j=1 (n −

,

1 j + 1)2

r = 1, 2, . . ., n,

,

1 ≤ r ≤ s ≤ n.

Order Statistics

21

One can write that Cov (Xr,n , Xs,n ) = σ2 cr ds, where

r

cr =

1 ≤ r ≤ s ≤ n,

1

∑ (n − j + 1)2 ,

1 ≤ r ≤ n,

j=1

and ds = 1,

1 ≤ s ≤ n.

Using Lemma 2.2.1, we obtain σ j j = (n − j)2 + (n − j + 1)2 , σ

j+1 j

=σ

j j+1

2

= (n − j) ,

j = 1, 2, . . ., n,

j = 1, 2, . . ., n − 1,

and σi j = 0,

if |i − j| > 1, i, j = 1, 2, . . ., n.

It can easily be shown that 10 ∑

−1


= n2, 0, 0, . . ., 0 ,

α0 ∑

−1

= (1, 1, . . ., 1)

and ∆ = n2 (n − 1) . The MVLUEs of the location and scale parameters µ and σ are respectively µˆ = and

nX1,n − X , n−1


n X − X1,n σˆ = , n−1

(2.2.7)

(2.2.8)

where X = ∑r=1n r,n . The corresponding variances and covariance of the estimators are n

X

Var [ˆµ] =

σ2 , n (n − 1)

(2.2.9)

22

M. Ahsanullah and G.G. Hamedani Var [ˆσ] =

σ2 , n−1

(2.2.10)

and

σ2 . (2.2.11) n (n − 1) The remainder of this section will be devoted to MVLUEs based on censored samples. We consider the case, when some smallest and largest observations are missing. In this situation we construct the MVLUEs for location and scale parameters. Suppose now that the smallest r1 and largest r2 of these observations are lost and we can deal with order statistics ˆ =− Cov (ˆµ, σ)

Xr1 +1,n ≤ · · · ≤ Xn−r2 ,n . We will consider here the MVLUEs of the location and scale parameters based on Xr1+1,n , . . ., Xn−r2,n . Suppose that X has an absolutely continuous cd f F of the form F ((x − µ) /σ) ,

−∞ < µ < ∞,

σ > 0.

Further, we assume that E [Xr,n ] = µ + αr σ, Var [Xr,n ] = vrr σ2 , Cov (Xr,n , Xs,n ) = vrs σ , 2

r1 + 1 ≤ r ≤ n − r 2 , r1 + 1 ≤ r,

s ≤ n − r2 .

Let X 0 = (Xr1 +1,n , . . ., Xn−r2,n ), then we can write   α, E X 0 = µ11 + σα with 1 = (1, 1, . . ., 1)0 , α = (αr1 +1 , . . ., αn−r2 )0 , and   Var X 0 = σ2 ∑, where ∑ is an (n − r2 − r1 ) × (n − r2 − r1 ) matrix with elements vrs , r1 < r, s ≤ n − r2 . The MVLUEs of the location and scale parameters µ and σ based on the order statistics X 0 are o 1 n 0 −1 −1 α10 − 1α 0 ) ∑ X, (2.2.12) µˆ = α ∑ (α ∆

Order Statistics and σˆ = where

23

o 1 n 0 −1 −1 1 ∑ (11 α0 − α10 ) ∑ X, ∆

(2.2.13)

   2  −1 −1 −1 ∆ = α0 ∑ α 10 ∑ 1 − α0 ∑ 1 .

The variances and covariance of these estimators are given as
  σ2 α 0 ∑−1 α ∗ Var µˆ = , ∆
  σ2 10 ∑−1 1 Var σˆ∗ = , ∆ and


2 α 0 −1 1 σ ∑ . Cov µˆ∗ , σˆ∗ = − ∆ Now, we consider the exponential distribution with cd f F as


F (x) = 1 − exp {− (x − µ) /σ} ,

−∞ < µ < x < ∞,

(2.2.14) (2.2.15)

(2.2.16)

0 < σ < ∞.

Assume that the smallest r1 and the largest r2 observations are missing. Then the MVLUEs of σ and µ, based on the order statistics Xr1+1,n , . . ., Xn−r2,n , are ( ) n−r2 1 σˆ∗ = X j,n − (n − r1 ) Xr1+1,n + r2 Xn−r2 ,n , (2.2.17) n − r2 − r1 − 1 j=r∑ 1 +1 µˆ∗ = Xr1 +1,n − αr1 +1 σˆ∗ , where αr1 +1 =

(2.2.18)

r1 +1 1 1 E [Xr1 +1,n − µ] = ∑ . σ n − j+1 j=1

If r1 = r2 = 0, then (2.2.17) and (2.2.18) coincide with (2.2.8) and (2.2.7) respectively. The variances and covariance of the estimators are: ( ) r1 +1   α2r1 +1 1 2 , + ∑ Var µˆ∗ = σ n − r2 − r1 − 1 j=1 (n − j + 1)2   Var σˆ∗ =

σ2 , n − r 2 − r1 − 1

24

M. Ahsanullah and G.G. Hamedani

and

αr1+1 σ2 . n − r2 − r 1 − 1 Sarhan and Greenberg (1967) prepared tables of the coefficients, Best Linear Unbiased Estimators (BLUEs), variances and covariances of these estimators for n up to 10.
Cov µˆ∗ , σˆ∗ = −

2.3. Minimum Variance Linear Unbiased Predictors (MVLUPs) Suppose that X1,n , X2,n , . . ., Xr,n are r (r < n) order statistics from a distribution with location and scale parameters µ and σ respectively. Then the best (in the sense of minimum variance) linear predictor Xˆs,n of Xs,n (r < s ≤ n) is given by X − µˆ 1 − σα ˆ α), W 0sV −1 (X Xˆs,n = µˆ + αs σˆ +W where µˆ and σˆ are MVLUEs of µ and σ respectively, based on X 0 = (X1,n, X2,n , . . ., Xr,n ) , αs = E [(Xs,n − µ)/σ] , and W 0s = (W1s,W2s, . . .,Wrs ), where W js = Cov (X j,n , Xs,n ) , V −1

Here V

j = 1, 2, . . ., r.

is the inverse of the covariance matrix of X 0 .

Suppose that for the exponential distribution with cd f −∞ < µ < x < ∞,

F (x) = 1 − exp {− (x − µ) /σ} ,

0 < σ < ∞,

all the observations were available. We recall that r

1 , n − j+1 j=1

E [Xr,n ] = µ + σ ∑ r

Var [Xr,n ] = σ2 ∑

j=1

1 (n − j + 1)2

,

r = 1, 2, . . ., n,

Order Statistics and

r

1

Cov (Xr,n , Xs,n ) = σ2 ∑

j=1 (n −

j + 1)2

25

,

1 ≤ r ≤ s ≤ n.

To obtain MVLUEs for the case, when r1 + r2 observations are lost, we need to deal with the covariance matrix ∑ of size (n − r1 − r2 ) × (n − r1 − r2 ), elements of which coincide with Cov (Xr,n , Xs,n ) = σ2 cr ds , where

r1 + 1 ≤ r ≤ s ≤ n − r 2 ,

r

cr =

1

∑ (n − j + 1)2 ,

j=1

and ds = 1. We can obtain the inverse matrix ∑−1 using Lemma 2.2.1 as

∑

−1

 − (n − r1 − 1)2 (n − r1 − 1)2 + 1/cr1 +1  2 (n − r1 − 2)2 + (n − r1 − 1)2 − (n − r1 − 1)   0 − (n − r1 − 2)2   0 0 =  .. ..  . .    0 0 0 0

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

0 0 0 0

     ,   ... 0   . . . − (r2 + 1)2  . . . (r2 + 1)2

where σ11 = (n − r1 − 1)2 + 1/cr1 +1, σn−r1 −r2 n−r1 −r2 = (r2 + 1)2 , σ j j = (n − r1 − j)2 + (n − r1 − j + 1)2 , σ

j+1 j

=σ

j j+1

2

= − (n − r1 − j) ,



j = 2, 3, . . ., n − r1 − r2 − 1, j = 1, 2, . . ., n − r1 − r2 − 1,

and σi j = 0, for |i − j| > 1, i, j = 1, 2, . . ., n − r1 − r2 .

26

M. Ahsanullah and G.G. Hamedani

Note also that we have α = (αr1+1 , . . ., αn−r2 )0 , where αr = E [(Xr,n − µ) /σ] =

r

1

∑ n− j+1. j=1

Simple calculations show that   αr1+1 −1 0 − (n − r1 − 1) , 1, 1, . . ., 1, r2 + 1 , α∑ = cr1+1 α2 −1 α0 ∑ α = r1 +1 + (n − r1 − r2 − 1) , cr1 +1 α0 ∑

−1

10 ∑

1 = αr1 +1/cr1 +1,

−1

1 = 1/cr1 +1,

1 0 ∑ α = αr1 +1 /cr1 +1 ,   αr1 +1 1 −1 −1 0 0 1 ∑ 1α ∑ = − (n − r1 − 1) , 1, 1, . . ., r2 + 1 , cr1+1 cr1 +1   1 αr1 +1 −1 −1 0 0 , 0, 0, . . ., 0 , 1 ∑ α1 ∑ = cr1+1 cr1 +1     2 −1 −1 −1 ∆ = α0 ∑ α 1 0 ∑ 1 − α ∑ 1 = (n − r1 − r2 − 1) /cr1 +1 . −1

Upon simplification, we obtain σˆ∗ =

1 n 0 −1 0 −1 0 −1 0 −1 o X 1 ∑ 1 α ∑ −11 ∑ α1 ∑ ∆ ) ( n−r2 1 X j,n − (n − r1 )Xr1 +1,n + r2 Xn−r2,n . = n − r2 − r1 − 1 j=r∑ +1 1

Analogously, from (2.2.12)   and (2.2.14)–(2.2.16) we have the necessary exˆ
  pressions for µˆ∗, Var µ∗ , Var σˆ∗ , and Cov µˆ∗, σˆ∗ .

Order Statistics

27

2.4. Limiting Distributions Let X1, X2, . . ., Xn be i.i.d. exponentially distributed rv0 s with cd f F (x) = 1 − e−x , x ≥ 0. Then with a sequence of real numbers an = ln n and bn = 1, we have  n P (Xn,n ≤ an + bn x) = P (Xn,n ≤ ln n + x) = 1 − e−(ln n+x) n  −x e−x = 1− → e−e as n → ∞. n Thus the limiting distribution of Xn,n with the constant an = ln n and bn = 1 when X 0j s are E (1) is type 1 extreme value distribution. The numbers an and bn are known as normalizing constants. Remark 2.4.1. We know that if Y has type 1 extreme value distribution, then E [Y ] = γ, the Euler constant. Thus E [Xn,n] − ln n → γ, as n → ∞. But E [Xn,n ] = ∑nj=1 n−1j+1 , so we have the known result, ∑nj=1 n−1j+1 − ln n → γ, as n → ∞. For the derivation of the limiting distribution of X1,n , we need the following lemma. Lemma 2.4.2. Let (Xn)n≥1 be a sequence of i.i.d. rv0 s with cd f F. Consider a sequence (en )n≥1 of real numbers. Then for any ξ, 0 ≤ ξ < ∞, the following two statements are equivalent: (a) limn→∞ (nF (en )) = ξ; (b) limn→∞ P (X1,n > en ) = e−ξ . x

Since limn→∞ n 1 − e− n = ex , if X 0j s are i.i.d. ∼ E (1), then  x x lim P X1,n > = e−e . n→∞ n Thus, the limiting distribution of X1,n with constants cn = 0 and dn = 1/n when X 0j s are i.i.d. ∼ E (1) is type 3 extreme value distribution. Here again the numbers cn and dn are normalizing constants. Let us now consider the asymptotic distribution of Xn−k+1,n for fixed k as n tends to ∞. It is given in the following theorem.

28

M. Ahsanullah and G.G. Hamedani

Theorem 2.4.3. Let X1, X2, . . ., Xn be n i.i.d. rv0 s with cd f F and Xn−k+1,n be their (n − k + 1)th order statistic. If for some stabilizing constants a n and bn (an + bn x → ∞ as n → ∞), F n (an + bn x) → T (x) as n → ∞, for all x, for some distribution T (x), then k−1

P (Xn−k+1,n ≤ an + bn x) →

∑ T (x) (− ln T (x)) j / j!

as n → ∞,

j=0

for any fixed k and all x. Proof. Let us consider a sequence (cn )n≥1 such that as n → ∞, cn → c. Then
n limn→∞ 1− cnn = e−c . Take cn (x) = n (1 − F (an + bn x)). Now, for every fixed x,   n n P (Xn−k+1,n ≤ an + bn x) = ∑ (F (an + bn x)) j (1 − F (an + bn x))n− j j j=n−k+1 k−1   n (cn (x) /n) j (1 − cn (x) /n)n− j . =∑ j j=0 Thus, for each fixed x, the RHS of the above equality can be considered as the value of a binomial cd f with parameters n and cn (x) /n at k − 1. Since F n (an + bn x) → T (x), as n → ∞, we have n ln [1 − (1 − F (an + bn x))] → T (x) , as n → ∞. Thus, for sufficiently large n, we have n ln [1 − (1 − F (an + bn x))] ∼ = −n (1 − F (an + bn x)) = −cn (x) → T (x) , as n → ∞, from which we obtain limn→∞ cn (x) = −T (x) uniformly in x. Now, using Poisson approximation to binomial, we arrive at k−1

P (Xn−k+1,n ≤ an + bn x) →

∑ T (x) (− ln T (x)) / j!, j=0

as n → ∞, for all x.

Order Statistics

29

For the special case of i.i.d. ∼ E (1) rv0 s with an = ln n and bn = 1, we have −e−x , as n → ∞ , for all x ≥ 0. We will then have n + bn x) → e

F n (a

k−1 − jx

P(Xn−k+1,n ≤ an + bn x) →

∑

j=0

e

j!

−x

e−e , as n → ∞, for all x ≥ 0.

The asymptotic distribution of Xk,n for fixed k as n → ∞ is given by the following theorem whose proof is similar to that of Theorem 2.4.3 and hence will be omitted. Theorem 2.4.4. Let X1 , X2, . . ., Xn be n i.i.d. rv0 s with cd f F and Xk,n be their kth order statistic. If for some stabilizing constants a n and bn n (an + bn x → 0 as n → ∞) , F (an + bn x) → G (x), as n → ∞, for all x, for some distribution G (x), then
j − ln G (x) P (Xk,n > an + bn x) → ∑ G (x) , as n → ∞, for any fixed k and all x. j! j=0 k−1

Again, for the special case of i.i.d. ∼ E (1) rv0 s with an = 0 and
bn = 1/n, n n we have F (an + bn x) → e−x as n → ∞. But, in this case F 0 + 1n x = e−x for all n, and hence we will have
j k−1 k−1 − ln G (x) xj = ∑ e−x , for all x and all n. P (Xk,n > an + bn x) = ∑ G (x) j! j! j=0 j=0

Chapter 3

Record Values 3.1. Definitions of Record Values and Record Times Suppose that (Xn)n≥1 is a sequence of i.i.d. rv0 s with cd f F . Let Yn = max (min){X j |1 ≤ j ≤ n} for n ≥ 1. We say X j is an upper (lower) record value of {Xn|n ≥ 1} , if X j > (<)Y j−1 , j > 1. By definition X1 is an upper as well as a lower record value. One can transform the upper records to lower records by replacing the original sequence of (Xn)n≥1 by (−Xn )n≥1 or (if P (Xn > 0) = 1 for all n) by (1/Xn)n≥1; the lower record values of this sequence will correspond to the upper record values of the original sequence. The indices at which the upper record values occur are given where U (n) =  by the record times {U (n) , n ≥ 1}, min j| j > U (n − 1) , X j > XU(n−1), n > 1 and U (1) = 1. The record times of the sequence (Xn)n≥1 are the same as those for the sequence (F (Xn))n≥1 . Since F (X) has a uniform distribution for rv X, it follows that the distribution of U (n), n ≥ 1 does not depend on F. We will denote L (n) as the indices where the lower record values occur. By our assumption U (1) = L (1) = 1. The distribution of L (n) also does not depend on F.

3.2. The Exact Distribution of Record Values Many properties of the record value sequence can be expressed in terms of the function R (x) = − ln F (x), 0 < F (x) < 1. If we define Fn (x) as the cd f of XU(n)

32

M. Ahsanullah and G.G. Hamedani

for n ≥ 1, then we have
F1 (x) = P XU(1) ≤ x = F (x) ,
Z x Z y ∞ F2 (x) = P XU(2) ≤ x = ∑ (F (u)) j−1 dF (u) dF (y) =

−∞ −∞ j=1

Z x Z y −∞

dF (u) dF (y) = −∞ 1 − F (u)

Z x

(3.2.1)

R (y) dF (y) ,

−∞

where R (x) = − ln (1 − F (x)), 0 < F (x) < 1. If F has a pd f f , then the pd f of XU(2) is f 2 (x) = R (x) f (x) .

(3.2.2)

The cd f
Z F3 (x) = P XU(3) ≤ x = =

x −∞

Z y ∞

(F (u)) j R (u) dF (u)dF (y) ∑ −∞ j=0

Z x Z y −∞

R (u) dF (u) dF (y) = 1 − F (u) −∞

Z x (R (u))2 −∞

2!

dF (u).

(3.2.3)

The pd f f 3 of XU(3) is (R (x))2 f (x) . 2! It can similarly be shown that the cd f Fn of XU(n) is f 3 (x) =

Fn (x) =

Z x (R (u))n−1 −∞

dF (u),

−∞ < x < ∞.

un−1 −u e du, (n − 1)!

−∞ < x < ∞,

(n − 1)!

This can be expressed as Fn (x) = and

Z R(x) −∞

n−1 (R (x)) j (R (x)) j = e−R(x) ∑ . j! j! j=0 j=0

n−1

F n (x) = 1 − Fn (x) = F (x) ∑

(3.2.4)

(3.2.5)

Record Values

33

The pd f f n of XU(n) is (R (x))n−1 fn (x) = f (x) , (n − 1)! Note that F n (x) − F n−1 (x) =

F(x) f (x)

−∞ < x < ∞.

(3.2.6)

f n (x), and for E (λ), F n (x) − F n−1 (x) =

λn−1 xn−1 −λx . Γ(n) e

A rv X is said to be symmetric about zero if X and −X have the same distribution function. If f is their pd f , then f (x) = f (−x) for all x. Two rv0 s X and Y with cd f 0 s F and G are said to be mutually symmetric if F (x) = 1 − G (x) for all x, or equivalently if their corresponding pd f 0 s f and g exist, then f (−x) = g (x) for all x. If a sequence of i.i.d. rv0 s are symmetric about zero, then they are also mutually symmetric about zero but not conversely. It is easy to show that for a symmetric or mutually symmetric (about zero) sequence (Xn)n≥1 of i.i.d. rv0 s, XU(n) and XL(n) are identically distributed. The joint pd f f (x1, x2 , . . ., xn) of the n record values XU(1), XU(2), . . ., XU(n) is given by f (x1 , x2 , . . ., xn ) n−1

= ∏ r (x j ) f (xn ), −∞ < x1 < x2 < · · · < xn−1 < xn < ∞, (3.2.7) j=1

where, as before, r (x) =

d f (x) R (x) = , dx 1 − F (x)

0 < F (x) < 1.

The joint pd f of XU(i) and XU( j) is fi j (xi , x j ) =

[R (x j ) − R (xi )] j−i−1 (R (xi ))i−1 r (xi ) f (x j ), (i − 1)! ( j − i − 1)! for − ∞ < xi < x j < ∞. (3.2.8)

In particular, for i = 1 and j = n we have f1n (x1 , xn ) = r (x1 )

[R (xn) − R (x1 )]n−2 f (xn ) , for − ∞ < x1 < xn < ∞. (n − 2)!

34

M. Ahsanullah and G.G. Hamedani

The conditional pd f of XU( j) |XU(i) = xi is f (x j |xi ) = =

fi j (xi , x j ) fi (xi )

f (x j ) [R (x j ) − R (xi )] j−i−1 · , for − ∞ < xi < x j < ∞. (3.2.9) ( j − i − 1)! 1 − F (xi )

For j = i + 1 f (xi+1 |xi ) =

f (xi+1 ) , for − ∞ < xi < xi+1 < ∞. 1 − F (xi )

(3.2.10)

For i > 0, 1 ≤ k < m, the joint conditional pd f of XU(i+k) and XU(i+m) given XU(i) is
f(i+k)(i+m) x, y|XU(i) = z f (y) r (x) 1 1 , · [R (y) − R (x)]m−k−1 [R (x) − R (z)]k−1 = Γ (m − k) Γ (k) F (z) for − ∞ < z < x < y < ∞. The marginal pd f of the nth lower record value can be derived by using the same procedure as that of the nth upper record value. Let H (u) = − ln F (u), d 0 < F (u) < 1 and h (u) = − du H (u), then
Z P XL(n) ≤ x =

(H (u))n−1 dF (u), −∞ (n − 1)! x

(3.2.11)

and corresponding pd f f (n) can be written as f (n) (x) =

(H (x))n−1 f (x) . (n − 1)!

(3.2.12)

The joint pd f of XL(1) , XL(2), . . ., XL(m) can be written as f(1)(2)...(m) (x1 , x2 , . . ., xm ) ( ∏m−1 j=1 h (x j ) f (xm ) , −∞ < xm < xm−1 < · · · < x1 < ∞, = 0, otherwise.

(3.2.13)

Record Values

35

The joint pd f of XL(i) and XL( j) is f(i)( j) (x, y) =

(H (x))i−1 [H (y) − H (x)] j−i−1 · h (x) f (y) , (i − 1)! ( j − i − 1)! j > i and − ∞ < y < x < ∞. (3.2.14)

Using the transformation U = H (y) and V = H (x) /H (y) in (3.2.14), it can easily be shown that V is distributed as Bi, j−i (x), where Bm,n (x) = B (m, n)xm−1 (1 − x)n−1 and B (m, n) is the Beta function. Proceeding as in the case of upper record values, we can obtain the conditional pd f 0 s of the lower record values. We will now consider the exponential distribution with pd f given, as before, by (
σ−1 exp −σ−1 (x − µ) , x ≥ µ, (3.2.15) f (x) = 0, otherwise, where µ and σ (σ > 0) are parameters. The corresponding cd f F and the hazard rate r of the rv X with pd f (3.2.15) are respectively
F (x) = 1 − exp −σ−1 (x − µ) , x ≥ µ, and r (x) = f (x) / (1 − F (x)) = σ−1 .

(3.2.16)

Again, as before, we will denote the exponential distribution with pd f (3.2.15) with E (µ, σ), the exponential distribution (µ = 0, σ = 1/λ) with E (λ), and the standard exponential distribution with E (1) . For E (µ, σ),the joint pd f of XU(m) and XU(n), m < n is

=

f m,n (x, y) ( −n n−m−1
m−1 (y−x) σ −1 Γ(m) (x − µ) Γ(n−m) exp −σ (y − µ) , µ ≤ x < y < ∞, 0,

(3.2.17)

otherwise.



It is easy to see that,in this case, XU(n) − XU(n−1) and XU(m) − XU(m−1) are i.d. for 1 < m < n < ∞.

36

M. Ahsanullah and G.G. Hamedani d

It can be shown that XU(m) = XU(m−1) +U, (m > 1) where U is independent of XU(m) and XU(m−1) and is identically distributed as X1 if and only if X1 ∼ E (λ). For E (1)with n ≥ 1,
Z P XU(n+1) > wXU(n) =

∞Z ∞

xn−1 −y e dydx Γ (n)

0

wx

0

Γ (n)

Z ∞ n−1 x e−wx dx = w−n . =

The conditional pd f of XU(n) given XU(m) = x is ( n−m−1
−1 (y − x) , µ ≤ x < y < ∞, exp −σ σm−n (y−x) Γ(n−m) (3.2.18) f (y|x) = 0, otherwise.

Thus, P XU(n) − XU(m) = y|XU(m) = x does not depend on x. It can be shown that if µ = 0, then XU(n) − XU(m) is identically distributed as XU(n−m), m < n. We take µ = 0 and σ = 1 and let Tn = ∑nj=1 XU( j) . Since
Tn = XU(n) − XU(n−1) + 2 XU(n−1) − XU(n−2) + · · ·
+ (n − 1) XU(2) − XU(1) + nXU(1) n

=

∑ jW j, j=1

where W j0 s are i.i.d.E (1), the characteristic function of Tn can be written as n

1 . j=1 1 − jt

ϕn (t) = ∏

(3.2.19)

Inverting (3.2.19), we obtain the pd f f Tn of Tn as (−1)n− j −u/ j n−2 1 · e fTn (u) = ∑ j . j=1 Γ ( j) Γ (n − j + 1) n

(3.2.20)

Theorem 3.2.1. Let (Xn )n≥1 be a sequence of i.i.d. rv0 s with the standard exponential distribution. Suppose ξ j = independent.

XU( j) XU( j+1) ,

j = 1, 2, . . ., m − 1, then ξ0 s are

Record Values

37

Proof. The joint pd f of XU(1), XU(2), . . ., XU(m) is m−1

f (x1, x2 , . . ., xm ) =

∏ xj

!

e−xm ,

0 < x1 < x2 < · · · < xm < ∞.

j=1

Let us now use the transformation ξ0 = XU(1) and ξ j =

XU( j) , j = 1, 2, . . ., m − 1. XU( j+1)

The Jacobian of the transformation is
∂ X , X , . . ., X ξm−1 U(1) U(2) U(m) J= . = m m−10 ξ1 ξ2 . . .ξ2m−1 ∂(ξ0, ξ1 , . . ., ξm−1 ) We can write the pd f of ξ j , j = 0, 1, . . ., m − 1, as f (e0 , e1 , . . ., em−1) =

αm em−1 0 m−1 em · · ·e2m−1 1 e2

e−(e0 /(e1 e2 ···em−1 )) .

Now, integrating the above expression with respect to e0 , we obtain the joint pd f of ξ j , j = 1, 2, . . ., m − 1, as f (e1 , e2, . . ., em−1) = Γ (m) e2 · · ·em−2 m−1 . Thus ξ j , j = 1, 2, . . ., m − 1 are independent and P (ξ j ≤ x) = x j ,

1 ≤ j ≤ m.

Since R (x) = x for the standard exponential distribution, the pd f of ξ j = XU( j) /XU( j+1), j = 1, 2, . . ., m − 1 can easily be obtained. Corollary 3.2.2. Let W j = (ξ j ) j , j = 1, 2, . . ., m − 1, then W j , j = 1, 2, . . ., m − 1 are i.i.d. U (0, 1) (uniform over the unit interval ) random variables. Finally, the Fisher’s Information for the nth record of E (λ) is n/λ2.

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M. Ahsanullah and G.G. Hamedani

3.3. Moments of Record Values Without any loss of generality we will consider in this section the standard exponential distribution E (1), with pd f f (x) = exp (−x), x > 0, for which we also have f (x) = 1 − F (x). We already know that XU(n) ,in this setting, can be written as the sum of n i.d. rv0 s V1,V2, ..,Vn with common distribution E (1). Further, we have also seen that   E XU(n) = n,   Var XU(n) = n, and


Cov XU(n), XU(m) = m,

For m < n, i Z h q p E XU(n) XU(m) =

∞Z u 0

0

m < n.

(3.3.1)

1 1 · uqe−x vm+p−1 (u − v)n−m−1 dvdu. Γ (m) Γ (n − m)

Substituting tu = v and simplifying we get i h p q XU(m) E XU(n) =

Z ∞Z ∞ 0

0

1 1 · un+p+q−1 e−xt m+p−1 (1 − t)n−m−1 dtdu Γ (m) Γ (n − m) Γ (m + p) Γ (n + p + q) . = Γ (m) Γ (n + p)

Using (3.2.19), it can be shown that for Tn = ∑nj=1 XU( j) , we have E [Tn ] = n (n + 1) /2 and Var [Tn] = n (n + 1) (2n + 1)/6. Some simple recurrence relations satisfied by single and product moments of record values are given by the following theorem. Theorem 3.3.1. For n ≥ 1 and k = 0, 1, . . . h i h i h i k+1 k+1 k E XU(n) = E XU(n−1) + (k + 1) E XU(n) ,

(3.3.2)

Record Values

39

and consequently, for 0 ≤ m ≤ n − 1 we can write h i h i k+1 k+1 E XU(n) = E XU(m) + (k + 1)

n

∑

h i k E XU( j) ,

(3.3.3)

j=m+1

i h i h k+1 0 = 0 and E XU(n) = 1. with E XU(0) Proof. For n ≥ 1 and k = 0, 1, . . ., we have i h k = E XU(n)

1 Γ (n) =

Z ∞ 0

1 Γ (n)

xr (R (x))n−1 f (x) dx Z ∞

xk (R (x))n−1 (1 − F (x)) dx, ( f (x) = 1 − F (x)) .

0

Upon integration by parts, treating xk for integration and the rest of the integrand for differentiation, we obtain h i k E XU(n) Z ∞  Z ∞ 1 n−1 n−2 k−1 k+1 x (R (x)) f (x) dx − (n − 1) x (R (x)) f (x) dx = (k + 1) Γ (n) 0 0  Z ∞ Z ∞ 1 1 n−1 n−2 k+1 1 k+1 x f (x) dx − x f (x) dx = (R (x)) (R (x)) k+1 0 Γ (n) Γ (n − 1) 0 h io 1 n h k+1 i k+1 , E XU(n) − E XU(n−1) = k+1 which, when rewritten, gives the recurrence relation (3.3.2). Then repeated application of (3.3.2) will derive the recurrence relation (3.3.3). Remark 3.3.2. The recurrence relation (3.3.2) can be used in a simple way to compute all the simple moments of all the record values. Once again, using property that f (x) = 1 − F (x), we can derive some simple recurrence relations for the product moments of record values. Theorem 3.3.3. For m ≥ 1 and p, q = 0, 1, 2, . . . h i h i h i p q+1 p+q+1 p q E XU(m) XU(m+1) = E XU(m) + (q + 1) E XU(m)XU(m+1) ,

(3.3.4)

40

M. Ahsanullah and G.G. Hamedani

and for 1 ≤ m ≤ n − 2, p, q = 0, 1, 2, . . . h i h i h i p q+1 p q+1 p q E XU(m) XU(n) = E XU(m) XU(n−1) + (q + 1) E XU(m) XU(n) ,

(3.3.5)

Proof. Let us consider 1 ≤ m < n and p, q = 0, 1, 2, . . . Z ∞ h i 1 p q E XU(m) XU(n) = x p (R (x))m−1 r (x) I (x) dx, Γ (m) Γ (n − m) 0

(3.3.6)

where I (x) =

Z ∞ x

=

yq [R (y) − R (x)]n−m−1 f (y) dy

Z ∞

yq [R (y) − R (x)]n−m−1 (1 − F (y))dy, since f (y) = 1 − F (y) .

x

Upon performing integration by parts, treating yq for integration and the rest of the integrand for differentiation, we obtain, when n = m + 1, that  Z ∞ 1 q+1 q+1 I (x) = y f (y) dy − x (1 − F (x)) , q+1 x and when n ≥ m + 2, that Z ∞ 1 I (x) = yq+1 {R (y) − R (x)}n−m−1 f (y) dy q+1 x − (n − m − 1)

Z ∞

q+1

y

n−m−2

{R (y) − R (x)}

 f (y) dy .

x

Substituting the above expression of I (x) in equation (3.3.6) and simplifying, we obtain, when n = m + 1 that h i i h io 1 n h p q q+1 p+q+1 p E XU(m) XU(m+1) − E XU(m) E XU(m) XU(m+1) = , q+1 and when n ≥ m + 2, that i h p q E XU(m) XU(n) =

i h io 1 n h p q+1 p q+1 E XU(m) XU(n) − E XU(m) XU(n−1) . q+1

The recurrence relations (3.3.4) and (3.3.5) follow readily when the above two equations are rewritten.

Record Values

41

Remark 3.3.4. By repeated application of the recurrence relation (3.3.5), with the help of the relation (3.3.4), we obtain, for n ≥ m + 1, that h i h i p q+1 p+q+1 E XU(m)XU(n) = E XU(m) + (q + 1)

n

∑

j=m+1

h

i p q XU(m) XU( j) .

(3.3.7)

Corollary 3.3.5. For n ≥ m + 1,
  Cov XU(m) , XU(n) = Var XU(m) . Proof. By setting p = 1 and q = 0 in (3.3.7), we obtain h i     2 E XU(m) XU(n) = E XU(m) + (n − m) E XU(m) .

(3.3.8)

Similarly, by setting p = 0 in (3.3.3), we obtain     E XU(n) = E XU(m) + (n − m) ,

(3.3.9)

n > m.

With the help of (3.3.8) and (3.3.9), we get for n ≥ m + 1
      Cov XU(m) , XU(n) = E XU(m) XU(n) − E XU(m) E XU(n) i h     2    2 − (n − m) E XU(m) + (n − m) E XU(m) − E XU(m) = E XU(m)   = Var XU(m) . Corollary 3.3.6. By repeated application of the recurrence relations (3.3.4) and (3.3.5), we also obtain for m ≥ 1 h i q+1 i h q+1 p+q+1− j p XU(m+1) = ∑ (q + 1)( j) E XU(m) , E XU(m) j=0

and for 1 ≤ m ≤ n − 2 i q+1 i h h p q+1 p q+1− j E XU(m) XU(n) = ∑ (q + 1)( j) E XU(m) XU(n−1) , j=0

where (q + 1)(0) = 1 and

(q + 1)( j) = (q + 1) q · · ·(q + 1 − j + 1) , for j ≥ 1.

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M. Ahsanullah and G.G. Hamedani

Remark 3.3.7. The recurrence relations (3.3.4) and (3.3.5) can be used in a simple way to compute all the product moments of all record values. Theorem 3.3.8. For m ≥ 2 and p, q = 0, 1, 2, . . ., h i h i h i p+1 q p+q+1 p q E XU(m−1)XU(m) = E XU(m) − (p + 1) E XU(m) XU(m+1) ,

(3.3.10)

and for 2 ≤ m ≤ n − 2 and p, q = 0, 1, 2, . . ., h i h i h i p+1 q p+1 q p q E XU(m−1)XU(n−1) = E XU(m)XU(n−1) − (p + 1) E XU(m) XU(m+1) . (3.3.11) Proof. For 2 ≤ m ≤ n and p, q = 0, 1, 2, . . ., i Z ∞Z ∞ h p q XU(n) x p yq fmn (x, y)dxdy = E XU(m) 0

0

1 = (m − 1)! (n − m − 1)!

Z ∞

yq f (y) J (y) dy,

(3.3.12)

0

where J (y) =

Z y

x p {− ln (1 − F (x))}m−1

0

× {− ln (1 − F (x)) + ln (1 − F (y))}n−m−1 =

Z ∞

f (x) dx 1 − F (x)

x p {− ln (1 − F (x))}m−1 {− ln (1 − F (x)) + ln (1 − F (y))}n−m−1 dx,

0

since f (x) = 1 − F (x). Upon integration by parts, treating x p for integration and the rest of the integrand for differentiation, we obtain, for n = m + 1, that J (y) =

i 1 h p+1 {− ln (1 − F (y))}m+1 y p+1 − (m − 1)

Z y 0

x p+1 {− ln (1 − F (x))}m−2

f (x) dx, 1 − F (x)

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43

and when n ≥ m + 2, that 1 J (y) = [(n − m − 1) p+1

Z y

x p+1 {− ln (1 − F (x))}m−1

0

f (x) 1 − F (x)

× {− ln (1 − F (y)) + ln (1 − F (x))}n−m−2 dx Z y f (x) x p+1 {− ln (1 − F (x))}m−2 − (m − 1) 1 − F (x) 0 × {− ln (1 − F (y)) + ln (1 − F (x))}m−2 dx]. Now, substituting the above expression of J (y) in equation (3.3.12) and simplifying, we obtain, for n = m + 1, that i h io h 1 n h p+q+1i p q p+1 q E XU(m) E XU(m) − E XU(m−1)XU(m) , XU(n) = p+1 and for n ≥ m + 2 that i h p q E XU(m) XU(n) =

h io 1 n h p+1 q i p+1 q E XU(m) XU(n) − E XU(m−1)XU(n−1) . p+1

The recurrence relations (3.3.10) and (3.3.11) follow readily when the above two equations are rewritten. Corollary 3.3.9. By repeated application of the recurrence relation (3.3.11), with the help of (3.3.10), we obtain for 2 ≤ m ≤ n − 1 and p, q = 0, 1, 2, . . . i h i i h n−1 h p+1 q p+q+1 p q E XU(m−1)XU(n−1) = E XU(n−1) − (p + 1) ∑ E XU( j) XU(n) . j=m

Corollary 3.3.10. By repeated application of the recurrence relations (3.3.10) and (3.3.11), we also obtain for m ≥ 2 that h i p+1 i h p+1 q p+q+1− j XU(m) = ∑ (−1) j (p + 1)( j) E XU(m+ , E XU(m−1) j) j=0

and for 2 ≤ m ≤ n − 2 that h h i p+1 i p+1 q p+1− j q E XU(m−1)XU(n−1) = ∑ (−1) j (p + 1)( j) E XU(m− j) XU(n+1− j) , j=0

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M. Ahsanullah and G.G. Hamedani

where (1 + p)( j) is as defined earlier. It is also important to mention here that this approach can easily be adopted to derive recurrence relations for product moments involving more than two record values.

3.4. Estimation of Parameters We shall consider here the linear estimations of µ and σ. (a) Minimum Variance Linear Unbiased Estimator (MVLUE) Suppose XU(1) , XU(2) , . . ., XU(m) are the m record values from an i.i.d.
sequence of rv0 s with common cd f E (µ, σ). Let Yi = σ−1 XU(i) − µ , i = 1, 2, . . ., m. Then E [Yi ] = i = Var [Yi ] ,

i = 1, 2, . . ., m,

and Cov (Yi ,Y j ) = min{i, j}. Let


X = XU(1), XU(2), . . ., XU(m) ,

then X ] = µL L + σδδ , E [X X ] = σ2V , Var [X where L = (1, 1, . . ., 1)0 ,

δ = (1, 2, . . ., m)0 ,

V = (Vi j ) , Vi j = min {i, j}, i, j = 1, 2, . . ., m.
The inverse V −1 = V i j can be expressed as   2 if i = j = 1, 2, . . ., m − 1,    1 if i = j = m, Vij =  −1 if |i − j| = 1, i, j = 1, 2, . . ., m,    0, otherwise.

Record Values

45

The MLVLUEs µ, ˆ σˆ of µ and σ respectively are
0 0 µˆ = −δδ V −1 L δ − δ L 0 V −1 X / ∆,
σˆ = L 0V −1 L δ 0 − δ L 0 V −1 X / ∆, where




2 ∆ = L0 V −1 L δ0V −1 δ − L0 V −1 δ ,

and Var [ˆµ] = σ2 L 0V −1 δ / ∆, Var [ˆσ] = σ2 L 0V −1 L / ∆, ˆ = −σ2 L 0V −1 δ / ∆. Cov (ˆµ, σ) It can be shown that L 0V −1 = (1, 0, 0, . . ., 0), δ 0V −1 δ = m

0 δ V −1 = (0, 0, . . ., 0, 1),

and ∆ = m − 1.

Upon simplification we get
µˆ = mXU(1) − XU(m) / (m − 1) ,
σˆ = XU(m) − XU(1) / (m − 1) ,

(3.4.1)

with Var [ˆµ] = mσ2 / (m − 1) ,

ˆ = σ2 / (m − 1) Var [σ]

ˆ = −σ / (m − 1) . Cov (ˆµ, σ) 2

and (3.4.2)

(b) Best Linear Invariant Estimator (BLIE) The best linear invariant (in the sense of minimum mean squared error and e of invariance with respect to the location parameter µ) estimators, BLIEs, e µ, σ µ and σ are   E12 , e µ = µˆ − σˆ 1 + E12

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M. Ahsanullah and G.G. Hamedani

and e = σ/ ˆ (1 + E12 ) , σ

where µˆ and σˆ are MVLUEs of µ and σ and     ˆ E11 E12 Var[ˆµ] Cov (ˆµ, σ) . = σ2 ˆ Var[σ] ˆ E21 E22 Cov (ˆµ, σ) The mean squared errors of these estimators are   2 (1 + E22 )−1 , MSE[e µ] = σ2 E11 − E12 and e ] = σ2 E22 (1 + E22 )−1 . MSE[σ We have e − σ)] = σ2 E12 (1 + E22 )−1 . E [(e µ − µ) (σ Using the values of E11 , E12 and E22 from (3.4.2), we obtain  e µ = (m + 1) XU(1) − XU(m) /m,
e = XU(m) − XU(1) /m, σ
Var [e µ] = σ2 m2 + m − 1 /m, and e ] = σ2 (m − 1) /m2 . Var [σ

3.5. Prediction of Record Values We will predict the sth upper value based on the first m record values for s > m. Let W 0 = (W1 ,W2, . . .,Wm), where and


σ2Wi = Cov XU(i) , XU(s) ,

i = 1, 2, . . ., m,

  α∗ = σ−1 E XU(s) − µ .

Record Values

47

The best linear unbiased predictor of XU(s) is XˆU(s) , where ˆ = µˆ + σα ˆ ∗ +W W 0V −1 (X X − µˆ L − σδ ˆ δ ), XU(s) X− µˆ and σˆ are MVLUEs of µ and σ respectively. It can be shown that W 0V −1 (X ˆ δ) = 0, and hence µˆ L − σδ
XˆU(s) = (s − 1) XU(m) + (m − s) XU(1) / (m − 1) , (3.5.1) E[XˆU(s) ] = µ + sσ,
Var[XˆU(s) ] = σ2 m + s2 − 2s / (m − 1) , MSE[XˆU(s) ] = E[(XˆU(s) − XU(s) )2 ] = σ2 (s − m) (s − 1) / (m − 1) . that

Let XeU(s) be the best linear invariant predictor of XU(s) . Then it can be shown eU(s) = XˆU(s) −C12 (1 + E22 )−1 σ, ˆ X

where

(3.5.2)




ˆ L −W W 0V −1 L µˆ + α∗ −W W 0V −1 δ σˆ C12 σ2 = Cov σ,

and ˆ σ2 E22 = Var[σ]. Upon simplification, we get eU(s) = m − s XU(1) + s XU(m) , X m  m  i h ms + m−s e σ, E XU(s) = µ + m i h
eU(s) = σ2 m2 + ms2 − s2 /m, Var X

2 i h eU(s) = MSE[XˆU(s) ] + (s − m) σ2 = s (s − m) σ2 . MSE X m (m − 1) m

It is well-known that the best (unrestricted) least square predictor of XU(s) is   Xˆ U(s) = E XU(s)|XU(1), . . ., XU(m) = XU(m) + (s − m) σ.

(3.5.3)

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M. Ahsanullah and G.G. Hamedani

But Xˆ U(s) depends on the unknown parameter σ. If we substitute the minimum variance unbiased estimate σˆ for σ, then Xˆ U(s) becomes equal to XˆU(s) . Now   E[Xˆ U(s) ] = µ + sσ = E XU(s) ,Var[Xˆ U(s) ] = mσ2 and

MSE[Xˆ U(s) ] = E[(Xˆ U(s) − XU(s) )2] = (s − m) σ2 .

We like to mention also that by considering the mean squared errors of eU(s) and Xˆ U(s) , it can be shown that XˆU(s) , X MSE[Xˆ U(s) ] = E[(Xˆ U(s) − XU(s) )2] = (s − m) σ2 .

3.6. Limiting Distribution of Record Values     We have seen that for µ = 0 and σ = 1, E XU(n) = n and Var XU(n) = n. Hence   √
XU(n) − n √ ≤ x = P XU(n) ≤ n + x n P n Z n+x√n n−1 −x x e dx, (3.6.1) = Γ (n) 0 = pn (x) , say. Let

1 Φ (x) = √ 2π

Z x

e−t

2 /2

dt.

−∞

The following table gives values of pn (x) for various values of n and x and values of Φ (x) for comparison.

Record Values

49

Table 1. Values of pn (x) 

n\x  5   10   15   25   45 Φ (x)

−2 0.0002 0.0046 0.0098 0.0122 0.0142 0.0226

−1 0.1468 0.1534 0.1554 0.1568 0.1575 0.1587

0 0.5575 0.5421 0.5343 0.5243 0.5198 0.5000

1 0.8475 0.8486 0.8436 0.8427 0.8423 0.8413

 2 0.9590  0.9601  0.9653  0.9684  0.9698 0.9774

Thus for large values of n, Φ (x) is a good approximation of pn (x). Finally, the entropy of nth upper record value XU(n) is n + ln Γ (n) − ln λ − (n − 1) ψ (n) , where ψ (n) is the digamma function, ψ (n) = Γ0 (n)/Γ (n). To see this we observe that pd f of XU(n) is given by fn (x) =

λn n−1 −λx x e , Γ (n)

x ≥ 0,

and its entropy is computed as follows

=

Z λ n λ

En = E [− ln XU (n)]

e−λx (ln Γ (n) − n ln λ + λx − (n − 1) ln x) dx Γ (n) = ln Γ (n) − n ln λ + n − (n − 1) {− ln λ + ψ (n)} 0

= n + ln Γ (n) − ln λ − (n − 1) ψ (n).

Chapter 4

Generalized Order Statistics In this chapter we will consider some of the basic properties of the generalized order statistics from exponential distribution. We shall present some inferences based on the distributional properties of the generalized order statistics.

4.1. Definition The concept of generalized order statistics (gos) was introduced by Kamps (1995) in terms of their joint pd f . The order statistics, record values and sequential order statistics are special cases of the gos. The rv0 s X (1, n, m, k), X (2, n, m, k), . . ., X (n, n, m, k), k > 0, m ∈ R, are n gos from an absolutely continuous cd f F with corresponding pd f f if their joint pd f f1,2,...,n (x1 , x2 , . . ., xn ) can be written as f 1,2,...,n (x1 , x2, . . ., xn ) ! =k

n−1

n−1

j=1

j=1

∏ γj

∏ (1 − F (x j ))

m

!

f (x j ) (1 − F (xn ))k−1 f (xn ),

F −1 (0+) < x1 < · · · < xn < F −1 (1), (4.1.1) where γ j = k + (n − j) (m + 1) ≥ 1 for all j, 1 ≤ j ≤ n, kis a positive integer and m ≥ −1. A more general form of (4.1.1), again due to Kamps, with a new notation for the joint pd f will be given in Chapter 5.

52

M. Ahsanullah and G.G. Hamedani

If k = 1 and m = 0, then X (s, n, m, k) reduces to the ordinary sth order statistic and (4.1.1) will be the joint pd f of the n order statistics X1,n ≤ X2,n ≤ · · · ≤ Xn,n. If k = 1 and m = −1, then (4.1.1) will be the joint pd f of the first n upper record values of the i.i.d. rv0 s with cd f F and pd f f . Integrating out x1, x2 , . . ., xs−1 , xs+1 , .., xn from (4.1.1) we obtain the pd f f s,n,m,k of X (s, n, m, k) fs,n,m,k (x) =

cs (1 − F (x))γs−1 f (x) gs−1 m (F (x)), (s − 1)!

(4.1.2)

where cs = ∏sj=1 γ j and gm (x) =

(

1 (m+1)



 1 − (1 − x)m+1 , m 6= −1,

− ln (1 − x) ,

m = −1, x ∈ (0, 1).

  1 Since limm→−1 m+1 1 − (1 − x)m+1 = − ln (1 − x) , we will write gm (x) =   m+1 1 1 − (1 − x) , for all x ∈ (0, 1) and for all m with g−1 (x) = m+1 limm→−1 gm (x) .

4.2. Generalized Order Statistics of Exponential Distribution Recall that pd f of X ∼ E (µ, σ) , is given by (
σ−1 exp −σ−1 (x − µ) , x > µ, σ > 0, f (x) = 0, otherwise.

(4.2.1)

Lemma 4.2.1. Let (Xi )i≥1 be a sequence of i.i.d. rv0 s from E (µ, σ), then d

γ1X (1, n, m, k) ∼ E (µγ1, σ) and X (s, n, m, k) = µ + σ ∑sj=1 W j ∼ E (0, 1) = E (1) for all j 0 s.

Wj γj ,

Proof. From (4.1.2), pd f f s,n,m,k of X (s, n, m, k), in this case, is  −1    cs −σ−1 (x−µ) fs,n,m,k (x) = 1 − e . · σ−1 e−σ (x−µ) γs gs−1 m (s − 1)!

where

(4.2.2)

Generalized Order Statistics

53

For s = 1, we obtain pd f of X (1, n, m, k) via (4.2.2) as  −1  f1,n,m,k (x) = γ1σ−1 e−σ (x−µ)γ1 , x > µ.
and γ1X (1, n, m, k) ∼ E (µγ1, σ) . Thus, X (1, n, m, k) ∼ E µ, σγ−1 1 The moment generating function of X (s, n, m, k), denoted by Ms,n,m,k is (using (4.2.2)) Z ∞

Ms,n,m,k (t)  −1

  −1 cs−1 1 − e−σ (x−µ) dt · σ−1 e−σ (x−µ)γs gs−1 m (s − 1)! µ s−1  Z ∞ cs−1 µt 1 −(γs −σt)y e dy. (4.2.3) = e (1 − exp (− (m + 1) y)) (s − 1)! m+1 0 =



etx

Using the following property (see, Gradsheyn and Ryzhik, (1980), p. 305) Z ∞  s−1 1 a  e−ay 1 − e−by dy = B , s , b b 0 where B (·, ·) is the Beta function, and the fact that γs + j (m + 1) = γs− j , we obtain from (4.2.3)  s  cs−1 µt (s − 1)! σt −1 µt   = e ∏ 1− . (4.2.4) Ms,n,m,k (t) = e (s − 1)! ∏s γ 1 − σt γj j=1 j=1 j γj Thus

s

Wj . j=1 γ j

X (s, n, m, k) = µ + σ ∑ d

(4.2.5)

Remarks 4.2.2. (a) For µ = 0, from Lemma 4.2.1, we have γ1X (1, n, m, k) ∼ E (0, σ). (b) For k = 1, m = 0, from (4.2.5), we obtain the following well-known result for order statistics for i.i.d. rv0 s from E (µ, σ) s

Xs,n = µ + σ ∑ W j / (n − j + 1) . d

j=1

(4.2.6)

54

M. Ahsanullah and G.G. Hamedani

(c) From (4.2.6) it follows that γs {X (s, n, m, k)− X (s − 1, n, m, k)} ∼ E (0, σ). This property can also be obtained by considering the joint pd f of X (s, n, m, k) and X (s − 1, n, m, k) and using the transformation U1 = X (s − 1, n, m, k) and Ts = γs {X (s, n, m, k)− X (s − 1, n, m, k)} . (d) For k = 1 and m = −1, we obtain XU(s) = µ + σ ∑sj=1 W j . For X ∼ E (µ, σ) , we have from (4.2.5), E [X (s, n, m, k)] = µ + σ ∑sj=1 γ1j and the recurrence relation E [X (s, n, m, k)] − E [X (s − 1, n, m, k)] =

σ . γs

Let D (1, n, m, k) = γ1X (1, n, m, k), D (s, n, m, k) = γsX (s, n, m, k)− X (s − 1, n, m, k),

2 ≤ s ≤ n,

then for X ∼ E (λ) all the D ( j, n, m, k), j = 1, 2, . . ., n are i.i.d. E (λ). Thus, we have the obvious recurrence relation E [D (s, n, m, k)] = E [D (s − 1, n, m, k)] . For k = 1 and m = 0, it coincides with the known results corresponding to order statistics. For k = 1 and m = −1, it coincides with the known results of upper record values. In the remainder of this section we would like to present two recurrence relations for the moments (single moments and product moments) of gos from the standard exponential distribution E (1). The joint pd f of X (r, n, m, k) and X (s, n, m, k), denoted by f r,s,n,m,k (x, y) is (see p. 68 of Kamps (1995)) f r,s,n,m,k (x, y) =

cs (1 − F (x))m f (x) (r − 1)! (s − r − 1)!

s−r−1 ×gr−1 (1 − F (y))γs f (y) , m (F (x)) [h (F (y)) − h (F (x))]

where h (x) =

(

(1 − x)m+1 , m 6= 1, − ln (1 − x) , m = −1. 1 m+1

(4.2.7)

Generalized Order Statistics

55

Theorem 4.2.3. For E (1) and s > 1 h i p+1   E (X (s, n, m, k)) p+1 = E (X (s − 1, n, m, k)) p+1 + E [(X (s, n, m, k))p ] , γs and consequently for s > r h i   E (X (s, n, m, k)) p+1 = E (X (r, n, m, k)) p+1 +

s

p+1 E [(X ( j, n, m, k))p ] . γ j j=r+1

∑

Proof. We have E [(X (s, n, m, k)) p ] =

Z ∞ 0

Z ∞


cs 1 − e−x dx e−γs x gs−1 m (s − 1)!
x p+1 e−γs x gs−1 1 − e−x dx m

xp

γs cs = 0 (p + 1) (s − 1)! Z ∞
cs (s − 1) − x p+1 e−γs x gs−2 1 − e−x e−(m+1) dx m 0 (p + 1) (s − 1)! io h  γs n  E (X (s, n, m, k)) p+1 − E (X (s − 1, n, m, k)) p+1 , = (p + 1) from which the result follows. For k = 1 and m = −1, Theorem 4.2.3 coincides with Theorem 3.3.1. For k = 1 and m = 0, we obtain h i h i p+1 E [(Xs,n ) p ] E (Xs,n ) p+1 = E (Xs−1,n)p+1 + n−s+1 and consequently i h i h E (Xs,n ) p+1 = E (Xs−1,n) p+1 +

s

p+1 E [(X j,n )p ] . n − j + 1 j=r+1

∑

Letting p = 0, in the last equation, we have s

1 , n − j+1 j=2

E [Xs,n ] = E [X1,n] + ∑

56

M. Ahsanullah and G.G. Hamedani

that is,

s

E [Xs,n ] =

1

∑ n− j+1. j=1

Theorem 4.2.4. For E (1), 1 ≤ r < s ≤ n and p, q = 0, 1, 2, . . . we have i h E (X (r, n, m, k)) p (X (s, n, m, k))q+1 h i = E (X (r, n, m, k)) p (X (s − 1, n, m, k))q+1 +

q+1 E [(X (r, n, m, k)) p (X (s, n, m, k))q ]. γs

Proof. We have E [(X (r, n, m, k)) p (X (s, n, m, k))q ] Z ∞
cs = e−(m+1)x gr−1 1 − e−x I (x) dx, (4.2.8) m 0 (r − 1)! (s − r − 1)! where  s−r−1 1  1  e−γs y dy 1 − e−(m+1)y − 1 − e−(m+1)x m + 1 m + 1 x  Z ∞  s−r−1 1  γs 1  yq+1 e−γs y dy = 1 − e−(m+1)y − 1 − e−(m+1)x q+1 x m+1 m+1  Z ∞  s−r−2 1  1 1  q+1 −(m+1)y −(m+1)x y e−γs−1 y dy. − − 1−e 1−e q+1 x m+1 m+1 I (x) =

Z ∞

yq



Upon substituting for I (x) in (4.2.8), we obtain E [(X (r, n, m, k)) p (X (s, n, m, k))q] h i γs = E (X (r, n, m, k)) p (X (s, n, m, k))q+1 q+1 i h −E (X (r, n, m, k)) p (X (s − 1, n, m, k))q+1 .

Generalized Order Statistics

57

Thus, h i E (X (r, n, m, k)) p (X (s, n, m, k))q+1 h i = E (X (r, n, m, k)) p (X (s − 1, n, m, k))q+1 +

q+1 E [(X (r, n, m, k)) p (X (s, n, m, k))q ]. γs

For k = 1 and m = −1, Theorem 4.2.4 coincides with Theorem 3.3.3. For k = 1 and m = 0, we obtain from Theorem 4.2.4 i i h h  p q+1 q p q+1 p q Xs,n = E Xr,n Xs−1,n + Xs−1,n . E Xr,n E Xr,n n−s+1

Estimation of µ and σ Minimum Variance Linear Unbiased Estimators (MVLUEs) †

†

Lemma 4.2.5. Let µ and σ be the MVLUEs of µ and σ respectively, based on n gos X (1, n, m, k), X (2, n, m, k), . . ., X (n, n, m, k) from an absolutely continuous cd f F with pd f f . Then †

†

µ = X (1, n, m, k)− (σ/γ1 ) and 1 σ= n−1 †

"

#

n

∑ (γ j − γ j+1 ) X ( j, n, m, k)− γ1 X (1, n, m, k)

, with γn+1 = 0,

j=1

    † † Var µ = nσ2 / (n − 1) γ21 ,Var σ = σ2 / (n − 1) ,   † † Cov µ, σ = −σ2 / (n − 1) γ1.

Proof. It is not hard to show that E [X (s, n, m, k)] = µ + αs σ, Var [X (s, n, m, k)] = σ2Vs,

for 1 ≤ s ≤ n,

58

M. Ahsanullah and G.G. Hamedani

where αs = ∑sj=1 γ1j , Vs = ∑sj=1 γ12 . Let j

X = (X (1, n, m, k), X (2, n, m, k), . . ., X (n, n, m, k)), then X ] = µ1 + σα α, E [X Cov (X ( j, n, m, k), X (i, n, m, k)) = σ2Vi,

1 ≤ i < j ≤ n,

X ] = σ2V , Var [X where, as in Chapter 3, 1 is an n × 1vector of units, α = (α1 , α2 , . . ., αn ), V = (Vi j ) and Vi j = Vi for 1 ≤
j ≤ n. Let Ω = V −1 = V i j , then V ii = γ2i + γ2i+1 ,

i = 1, 2, . . ., kn,

γn+1 = 0,

V i+1i = V ii+1 = −γi+1, V i j = 0,

for |i − j| > 1.

†

†

The MVLUEs µ and σ respectively are (see, David, (1981)) †

α0V −1 (11α 0 − α 1 0 )V V −1 X / ∆, µ = −α †

V −1 X / ∆, σ = 1 0V −1 (11α 0 − α 10 )V where

α 0V −1 α) − (110V −1 α ) . ∆ = (11 0V −1 1 )(α 2

We also have   † Var µ = σ2 α 0V −1 α, and

  † Var σ = σ2 1 0V −1 1 /∆,

  † † Cov µ, σ = −σ2 1 0V −1α /∆.

Generalized Order Statistics

59

It can easily be shown that 1 0V −1 = (γ1 , 0, 0, . . ., 0), α 0V −1 = (γ1 − γ2 , γ2 − γ3 , . . ., γn−1 − γn , γn ) , α 0V −1 α = n, 10V −1 1 = γ21 , 10V −1α = γ1 and ∆ = (n − 1) γ21. Now, 1 0 −1 0 V −1 X (11 V 1 α − 1 0V −1 α1 0 )V ∆ 1 1 α0V −1 X − γ1 X (1, n, m, k)). = (γ21α0V −1X − γ1 1 0V −1 X ) = (α ∆ n−1 V −1X / ∆ = 1 0V −1 (11α 0 − α 1 0 )V

Hence 1 σ= n−1 †

"

#

n

∑ (γ j − γ j+1) X ( j, n, m, k)− γ1 X (1, n, m, k)

.

j=1

0 0 We can write α0 = γ−1 1 1 + c , where n 1 1 1 1 1 1 1 0, , + , + + , . . ., ∑ γ 2 γ 2 γ 3 γ 2 γ3 γ4 γ j=2 j

c0

c = Thus

!

.

†

†

V −1 X /∆ − µ = −cc 0V −1 (11α0 − α 10 )V

σ . γ1

We have †

0

cV

−1

0

1 = 0, c V

−1

σ α = n − 1 and hence µ = X (1, n, m, k)− . γ1 †

†

†

If k = 1 and m = 0, then γ j = n − j + 1 and µ and σ coincide with MVLUEs given by order statistics (see, Arnold et al., (1992), p. 176). If k = −1 and †

†

m = 0, then γ j = 1 and µ and σ coincide with MVLUEs given by Ahsanullah, ((1980), p.466).

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M. Ahsanullah and G.G. Hamedani †

†

The variances and covariance of µ and σ are   σ2 0 −1 † α V α ) = nσ2 / (n − 1) γ21 , Var µ = (α ∆   † σ2 0 −1 α V 1) = σ2 / (n − 1) , Var σ = (α ∆   σ2 0 −1 † † α V 1 ) = −σ2 / (n − 1) γ1 . Cov µ, σ = − (α ∆

Best Linear Invariant Estimators (BLIEs) The best linear invariant (in the sense of minimum mean squared error and in‡

‡

variance with respect to the location parameter µ) µ and σ of µ and σ are   † ‡ † E12 ‡ † and σ = σ/ (1 + E22 ) , µ = µ−σ 1 + E22 †

†

where µ and σ are MVLUEs of µ and σ and    ‡ ‡ ‡ Var[µ] Cov µ,σ   ‡ ‡ ‡ Cov µ,σ Var[σ]



 = σ2

h

E11 E12 E21 E22

i

.

The mean squared errors of these estimators are   n o ‡ 2 (1 + E22 )−1 MSE µ = σ2 E11 − E12 and

  ‡ MSE σ = σ2 E22 (1 + E22 )−1 .

Substituting the values of E12 and E22 in the above equations, we have, on simplification, that ‡ 1 † n−1 † ‡ † µ = µ+ σ and σ = σ, nγ1 n     ‡ n + 1 σ2 1 ‡ MSE µ = · 2 and MSE σ = σ2 . n n γ1

Generalized Order Statistics

61

Prediction of X (s, n, m, k) We shall assume that s > n. Let η = (η1 , η2 , . . ., ηn ) where η j = Cov(X (s, n, m, k), X ( j, n, m, k)), j = 1, 2, . . ., n and α∗ = The best linear unbiased predictor (BLUP) σ−1 E [X (x, n, m, k) − µ]. † † † † X − µ11 − σα α), Xˆ (s, n, m, k) of X (s, n, m, k) is Xˆ (x, n, m, k) = µ+ α∗ × σ + ηV −1 (X †

†

where µ and σ are the MVLUEs of µ and σ respectively. But α∗ = αs and η 0 = (V1,V2, . . .,Vn) . It can be shown that η 0V −1 = (0, 0, . . ., 0, 1) and hence †

†

† † Xˆ (s, n, m, k) = µ + αs σ + X (n, n, m, k)− µ − αn σ †

= X (n, n, m, k) + (αs − αn ) σ.

(4.2.9)

ˆ

If k = 1 and m = 0, then γ j = n − j + 1 and X (s, n, m, k) coincides with the BLUP based on the order statistics (see, Arnold et al. (1992), p. 181). If k = 1 and m = −1, then γ j = 1 and Xˆ (s, n, m, k) coincides with the BLUP based on record values (see, Ahsanullah (1980), p. 467). We have E[Xˆ (x, n, m, k)] = µ + (αs − αn ) σ, † σ2 + 2 (αs − αn )Cov(Xˆ (n, n, m, k), σ) Var[Xˆ (x, n, m, k)] = σ2Vn + (αs − αn )2 n−1     1 1 1 1 1 2 2 =σ +···+ 2 + +···− γn n − 1 γn+1 γs γ21    2 1 1 1 1 + +···+ +···+ , n − 1 γn+1 γs γ2 γn MSE[Xˆ (x, n, m, k)] = E[(Xˆ (x, n, m, k)− X (s, n, m, k))2 ] †

= E[(X (n, n, m, k)− X (x, n, m, k) + (αs − αn ) σ)2]   2 1 2 = σ Vn +Vs − 2Vn + (αs − αn ) n−1   = σ2 Vs −Vn + (αs − αn )2 (n − 1)−1 . If k = 1 and m = −1, then the BLUP XˆU(s) of the sth upper record value from

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(4.2.9) is (x − 1) XU(n) − (s − n) XU(1) , (n − 1)

(4.2.10)


E[XˆU(n)] = σ2 m + s2 − 2s / (m − 1) .

(4.2.11)

XˆU(s) = and

Let Xe (x, n, m, k) be the best linear invariant predictor of X (s, n, m, k). Then e (s, n, m, k) = Xˆ (s, n, m, k) − X

c∗12 † σ, 1 + c22

(4.2.12)

where
†
† † † c∗12 σ2 = Cov(σ, 1 − η 0 V −1 1 µ + α∗ − ηV −1 α σ) and c22 σ2 = Var[σ]. It can easily be shown that c∗12 = (αs − αn ) / (n − 1) and since c22 = 1/ (n − 1), c∗ n we have 1+c1222 = αs −α n . Thus e (s, n, m, k) = Xˆ (s, n, m, k)− αs − αn σ† X n † n−1 = X (n, n, m, k)+ (αs − αn ) σ  n αs − α n e σ E[X (s, n, m, k)] = µ + αs + n

(4.2.13) (4.2.14)

and (



n−1 Vn + n

2

1 Var[Xe (s, n, m, k)] = σ2 (αs − αn ) n−1   n−1 = σ2 γs − γn + 2 (αs − αn )2 . n 2

)

(4.2.15)

The bias term is e n, m, k) − X (s, n, m, k)] = (αn − αs )σ + E[X(s, 1 = − (αs − αn ) σ. n

n−1 (αs − αn )σ n

Generalized Order Statistics Thus, MSE[Xe (s, n, m, k)] = Var[Xe (s, n, m, k)] + (bias)2    2 1 n−1 2 2 (αs − αn ) σ = σ γs − γn + 2 (αs − αn ) + n n   1 2 2 = σ γs − γn + (αs − αn ) n 1 = MSE[Xˆ (s, n, m, k)] − (αs − αn )2 . n (n − 1) For k = 1 and m = 0, we obtain   s−n ˆ σ, E[XU(s) ] = µ + s + n  2 2 2 n + ns − s ˆ Var[XU(s) ] = σ . n2

63

Chapter 5

Characterizations of Exponential Distribution I 5.1. Introduction The more serious work on characterizations of exponential distribution based on the properties of order statistics, as far as we have gathered, started in early sixties by Ferguson (1964,1965), Tanis (1964), Basu (1965), Crawford (1966) and Govindarajulu (1966). Most of the results reported by these authors were based on the independence of suitable functions of order statistics. Chan (1967) reported a characterization result based on the expected values of extreme order statistics. The goal of this chapter is first to review characterization results related to the exponential distribution based on order statistics (Section 5.2) and then based on generalized order statistics (Section 5.3). We will discuss these results in the chronological order rather than their importance. We apologize in advance if we missed to report some of the existing pertinent results. Let X1and X2 be two i.i.d. random variables with common cd f F (x) and let X(1) = min {X1, X2 } and X(2) = max {X1 , X2 } . Basu (1965) showed that if F (x) is absolutely continuous with F (0) = 0, then a necessary and sufficient condition for F to be the cd f of an exponential random
variable with parameter λ, is that the random variables X(1) and X(2) − X(1) are independent. Freguson (1964) and Crawford (1966) also used the property of independence of X(1) and (X1 − X2 ) to characterize the exponential distribution. Puri and Rubin (1970)

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M. Ahsanullah and G.G. Hamedani

showed that X(2) − X(1) ∼ X1 (∼ means having the same distribution) characterizes the exponential distribution among the class of absolutely continuous distributions. Seshardi et al. (1969) reported a characterization of the exponential distribution based on the identical distribution of an (n − 1)-dimensional random vector of random variables Vr = Sr /Sn, r = 1, 2, . . ., n − 1, where Sr is the rth partial sum of the random sample, and vector of order statistics of n − 1 i.i.d. U (0, 1) random variables. Cs¨org¨o et al. (1975) and Menon and Seshardi (1975) pointed out that the proof given in Seshardi et al. was incorrect and presented a new proof. Puri and Rubin (1970) established a characterization of the exponential distribution based on the identical distribution of Xs,n − Xr,n and Xs−r,n−r (these rv0 s will be defined in the next paragraph) . Rossberg (1972) gave a more general result when s = r + 1, which will be stated in the following section. A different type of result characterizing the exponential distribution based on a function of the order statistics having the same distribution as the one sampled was established by Desu (1971), which is stated in the following section as well. Let X1 , X2, . . ., Xn be a random sample from a random variable X with cd f F. Let X1,n ≤ X2,n ≤ · · · ≤ Xn,n , be the corresponding order statistics. As pointed out by Gather et al. (1997), ”the starting point for many characterizations of exponential distribution via identically distributed functions of order statistics is the well-known result of Sukhatme (1937): The normalized spacings D1,n = nX1,n and Dr,n = (n − r + 1) (Xr,n − Xr−1,n), 2 ≤ r ≤ n

(5.1.1)

from an exponential distribution with parameter λ, i.e., F (x) = 1 − e−λx , x ≥ 0, λ > 0, are again independent and identically exponentially distributed. Thus, we have F ∼ E (λ) implies that D1,n, D2,n, . . ., Dn,n are i.i.d. ∼ E (λ).

5.2. Characterizations Based on Order Statistics We start this section with the following result due to Desu (1971).

(5.1.2)”

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67

Theorem 5.2.1. If F is a nondegenerate cd f , then for each positive integer k, kX1,k and X1 are identically distributed if and only if F (x) = 1 − e−λx for x ≥ 0, where λ is a positive parameter. Arnold (1971) proved that the characterization is preserved if in Theorem 5.2.1 the assumption ’ for all k ’ is replaced with the assumption ’ for two relatively prime positive integers k1, k2 > 1’ . Here is his theorem: Theorem 5.2.2. Let supp(F) = (0, ∞). Then X1 ∼ E (λ) if and only if ni X1,ni ∼ E (λ) for 1 < n1 < n2 with ln n1/ ln n2 irrational. Ahsanullah and Rahman (1972) presented the following characterization of the exponential distribution. Theorem 5.2.3. A necessary and sufficient condition that a non-negative random variable X with an absolutely continuous cd f F has an exponential distribution is that its k th order statistic, X k,n , can be expressed as k

Xk,n =

Yj

∑ (n − j + 1) , j=1

for any k ∈ N (the set of all positive integers ) such that 1 ≤ k ≤ n, where the Y j0 s ( j = 1, 2, . . ., k) are i.i.d. with cd f F. Galambos (1972) and Rossberg (1972) presented characterizations of the exponential distributions by the independence of certain functions of order statistics. Here we state Rossberg’s result which has an interesting proof based on complex analysis. Theorem 5.2.4. Let α (F) = inf{x : F (x) > 0} > −∞. Suppose that for R ∞ −sx r−1 some fixed r, n ∈ N, 2 ≤ r ≤ n, the Laplace transform α e dF (x) is nonzero for all s ∈ C (the set of all complex numbers ) with Re(s) > 0. Then Dr,n ∼ E (1) if and only if X 1 − α (F) ∼ E (1) . Rossberg pointed out that the assumption concerning the Laplace transform cannot be dropped since Dr,n ∼ E (1) is satisfied by underlying distributions

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other than exponential. He then gave the following example to demonstrate this point. Let   √ 4 F (x) = 1 − e−x 1 + 2 (1 − cos (αx)) , α ≥ 2 2, x ≥ 0, α then Dr,n ∼ E (1) , but the corresponding Laplace transform has zeros in {s ∈ C : Re (s) > 0}. We would like to point out that Arnold’s result (Theorem 5.2.2 above) requires kX1,k ∼ E (λ) for two different values of k and Gupta (1973) requires “limx→0 (F (x) /x) = λ for some 0 < λ < ∞”. Furthermore, Huang (1974) showed that Desu’s (1971) result is readily improved by special cases of characterization theorems in Chan (1967) and Huang (1974) as follows: Theorem 5.2.5. If F does not degenerate at the origin, and if E [nX1,n ] = E [X1 ] < ∞,

for n = 2, 3, . . .,

then F is an exponential distribution function. Remark 5.2.6. Huang (1974) showed that despite his assumption of E [X1 ] < ∞, this assumption is less restrictive than Desu’s (Theorem 5.2.1) condition. He then presented two stronger versions of Theorem 5.2.5 as stated below in Theorems 5.2.5∗ and 5.2.5∗∗. Theorem 5.2.5∗ . If F does not degenerate at the origin and if (i) nX1,n and X1are i.d. for some n ≥ 2, and (ii) E [nX1,n] = E [X1 ] for the other n 0 s, then F is exponential. Theorem 5.2.5∗∗. If F does not degenerate at the origin, and if there exists a real number δ and a positive integer k such that k

E [Xk,n] = δ ∑ (n − j + 1)−1 j=1

−1 for n = n1 , n2, . . ., where n0i s are distinct positive integers with ∑∞ i=1 ni = ∞, then F is exponential.

Characterizations of Exponential Distribution I

69

Kotz (1974) and Galambos (1975) discussed rather extensively the characterizations of distributions including the exponential distribution by order statistics. Ahsanullah (1975) reported a characterization of the exponential distribution based on identical distribution of nX1,n and the spacing Xn,n − Xn−1,n and then generalized his result in his (1976) paper as follows. Theorem 5.2.7. Let X be a non-negative random variable with an absolutely continuous (with respect to the Lebesgue measure) strictly increasing distribution F (x) for all x > 0, and F (x) < 1, for all x. Then the following properties are equivalent: (a) X has an exponential distribution; (b) X has a monotone hazard rate and for one i and one n with 2 ≤ i ≤ n, the random variables D 1,n and Di,n are i.d. Arnold and Ghosh (1976) presented characterizations of exponential distribution based on the conditional distributions of the spacings. These results were later extended by Rao and Shanbhag (1994), which will be stated later in this section. Let F be the cd f of a non-negative random variable and let, as before, F (x) = 1 − F (x) ,for x ≥ 0. We will call F “new better than used” (NBU), if F (x + y) ≤ F (x) F (y), x, y ≥ 0, and F is “new worse than used” (NWU) if F (x + y) ≥ F (x) F (y) , x, y ≥ 0. We will say that F belongs to the class C, if F is either NBU or NWU. Using these concepts, Ahsanullah (1977) established the following characterization of the exponential distribution. Theorem 5.2.8. Let X be a non-negative random variable with an absolutely continuous (with respect to the Lebesgue measure) cd f F which is strictly increasing on [0, ∞). Then the following properties are equivalent: (a) X has an exponential distribution; (b) for some i and n, 1 ≤ i < n, the random variables D i+1,n and X are i.d. and F belongs to class C. Gather (1989) pointed out that the weaker condition of NBU or NWU were assumed in Theorem 5.2.8. However, there is a gap in the proof and the desired result can be obtained for the special case r = n.

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Let F be cd f of a non-negative random variable X. Then F has increasing failure rate (IFR) (decreasing failure rate (DFR)) if [1−F(x+y)] [1−F(x)] is decreasing (increasing) on the support of Ffor all y ≥ 0. For the class of IFR (DFR) distributions, Ahsanullah (1978a) showed that D1,n ∼ Dr+1,n is sufficient to characterize exponential distribution. If F is absolutely continuous with pd f f , then F is IFR f (x) (DFR) if its failure rate F(x) is increasing (decreasing) on the support of F. In Ahsanullah (1978a), below, we find that Dr,n ∼ Ds,n is a characteristic property of the exponential distribution if F is absolutely continuous. Theorem 5.2.9. Let F be absolutely continuous, strictly increasing on (0, ∞), and IFR or DFR. Then F ∼ E (λ) if and only if there exists a triple (r, s, n), 2 ≤ r < s ≤ n, with Dr,n ∼ Ds,n . Again let F be an absolutely continuous cd f of a non-negative random varif (x) able with pd f f , and with hazard rate r (x) = F(x) , for x ≥ 0, and F (x) > 0. We will say that F has increasing hazard rate (IHR) if r (x) ≤ r (x + y), x, y ≥ 0 and F has decreasing hazard rate (DHR) if r (x) ≥ r (x + y), x, y ≥ 0. We say that F belongs to class D if F is either IHR or DHR. Using these concepts, Ahsanullah (1978b) presented the following characterization of the exponential distribution. Theorem 5.2.10. Let X be a non-negative random variable with an absolutely continuous (with respect to the Lebesgue measure) cd f F which is strictly increasing on [0, ∞). Then the following properties are equivalent: (a) X has an exponential distribution; (b) for some i, j and n, 1 ≤ i < j < n, the random variables D i,n and D j,n are i.d. and F belongs to class D. Property nX1,n ∼ X1 is studied in more general setting by Shimizu (1979) as stated below. Theorem 5.2.11. Let m, n1, . . ., nm ∈ N, c, a1, . . ., am > 0 with ( c > max1≤k≤m ak , if m > 1, if m = 1 c = a1 ,

Characterizations of Exponential Distribution I

71

and let ( α α α be the uniquely determined positive real number with ∑m k=1 ak = c , if m > 1, α > 0 arbitrary, if m = 1.   (k) Let the trivial case m = n1 = a1 = 1 be excluded. Let X j 1≤ j≤nk , 1≤k≤m

be i.i.d. random variables with distribution function F satisfying 0 = F (0) < (k) (k) F (x) < 1 for some x > 0, and X1,nk = min1≤ j≤nk X j . Then ! 1/α cnk (k) min X ∼F 1≤k≤m ak 1,nk if and only if F (x) =

(

1 − exp (−xα H (− ln x)) , x ≥ 0, 0, x < 0, 1/α

where H is a positive, bounded function with periods A k = ln(cnk /a), 1 ≤ k ≤ m. As pointed out by Gather et al. (1997), the above general settingreduces to (k)

(k) α

. Davies the case c = α = 1 via the monotone transformation X j → cX j and Shanbhag (1987) pointed out that the proof can be simplified by applying the integrated Cauchy functional equation. As corollaries to Theorem 5.2.11, characterizations of exponential distribution result without assuming continuity of F. One of these corollaries is due to Shimizu (1979) which is stated below. Corollary 5.2.12. Let F be as in Theorem 5.2.11, a1 , a2, . . ., an > 0 with = 1 and ln ai / ln a j irrational for some i , j ∈ {1, 2, . . ., n}. Then F ∼ E (λ) if and only if min1≤k≤n Xk /ak ∼ F. ∑nk=1 ak

Huang et al. (1979) restated Seshardi et al. (1965) result as follows. Theorem 5.2.13. Let F have mean 1/λ, 0 < λ < ∞, and X1 > 0, n ≥ 3. Let Sn = ∑ni=1 Xi and V1 = X1/Sn ,V2 = (X1 + X2 ) /Sn. Then F ∼ E (λ) if and only if (V1,V2) ∼ (U1,n−1,U2,n−1) , where Ui,n−1 , i = 1, 2 are order statistics from U (0, 1).

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Bell and Sarma (1980), presented a characterization of the exponential distribution based on the properties of linear transformations of order statistics and mentioned some statistical applications of their result, (see Theorems 5.2.14 and 5.2.15 below). Let Y1,Y2, . . .,Yn be i.i.d. with common exponential distribution E (λ) 0 and let Y1,n,Y2,n , . . .,Yn,n be the associated order statistics, and Y ∗ = (Y1,n,Y2,n, . . .,Yn,n ) . Let A denote the (n × n) matrix with elements   − (n − j + 1) for i = j − 1, i > 1, si j = (n − j + 1) for i = j, i = 1, 2, . . ., n,   0 otherwise. Finally, P is called an elementary matrix if it is obtained by interchanging some rows of the identity matrix. 0

Y ∗ = (X1 , X2, . . ., Xn) where M = ((mi, j )) is an Theorem 5.2.14. Let X = MY (n × n) nonsingular matrix. Then X i0 s are i.i.d. with common distribution as Y 1 if and only if M = PA for some elementary matrix P. Theorem 5.2.15. Let Z1 , Z2, . . ., Zn be non-negative i.i.d. with cd f F. Let 0 Z ∗ = (X1, X2, . . ., Xn ) . F (z) be continuous and strictly increasing. Let XX = AZ Then Xi0 s are i.i.d. with common cd f F for all n > 1 if and only if ( 1 − e−λx for x ≥ 0, F (x) = 0 for x < 0 for some constant λ > 0. Using failure rates, Ahsanullah (1981) presented the following two characterizations of exponential distribution. Theorem 5.2.16. Let F be absolutely continuous, F (0) = 0 and strictly increasing on (0, ∞) . Let the failure rates r X1 and rDr,n of X1 and Dr,n, respectively, be continuous from the right at zero. Moreover, assume that r X1 attains its maximum or minimum at zero. Then F ∼ E (λ) if and only if there exists a pair (r, n), 2 ≤ r ≤ n, such that r Dr,n (0+) = rX1 (0+) .

Characterizations of Exponential Distribution I

73

Theorem 5.2.17. Let F be absolutely continuous, supp(F) = (0, ∞), and IFR or DFR. Moreover, let E [X1] < ∞. Then F ∼ E (λ) if and only if there exists a pair (r, n), 2 ≤ r ≤ n, such that E [Dr,n ] = E [Dr−1,n ]. Galambos and Kotz (1983) pointed out that Theorem 5.2.1 is related to an assertion for the integer part of a random variable as follows. (t)

Theorem 5.2.18. Let F −1 (0+) ≥ 0, Xi = [Xi /t] + 1, t > 0, i = 1, 2, where [x] denotes the integer part of x ∈ R. If the distribution of   (t) (t) (t) (t) min X1 , X2 |X1 + X2 = 2m + 1 is uniform on 1, 2, . . ., m for every m ∈ N, then F ∼ E (λ) for some λ > 0. For certain characterizations of exponential distribution by the properties of order statistics associated with non-linear statistics, we refer the reader to Chapter 3.1 of the book by Kakosyan et al. (1984). Here is one of their results. Theorem 5.2.19. Let (Xi )i∈N be a sequence of i.i.d. random variables with cd f F, X1 > 0, F continuous for x ≥ 0, and let limx→0 F (x) /x exist and be finite. Moreover, let N ≥ 2 be an integer-valued random variable independent of (Xi )i∈N . Then F ∼ E (λ) for some λ > 0 if and only if NX 1,N ∼ X1. The following characterization is based on a moment equation reported by Iwi´nska (1986). Theorem 5.2.20. Let F be absolutely continuous, F −1 (0+) = 0, strictly increasing on (0, ∞) and NBU or NWU. Then F ∼ E (λ) if and only if there exists a triple (r, s, n), 1 ≤ r < s ≤ n, with E [Xs,n ] − E [Xr,n ] = E [Xs−r,n−r ]. Remark 5.2.21. It is worth mentioning that without additional assumption, the moment equation given in the above theorem with s = r + 1 for one pair (r, n), 1 ≤ r ≤ n − 1, does not characterize exponential distribution. For every

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M. Ahsanullah and G.G. Hamedani

choice of r and n there is a distribution different from exponential which possesses the above property (see, Kamps, 1991 and 1992a). For an example, the cd f given by  −1 F (x) = 1 + ecd x−d , c > 0, d =

n and x > 0 n−1

satisfies the moment relation for r = 2. We would like to mention that under conditions of Theorem 5.2.20 and based on stronger condition Xs,n − Xr,n ∼ Xs−r,n−r , Iwi´nska (1985) characterized exponential distribution. Ahsanullah (1984) proved result similar to Theorem 5.2.20 using stronger condition which we will mention in more detail later on in this section (see Remarks 5.2.28). The following characterization is due to Ahsanullah (1987) which is based on the distributions of random sums. Theorem 5.2.22. Let F be absolutely continuous with pd f f , strictly increasing on (0, ∞) and let f be continuous from the right at zero. Then F ∼ E (λ) if and only if there exists a pair (r, n), 2 ≤ r ≤ n − 1, such that r

∑ Di,n ∼

i=1

r

∑ Yj

with Y1 ,Y2, . . .,Yr i.i.d. ∼ F.

j=1

The following result which is in the spirit (and somewhat stronger than) of that of Ahsanullah (1984) and of Iwi´nska (1985) is due to Gather (1988). Theorem 5.2.23. Let F be continuous and strictly increasing on (0, ∞). Then F ∼ E (λ) if and only if there exists a quadruple (r, s1, s2, n), 1 ≤ r < s1 < s2 ≤ n, such that Xsi,n − Xr,n ∼ Xsi −r,n−si

for i = 1, 2.

Gather (1988) pointed out that the above result was already stated in Ahsanullah (1975). However, in the proof the NBU/NWU property of F was implicitly used.

Characterizations of Exponential Distribution I

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Kakosyan et al. (1984) stated the following conjecture: The distributional identity N

(1 − p) ∑ Xi ∼ nX1,n i=1

for some fixed n ∈ N where N has a geometric distribution with parameter p, characterizes exponential distribution. Ahsanullah (1988a) discussed this conjecture and pointed out that under additional assumption that F is IFR or DFR, nX1,n can be replaced by NX1,N , see the next theorem. Theorem 5.2.24. Let (Xi )i∈N be a sequence of i.i.d. random variables with cd f F, F −1 (0+) ≥ 0, F (x) < 1 for all x > 0, F IFR or DFR, E [X1] < ∞, and 0 < limx→0+ F (x) /x = λ < ∞. Moreover, let N be a geometrically distributed random variable independent of (Xi )i∈N . Then F ∼ E (λ) if and only if N

(1 − p) ∑ Xi ∼ NX1,N . i=1

The following more general result (with respect to r ) is given in Ahsanullah (1988b). Theorem 5.2.25. Under the assumptions of Theorem 5.2.24, F ∼ E (λ) if and only if there exists a pair (r, n), 1 ≤ r ≤ n, such that N

(1 − p) ∑ Xi ∼ NX1,N . i=1

Gajek and Gather (1989) improved Theorem 5.2.9 in the sense of not using Dr,n ∼ Ds,n as a distributional identity, but only required the equality of the corresponding pd f 0 s or the failure rates at zero. Here is their result. Theorem 5.2.26. Let F be absolutely continuous, F −1 (0+) = 0, strictly increasing on (0, ∞), and IFR or DFR. Moreover, the pd f 0 s, f Dr,n and f Ds,n of Dr,n and Ds ,n are assumed continuous from the right at zero. Then F ∼ E (λ) if and only if there exists a triple (r, s, n), 1 ≤ r < s ≤ n, such that f Dr,n (0) = f Ds,n (0).

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In Theorem 5.2.16, the equation rDr,n (x) = rX1 (x) is required for all x ≥ 0, whereas the validity for x = 0 is sufficient, see Gajek and Gather (1989). Gajek and Gather (1989) also presented characterizations of the exponential distribution based on identical distributions of Dr,n and Ds,n as well as weaker conditions for some integers r and s with 1 ≤ r < s ≤ n. These results were later improved by Kamps and Gather (1997) which we will discuss in more details later in this section. Gajek and Gather (1989) presented another result using expectations of Dr,n and Ds,n which extended Ahsanullah’s result (Theorem 5.2.17). Gather (1989) established the following characterization of the exponential distribution extending the results reported by Rossberg (1972) and Ahsanullah (1984). Theorem 5.2.27. Let F be a continuous cd f and let F (x) be strictly increasing for all x > 0. Then F is exponential if and only if X j,n − Xi,n ∼ X j−i,n−i ,

1 ≤ i < j ≤ n,

holds true for a sample from F with two distinct values j 1 and j2 of j and some i, n, 1 ≤ i < j1 < j2 ≤ n, n ≥ 3. Remarks 5.2.28. (a) If Xi+1,n − Xi,n ∼ X1,n−i for some 1 ≤ i ≤ n, n ≥ 2, this characterizes the exponential distribution in the class of all continuous distributions without further assumptions (Rossberg, 1972). (b) If for one arbitrary j, i, n, 1 ≤ i < j ≤ n, X j,n − Xi,n ∼ X j−i,n−i , this together with F being continuous and IRF is a characteristic property of the exponential distribution (Ahsanullah, 1984). (c) Gather mentions that she has not been able to drop the condition IRF in (b) and no counterexamples,as far as she had gathered, are known. (d) Similar characterization results based on the property given in Theorem 5.2.10 are discussed in Iwi´nska (1986), Gather (1988) and Gajek and Gather (1989). As pointed out by Gather et al. (1998), for specific distributions it may be difficult to verify the assumption concerning the Laplace transformation of Theorem 5.2.4. For this reason, in Pudeg (1990) it is replaced by an aging property as follows. Theorem 5.2.29. Let F be IFR (or DFR) and let α (F) = inf {x : F (x) > 0} > −∞. Then Dr,n ∼ E (λ) for a pair (r, n), 2 ≤ r ≤ n, if and only if X1 − α (F) ∼ E (λ) for some λ > 0.

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F is called decreasing (increasing) mean residual life DMRL (IMRL) if 1 F (x)

Z ∞

F (y) dy decreases (increases) with respect to x on supp(F) .

x


n−r+1 Clearly, the assumption ‘F is IFR or DFR’ can be replaced by ‘1 − F is IFR or DFR’. Pudeg (1990) pointed out that this condition in Theorem 5.2.29 can be weakened to ‘1 − (F)n−r+1 is DMRL or IMRL’. She also showed that Rossberg’s (1972) example still serves as a counterexample. FR does not have DMRL or IMRL property and hence is neither IFR nor DFR. Riedel and Rossberg (1994) studied characterization of exponential distribution via distributional property of a contrast Xr+s,n −Xr,n . Their main assumption is the asymmetric behavior of the survival function of the contrast. Here is their result. Theorem 5.2.30. Let F be absolutely continuous with a continuous and bounded pd f f on [0, ∞) , and let λ > 0. Then F ∼ E (λ) if one of the following conditions is satisfied. a) There exists a triple (r, s, n), 1 ≤ s ≤ n − r, such that P (Xr+s,n − Xr,n ≥ x) − e−λ(n−r)x = o (xs ), x → 0,
and f (x) /F (x) − λ does not change its sign for any x ≥ 0. b) There exists a quadruple (r, s1, s2, n), 1 ≤ s1 < s2 ≤ n − r, such that P(Xr+si,n − Xr,n ≥ x) − e−λ(n−r)x = o (xsi ) , x → 0, for i = 1, 2. The following result is due to Rao and Shanbhag (1994) based on strong memoryless property characterization of exponential and geometric distributions, which is an extended version of Ferguson-Crawford result which is stated as: If X and Y are independent nondegenerate random variables, then min {X,Y } is independent of X −Y if and only if for some α > 0 and β ∈ R, we have α (X − β) and α (Y − β) to be either both exponential or both geometric (in usual sense). Theorem 5.2.31. Let X and Y be as in Ferguson-Crawford result and y 0 be a point such that there are at least two support points of the distribution of

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min {X,Y } in (−∞, y0 ] . Let φ be a real-valued Borel measurable function on R such that its restriction to (−∞, y0 ] is nonvanishing and strictly monotonic. Then X − Y and φ (min {X,Y }) I{min{X,Y }≤y0 } are independent if and only if for some α ∈ (0, ∞) and β ∈ R, α (X − β) and α (Y − β) are both exponential, or geometric on N, in which case X −Y and min {X,Y } are independent. Rao and Shanbhag (1994), gave the following two corollaries of Theorem 5.2.31. Corollary 5.2.32. If in Theorem 5.2.31, X and Y are additionally assumed to be i.d., then the assertion of the theorem holds with |X −Y | in place of X −Y. Corollary 5.2.33. Let X and Y be two i.i.d. nondegenerate positive random variables and y0 be as defined in Theorem 5.2.31. Then min {X,Y } / max {X,Y } and min {X,Y } I{min(X,Y )≤y0 } are independent if and only if for some α > 0 and β ∈ R, α (ln X − β) is either exponential or geometric. The following interesting remarks are given in Rao and Shanbhag (1998) concerning Theorem 5.2.31 and Corollaries 5.2.32 and 5.2.33, which are copied here from theirs. Remarks 5.2.34. (i) If we replace in Corollary 5.2.33, the condition on the 0 existence of y0 by that there exists a point y0 such h thatthere are at least two

support points of the distribution of max {X,Y } in y0, ∞ , then the assertion of corollary with min {X,Y } I{min{X,Y }≤y0 } replaced by max {X,Y }I{max{X,Y }≥y0 } 0  and ln X replaced by − ln X holds. This follows because min X −1 ,Y −1 =  (max {X,Y })−1 and max X −1 ,Y −1 = (min {X,Y })−1 . The result that is observed here is indeed a direct extension of Fisz’s (1958) result, and it is yet another result mentioned in Rao and Shanbhag (1994). (Fisz characterizes the distribution in question via independence of max {X,Y } / min{X,Y } and max {X,Y }.) (ii) Under the assumptions in Theorem 5.2.31, the condition that X −Y and φ (min {X,Y }) I{min{X,Y }≤y0} be independent is clearly equivalent to that for each y ∈ (−∞, y0 ] , X − Y be independent of I{min{X,Y }≤y}. (The remark with X − Y replaced by |X −Y | applies to Corollary 5.2.32, i.e. when we have the assumptions as in the corollary.) 0

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(iii) Theorem 5.2.31 remains valid if the “independence” condition appearing in the assertion is replaced by that conditional upon min {X,Y } ∈ (−∞, y0 ], X − Y and min {X,Y } are independent. (The corresponding remark in the case of Corollary 5.2.32 is now obvious.) (iv) If the assumptions in Theorem 5.2.31 are not met with P {X ≥ Y } > 0, then on modifying slightly, the Rao-Shanbhag argument that we have referred to in the proof of the theorem proves that conditionally upon min {X,Y } ∈
(−∞, y0 ] , I{X=Y }, (X −Y )+ and min {X,Y } are independent only if l ∈ (−∞, y0 ) , the conditional distribution of X − l given that X ≥ l is exponential if the conditional distribution of Y − l given that Y ≤ y0 is nonarithmetic, and   given that X ≥ l is geometric on {0, λ, 2λ, . . .} if the conditional that λ X−l λ distribution of Y − l given that Y ≤ y0 is arithmetic with span λ, where l is the left extremity of the distribution of Y and [·] denotes the integral part. (v) The version of Theorem 5.2.31 with min {X,Y } in place of φ (min {X,Y }) holds if in place of “two support points” we take “two nonzero support points” or in place of “there are . . . in (−∞, y0 ]” we take “the left extremity of the distribution of min {X,Y } is nonzero and is less than y0 ”. The result in (iii) above and that mentioned here are essentially variations of Theorem 8.2.1 of Rao and Shanbhag (1994). (Incidentally, the cited result of Rao and Shanbhag requires a minor notational alteration such as the one where “(1, min{X,Y })I{min{X,Y }≤y0 }” appears in place of “min {X,Y } I{min{X,Y }≤y0 } ”.) The following theorem is due to Rao and Shanbhag (1994) and extends a result of Shimizu. It is pointed out in Rao and Shanbhag (1998) that Theorem 5.2.35 below is an obvious consequence of their Theorem 2, (1998). Theorem 5.2.35. Let X1 , X2 , . . ., Xn, n ≥ 2, be i.i.d. positive random variables and a 1 , a2, . . ., an be positive real numbers not equal to 1, such that the smallest closed subgroup of R containing ln a1, ln a2, . . ., ln an equals R itself. Then, in obvious notation, for some m ≥ 1 min {X1a1 , X2a2 , . . ., Xnan } ∼ X1,m if and only if the survivor function of X 1 is of the form F (x) = exp {−λ1 xα1 − λ2 xα2 } ,

x ∈ R+,

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with λ1, λ2 ≥ 0, λ1 + λ2 > 0 and αr (r = 1, 2) as positive numbers such that −α ∑ni=1 ai i = m. (If α1 = α2, the distribution on the above line is Weibull. ) Remark 5.2.36. Franco and Ruiz (1995) defined the “order mean function” between the adjacent order statistics Xk,n and Xk+1,n by ξ (x) = E [h (Xk,n) |Xk+1,n = x] =

1 (F (x))k

Z x

h (y) d (F (y))k ,

−∞

whose domain of definition is the set (α, ∞), where α may be −∞, h is a given real, continuous and strictly monotone function and F ∈ F , the set of continuous distributions for which the integral on the RHS of the above equation is finite for all x ∈ R. They used this concept to characterize certain continuous distributions of which, in the special case of h (x) = x and ξ (x) = x + b ( ξ (x) corresponds to F (x) in the above equation) is the exponential distribution. Set X0,n = 0 and define i

Si,n =

∑ (n − j + 1) (X j,n − X j−1,n) ,

i = 1, 2, . . ., n,

j=1

Wr,n = (S1,n/Sr,n , S2,n/Sr,n , . . ., Sr−1,n /Sr,n ) . A conjecture stated by Dufour (1982) is that if Wr,n ∼ U(·) (r − 1) = (U1,r−1,U2,r−1, . . .,Ur−1,r−1), where Ui,r−1, i = 1, 2, . . ., r −1 are the order statistics of r −1 i.i.d. random variables with a uniform distribution on (0, 1), then X1 has an exponential distribution. This result, if true, is a characteristic property of the exponential distribution. Rao and Shanbhag (1995b), dealt, among other things, with Dufour’s conjecture, mentioned above, in the following theorem. As pointed out by Rao and Shanbhag (1995b) below, a special case of their result was established earlier by Leslie and van Eeden (1993). Theorem 5.2.37. Let r an n be positive integers greater than or equal to 3 such that either r, n ∈ {3, 4} with r = n, or r, n ≥ 5. Also let X1 , X2, . . ., Xn be i.i.d. positive random variables. If W r,n ∼ U(·) (r − 1) , then X1 has an exponential distribution.

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Rao and Shanbhag (1998) pointed out that they proved the above theorem when r, n ≥ 5 in their (1995b) paper and the result for r = n = 3, 4 follows from certain uniqueness theorem on the problem, established in their (1995b) paper. They also mentioned that Theorem 5.2.37 for r, n ≥ 5 was proved independently via a different argument by Xu and Young (1995). This will be pointed out below. Following Xu and Yang (1995)’s Introduction, the characterization problem for the case r = n, has been studied by Seshardi et al. (1969) and Dufour et al. (1984). Dufour’s conjecture has been partially answered by Leslie and van Eeden (1993), who showed that the conjecture is true for (2/3) n+1 ≤ r ≤ n−1, but the case r < (2/3)n = 1 has not been determined as of 1995. Xu and Yang (1995) showed that the conjecture is true when r ≥ 5 indicating that their lower bound is independent of the sample size n, whereas in Leslie and van Eeden (1993) the lower bound increases with n. The cases r = 2, 3 and 4 are, as of 1995, still not determined. Menon and Seshardi (1975) have shown that for r = n = 2 the conjecture is false. So, it is assumed that n ≥ 3. If, however, cd f of X1 belongs either to the class of NBU or NWU distributions, then Dufour’s conjecture is true for r ≥ 2. It seems that Xu and Young were not aware of Rao and Shanbhag (1995b) paper which has dealt with the case r = n = 3, 4. Here is Xu and Young’s result. Theorem 5.2.38. Suppose that n ≥ r ≥ 5. Then X1 has an exponential distribution if and only if W r,n ∼ U(·) (r − 1) . Remark 5.2.39. Xu and Yang (1995) mentioned an alternative statement for Theorem 5.2.38: “If n ≥ 5, X1 has an exponential distribution if and only if W5,n ∼ U(·) (4).” Then the general case follows immediately. This alternative statement indicates that a sample size of 5 is large enough to characterize the exponential distribution. Employing the concepts of NBU and NWU, Xu and Yang presented the following characterization of the exponential distribution. Theorem 5.2.40. Suppose that cd f F of X 1 is either NBU or NWU and that {F −1 (u) : 0 < u < 1} = (0, ∞) . Let r for 2 ≤ r ≤ n, n ≥ 3, be fixed. Then X1 is exponentially distributed if and only if W r,n ∼ U(·) (r − 1) .

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Lo´ pez-Bl´aquez and Moreno-Rebollo (1997) start their Introduction section by stating that Ferguson (1967) characterized distributions for which the regression of an order statistic on an adjacent one is linear. Ferguson pointed out that it is not known which distributions would be characterized if non-adjacent order statistics are considered. Nagaraja (1988) affirms that the problem remained unsolved. Similar reference to this problem was given by Arnold et al. (1992). Lo´ pez-Bl´aquez and Moreno-Rebollo (1997) established the following two characterizations of the distributions when the regression of two order statistics, not necessarily adjacent, is linear. Theorem 5.2.41. Let X1 have a cd f which is k times differentiable in D F = {x ∈ R : 0 < F (x) < 1}, such that E [Xi+k,n|Xi,n] = bXi,n + a. Then, except for location and scale parameters, F (x) = 1 − |x|δ , F (x) = 1 − exp (−x) ,

for x ∈ [−1, 0], for x ∈ [0, ∞) ,

if 0 < b < 1, if b = 1,

F (x) = 1 − xδ ,

for x ∈ [1, ∞) ,

if b > 1,

where, δ = (r − (n − i))−1 and r is the unique real root greater than k − 1 of the polynomial equation 1 Pk (x) = Pk (n − i) . b Theorem 5.2.42. Let X1 have a cd f F which is k times differentiable in D F , such that E [Xi,n |Xi+k,n] = cXi+k,n + d, then, except for location and scale parameters, F (x) = xθ ,

for x ∈ [0, 1],

if 0 < c < 1,

F (x) = exp (x) ,

for x ∈ (−∞, 0] ,

if c = 1,

F (x) = |x|θ ,

for x ∈ (−∞, 1] ,

if c > 1,

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where, θ = (r − (i + k − 1))−1 and r is the unique real root greater than k − 1 of the polynomial equation 1 Pk (x) = Pk (i + k − 1) . c Remark 5.2.43. Most of the characterizations mentioned so far have already appeared in two excellent survey papers by Gather et al. (1998) and by Rao and Shanbhag (1998). The latter contains new characterizations which had not appear prior to 1998 of which some have been mentioned in this section already and some will be given below. Rao and Shanbhag (1998) pointed out that: There is an interesting variant of Theorem 5.2.31; Rossberg (1972), Ramachandran (1980), Rao (1983), Lau and Ramachandran (1991), and Rao and Shanbhag (1994) among others have produced versions of Theorem 5.2.31. A special case of this result for n = 2 was given in a somewhat restricted form by Puri and Rubin (1970). Then Rao and Shanbhag gave their variant (below) and showed that it is also linked with the strong memoryless property characterization of the exponential and geometric distributions. Theorem 5.2.44. Let n ≥ 2 and X1 , X2, . . ., Xn be i.i.d. random variables with cd f F that is not concentrated on {0}. Then, for some 1 ≤ i < n, Xi+1,n − Xi,n ∼ X1,n−i, where X1,n−i = min{X1, X2, . . ., Xn−i} , if and only if one of the following two conditions holds: (i) F is exponential. (ii) F is concentrated on some semilattice of the form {0, λ, 2λ, . . .} with j−1 for j = 1, 2, . . . for F (0) = α and  F
( jλ)i− F (( j − 1) λ) = (1 − α) (1 − β) β −1/i n some α ∈ 0, i and β ∈ [0, 1) such that P {Xi+1,n > Xi,n } = (1 − α)n−i
−1/i
−1/i (which holds with α = ni or β = 0 if and only if F (0) − F (0−) = ni
−1/i and F (λ)− F (λ−) = 1 − ni for some λ > 0). (The existence of cases β > 0 can easily be verified.)

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The following two remarks ( Remarks 2 & 3, p. 240, 1998) are taken from Rao and Shanbhag (1995a) which explain respectively as how the existence of β > 0 in Theorem 5.2.44 follows and how the result of Stadje (1994) is a corollary to this theorem. Remarks 5.2.45. (a) Suppose we consider a family of distributions of the form in (ii) (of Theorem 5.2.44), but not necessarily satisfying the condition that P {Xi+1,n > Xi,n} = (1 − α)n−i . Then, if we take a fixed β ∈ (0, 1) and allow α to vary, we get for a sufficiently small α, P{Xi+1,n > Xi,n } < (1 − α)n−i ,
−1/i and for α = ni , we get P{Xi+1,n > Xi,n } > (1 − α)n−i ; since we have P {Xi+1,n > Xi,n } to be a continuous function of α, we have the existence of an α value such that P{Xi+1,n > Xi,n } = (1 − α)n−i . This proves the last statement (in the parentheses) of Theorem 5.2.44.
−1/i = 1/2. In this case, if nei(b) If n = 2 and i = 1, we get ni ther α = 1/2 nor β = 0, we get P {Xi+1,n > Xi,n } = 1 − P{X1 = X2} = {2 (1 − α) (α + β)} / (1 + β) ; consequently we have here P {Xi+1,n > Xi,n } = (1 − α)n−i , i.e. the probability to be equal to 1 − α, if and only if β = 1 − 2α. One can hence see as to how Stadje’s result follows as a corollary to Theorem 5.2.44. The following theorem is due to Rao and Shanbhag (1998) special versions of which have been dealt with by Arnold and Ghosh (1976) and Arnold (1980). Zijlstra (1983) and Fosam et al. (1993) have reported further specialized versions of Theorem 5.2.44. Theorem 5.2.46. Let n ≥ 2 and X1, X2, . . ., Xn be nondegenerate i.i.d. random variables with cd f F. Then, for some i ≥ 1, the conditional distribution of Xi+1,n − Xi,n given that Xi+1,n − Xi,n > 0 is the same as the distribution of X 1,n−i, where X1,n−i is as defined in Theorem 5.2.44, if and only if F is either exponential, for some λ > 0, or geometric on {λ, 2λ, . . .} . The following characterization of exponential distribution is due to Rao and Shanbhag (1998) which is in terms of certain function of spacings. Theorem 5.2.47. Let X1, X2, . . ., Xn (n ≥ 2) be i.i.d.with a continuous cd f F. Further, let i be a fixed positive integer less than n and φ be a nonarithmetic (or

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nonlattice) real monotonic function on R+ such that E [|φ (Xi+1,n − Xi,n )|] < ∞. Then, for some constant c 6= φ (0+) , E [φ (Xi+1,n − Xi,n ) |Xi,n ] = c a.s. if and only if F is exponential, within a shift. The following characterization of the exponential distribution in terms of the independence of spacing Xn,n − Xn−1,n and order statistic Xn−1,n is due to Lee et al. (2002). Theorem 5.2.48. Xn,n − Xn−1,n and Xn−1,n are independent if and only if F is exponential. Remark 5.2.49. Bairamov et al. (2002) showed that each of the following two conditions is a characteristic property of the exponential distribution: P (Xn,n > t + x|X1,n > t) = P (Xn,n > x) and E [Xn,n − t|X1,n > t] = E [Xn,n ] . ¨ Bairamov and Ozkal (2007) presented characterizations of certain distributions based on the conditional expectations of order statistics. One of the distributions considered is Weibull. We shall state their result for the special case of Weibull, namely exponential. Proposition 5.2.50. The absolutely continuous non-negative random variable X with strictly increasing cd f F has exponential distribution with parameter λ > 0, if and only if, the representation  1 k  E λX j,n |X j−p,n = x, X j+k+1−p,n = y ∑ k p=1 =

exp (−λx) (λx + 1) − exp (−λy) (λy + 1) , exp (−λx) − exp (−λy)

holds for all 0 ≤ x < y < ∞. Here k + 1 ≤ j ≤ n − k. Hamedani et al. (2008) characterized various distributions based on truncated moment of the first order statistic, one of which is Weibull. Here we state the result for the special case of exponential distribution.

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Proposition 5.2.51. Let X be a non-negative continuous random variable with cd f F such that limx→∞ x (1 − F (x))n = 0. Then X has an exponential distribution with parameter λ > 0 if and only if E [X1,n|X1,n > t] =

nλt + 1 , nλ

t > 0.

5.3. Characterizations Based on Generalized Order Statistics In this section we review characterizations of exponential distribution based on generalized order statistics. Kamps (1995), introduced a concept of generalized order statistics as a unified approach to order statistics and record values. Kamps defined generalized order statistics as follows. Let, for simplicity, F denote an absolutely continuous cd f with pd f f . The random variables X (1, n, m, k), X (2, n, m, k), . . ., X (n, n, m, k) are called generalized order statistics based on F, if their joint pd f is given by f X(1,n,m,k),X(2,n,m,k),...,X(n,n,m,k) (x1 , x2, . . ., xn ) ! ! n−1 n−1
m
k−1 = k ∏ γj ∏ F (xi) f (xi ) F (xn ) f (xn) , j=1

F

−1

i=1

(0+) < x1 ≤ x2 ≤ · · · ≤ xn < F −1 (1),

with n ∈ N, k > 0, m ∈ R such that γ j = k + (n − j) (m + 1) > 0 for all 1 ≤ j ≤ n. As pointed out by Kamps, in the case m = 0 and k = 1 this model reduces to the joint pd f of ordinary order statistics, and in the case m = −1 and k ∈ N we obtain the joint pd f of the first n kth record values based on a sequence X1, X2, . . . of i.i.d. rv0 s with cd f F. The marginal pd f of the rth generalized order statistic is given by f X(r,n,m,k) (x) =


γ −1 cr−1 F (x) r f (x) gr−1 m (F (x)), (r − 1)!

(see Kamps 1995, p. 64) and the pd f of the spacings Wr−1,r,n = X (r, n, m, k) − X (r − 1, n, m, k),

2 ≤ r ≤ n,

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has the following representation: f Wr − 1 , r , n (y) =

cr−1 (r − 2)!

Z ∞ −∞


m
γr −1 F (x) f (x) gr−2 f (x + y) dx m (F (x)) F (x + y)

with (see Kamps 1995, p. 69) r

cr−1 = ∏ γ j , j=1

hm (x) =

Z

(1 − x)m dx =

1 ≤ r < n,

(

1 − m+1 (1 − x)m+1 , m 6= 1,
1 , m = 1, ln 1−x

gm (x) = hm (x) − hm (0),

x ∈ [0, 1).

Kamps (1995, p. 81) generalized Sukhatme’s result that the normalized spacings D (1, n, m, k) = γ1X (1, n, m, k), D (r, n, m, k) = γr (X (r, n, m, k)− X (r − 1, n, m, k)) ,

2 ≤ r ≤ n,

based on exponential distribution with parameter λ are i.i.d. E (λ) . Again following Kamps (1995), Wr−1,r,n and X (1, n − r + 1, m, k) are i.d., since γr Wr−1,r,n ∼ X (1, n, m, k) = Z γ1 and     γ1 −1   γr X(1,n,m,k) γr γr γr z = γr F z f z f (z) = f γ1 γ1 γ1 γ1 Z

= γr exp {−λγr z} = f X(1,n−r+1,m,k) (z) . Hence, it remains to be seen that the above properties will indeed characterize exponential distribution uniquely. In the following theorem, Kamps and Gather (1997) show that, under certain regularity conditions, a weaker assumption than D (r, n, m, k) ∼ D (s, n, m, k) is sufficient to characterize exponential distribution within the class of distributions with IFR or DFR property. A special case of this result is that of Gajek and Gather (1989) established for ordinary order statistics.

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As in Kamps and Gather (1997), let rY (x) = g (x) /G (x) denote the failure rate of a random variable Y with cd f G and pd f g. The failure rates as well as the pd f f in Theorem 5.3.1 and Remark 5.3.2 below ( both due to Kamps and Gather, 1997) are assumed to be continuous from the right. If rY is monotone, then the limit rY (0) = limx→0 rY (x) is assumed to be finite (cf. Gajek and Gather, 1989). Theorem 5.3.1. Let F be absolutely continuous with pd f f , F (0) = 0, and suppose that F is strictly increasing on (0, ∞), and either IFR or DFR. Then F ∼ E (λ) for some λ > 0 if and only if there exist integers r, s and n, 1 ≤ r < s ≤ n, such that r D(r,n,m,k) (0) = rD(s,n,m,k) (0) . Remark 5.3.2. Again, Kamps and Gather (1997) pointed out that it is easily seen that the property rD(s,n,m,k) (0) = r (0) (r is failure rate of F) for some 2 ≤ s ≤ n is also a characteristic property of exponential distribution. This assertion corresponds to Remark 2.1 in Gajek and Gather (1989) and generalizes Theorem 2.2 in Ahsanullah (1981b) for ordinary order statistics. As in the case r = 1 in Theorem 5.3.1, it is obvious that the IFR or DFR assumption can be replaced by the condition that zero is an extremal point of the failure rate of F. Kamps and Gather (1997) presented the following theorem generalizing Theorem 2.1 of Ahsanullah (1981b), which in the case of ordinary order statistics, characterizes exponential distribution based on the equality of the expectations of successive (normalized) spacings. Theorem 5.3.3. Let F be absolutely continuous with pd f f , F (0) = 0, F (x) < 1 for all x > 0, and suppose that F is IFR or DFR. Moreover, let m ≥ −1. Then F ∼ E (λ) for some λ > 0 if and only if there exist integers r and n, 1 ≤ r ≤ n − 1, such that E [D (r, n, m, k)] = E [D (r + 1, n, m, k)] . In the following theorem, Kamps and Gather, (1997) show that under an NBU or NWU assumption, characterizations of exponential distribution based on the equality of the expectations of Xs,n − Xr,n and Xs−r,n−r established by Iwi´nska (1986) and Gajek and Gather (1989), can also be extended to generalized order statistics in the special case of s = r + 1.

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Theorem 5.3.4. Let F be absolutely continuous with pd f f , F (0) = 0, and suppose that F is strictly increasing on (0, ∞), and either NBU or NWU. Moreover, let the expected values involved be finite. Then F ∼ E (λ) for some λ > 0 if and only if there exist integers r and n , 1 ≤ r ≤ n − 1, such that E [X (r + 1, n, m, k)] − E [X (r, n, m, k)] = E [X (1, n − r, m, k)]. The following remark is also due to Kamps and Gather (1997) involving a counterexample of Kamps (1995, p. 128). Remark 5.3.5. Without any further assumption, the equation E [X (r + 1, n, m, k)] − E [X (r, n, m, k)] = E [X (1, n − r, m, k)] for just one pair (r, n), 1 ≤ r ≤ n − 1, does not characterize exponential distribution. For every choice of r, n and m 6= −1 there are distributions different from exponential with the above property. For example, the distributions given by (  −1/(m+1) c > 0, x ∈ (0, ∞) , m > −1,   F (x) = 1 − 1 + cxd c < 0, x ∈ 0, (−1/c)1/d , m < −1, with

k + (n − 1) (m + 1) d= k + (n − 2) (m + 1)



γ1 = γ2



satisfying the moment condition for r = 1. The definition of the generalized order statistics given by Kamps in (1995) and in Kamps and Gather (1997) (also given before in this section) involved a single real number m. Kamps and Cramer (1999) extended this definition slightly by replacing m with a vector m e = (m1, m2, . . ., mn−1) in Rn−1 as given below. Then a characterization of exponential distribution based on this extended version of the generalized order statistics was presented by Kamps and Cramer, which is given in Theorem 5.3.6 below. Again let, for simplicity, F denote an absolutely continuous cd f with corresponding pd f f . The random variables

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X (1, n, m, e k) , X (2, n, m, e k) , . . ., X (n, n, m, e k) are called generalized statistics based on F, if their joint pd f is given by

order

e m,k),...,X(n,n, e m,k) e (x1 , x2, . . ., xn ) f X(1,n,m,k),X(2,n, ! ! n−1 n−1
m
k−1 = k ∏γj ∏ F (xi ) i f (xi ) F (xn ) f (xn ) , j=1

i=1

on the cone F −1 (0) < x1 ≤ x2 ≤ · · · ≤ xn < F −1 (1−) of R, with parameters n ∈ N, n ≥ 2, k > 0, m e = (m1, m2 , . . ., mn−1) ∈ Rn−1, Mr = ∑n−1 j=r m j , such that γr = k + n − r + Mr > 0 for all r ∈ {1, 2, . . ., mn−1} (cf. Kamps, 1995, 1998a). Moreover, let cr−1 = ∏rj=1 γ j , r = 1, 2, . . ., n − 1, and γn = k. Kamps and Cramer (1999) pointed out that in the context of ordinary order statistics X1,n , X2,n , . . ., Xn,n based on F it is well-known (cf. Ahsanullah 1984; Iwi´nska 1986; Gajek and Gather 1989; Gather et al. 1998, p. 266/7, Kamps 1998a; see also Gather 1988) that, under NBU or NWU assumption Xs,n −Xr,n ∼ Xs−r,n−r , r < s as well as the equality E [Xs,n ]−E [Xr,n ] = E [Xs−r,n−r] are characteristic properties of exponential distribution. We would like to mention that except Kamps (1998a) result, the rest of the results mentioned in the parentheses above have already been mentioned in this chapter. In what follows (Theorem 5.3.6), Kamps and Cramer (1999) extend the above mentioned results to generalized order statistics restricting on pairwise different parameters γ1, γ2, . . ., γn. For the case m1 = m2 = · · · = mn−1 see Theorems 5.3.1, 5.3.3 and 5.3.4 above and for more details see Kamps (1998b). Theorem 5.3.6. Let the appearing generalized order statistics be welldefined and be based on an absolutely continuous cd f F. Let F be strictly increasing on (0, ∞) with F (0) = 0, and let F be NBU or NWB. Then F ∼ E (λ) for some λ > 0 if and only if there exist integers r, s and n, 1 ≤ r < s ≤ n, such that i) X (s, n, m, e k) − X (r, n, m, e k) ∼ X (s − r, n − r,e µ, k), or

ii)

E [X (s, n, m, e k)] − E [X (r, n, m, e k)] = E [X (s − r, n − r,e µ, k)] , assuming that the expected values are finite,

where e µ = (µ1, µ2, . . ., µn−r−1) = (mr+1, mr+2 , . . ., mn−1) .

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Keseling (1999) considered, among other things, X (i, n, 0, 1), 1 ≤ i ≤ n, based on a continuous cd f F and presented the following characterization of the exponential distribution. Theorem 5.3.7. F ∼ E (1) if and only if E [X (r + 1, n, 0, 1)|X (r, n, 0, 1) = x] = x + b

a.s.

Remark 5.3.8. As we mentioned in Section 5.2, Rao and Shanbhag (1994) proved that exponential distribution is the only continuous distribution with constant regression E [φ (Xr+1,n − Xr,n )|Xr,n = x] with φ nonarithmetic and monotonic and E [|φ (Xr+1,n − Xr,n ) |] < ∞. Rao and Shanbhag (1986) presented similar result for record values. In the following corollary, Keseling (1999) showed the same result for gos. Corollary 5.3.9. Let X (1, n, m, e k), . . ., X (n, n, m, e k) be a sequence of gos based on a continuous cd f F. Further, let r ∈ {1, 2, . . ., n − 1} be fixed and φ a nonarithmetic real monotonic function on R+ such that E [|φ(X (r + 1, n, m, e k) − X (r, n, m e , k))|] is finite. Then, for some constant c 6= φ (0+ ) , E [φ(X (r + 1, n, m, e k) − X (r, n, m, e k))|X (r, n, m, e k) = x] = c a.s. if and only if there exists λ > 0 and µ ∈ R with F (x) = 1 − exp (−λ (x − µ)) , x ≥ µ. Remark 5.3.10. It is known that within a suitable class of continuous distributions, the exponential distribution is the only distribution for which Var [Xi+1,n|Xi,n = x] is constant. For gos Keseling (1999) presented the following corollary. Corollary 5.3.11. Let X (1, n, m, e k), . . ., X (n, n, m), e n ≥ 2, be a sequence 0 of gos 0s based on the differentiable cd f F with supp(F) = (0, ∞) , F > 0 on  supp(F) and for some r ∈ {1, 2, . . ., n − 1} , E X 2 < ∞ for a rv X ∼ 1 − (1 − F)γr+1/(n−r) . Then Var [X (r + 1, n, m, e k)|X (r, n, m e , k) = x] = c > 0 a.s.

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if and only if there exists λ > 0 with F (x) = 1 − exp (−λx) , x ≥ 0. Remark 5.3.12. In view of the results reported in this section, the similarities in some characterization results based on order statistics and based on record values (which will be discussed in details in Chapter 6) are no longer astonishing. It clearly shows that the concept of generalized order statistics presents a unified approach to characterizations of distributions and well-known characterization results based on order statistics and record values can be deduced as special cases of gos. Bieniek and Szynal (2003) presented various characterizations of distributions via linearity of regression of gos extending results of Dembi´nska and Wesolowski (1998, 2000). We state here their characterization of the exponential distribution. We need to define lF = inf {x ∈ R : F (x) > 0} and rF = sup {x ∈ R : F (x) < 1} for the following theorem. Theorem 5.3.13. Let X (i, n, m, e k), 1 ≤ i ≤ n, be gos based on an absolutely continuous distribution function F , and let γi 6= γ j ,

for all

i 6= j, 1 ≤ i, j ≤ n.

Suppose that E [|X (r + l, n, m, e k)|] < ∞, where r + l ≤ n. If the following linearity of regression holds E [X (r + l, n, m, e k) |X (r, n, m, e k) = x] = x + b,

x ∈ (lF , rF ) ,

for some b ∈ R, then F ∼ E (µ, λ), where µ ∈ R and λ > 0 is determined by λ = 1b ∑r+l i=r+1 . For further studies regarding the characterizations of distributions (in particular the exponential distribution) via linear regression of gos0 s, we refer the interested reader to an excellent paper by Cramer, Kamps and Keseling (2004). Ahsanullah (2006) presented the following characterization of exponential distribution based on the generalized order statistics, in different direction than the ones mentioned earlier. Theorem 5.3.14. Let X be a non-negative random variable having an absolutely continuous (with respect to Lebesgue measure) cd f F with F (0) = 0 and 0 < F (x) < 1 for all x > 0. Then the following properties are equivalent:

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a) X has an exponential distribution with pd f , f (x) = σ−1 e−σ x ;
b) X (r + 1, n, m, d) ∼ X (r, n, m, k)+ σ γW , r > 1, where W ∼ E σ−1 is inr+1 dependent of X (r + 1, n, m, k) and X (r, n, m, k) and m ≥ −1. −1

For k = 1 and m = 0, a characterization of exponential distribution via relaW tion Xr+1,n ∼ Xr,n + σ n−r is obtained. We will now present some new characterizations of the exponential distribution based on generalized order statistics. Theorem 5.3.15. Let X be a non-negative rv with an absolutely continuous (with respect to Lebesgue measure) and strictly increasing cd f F with F (0) = 0 and F (x) < 1 for all x > 0. Then the following properties are equivalent: (a) X ∼ E (λ); (b) For 1 < r ≤ n, the rv0 s (X (r, n, m, k)− (X (r − 1, n, m, k))) and X (r − 1, n, m, k) are independent. Proof. Let U = X (r − 1, n, m, k) and V = (X (r, n, m, k)− X (r − 1, n, m, k)). We derive the joint pd f of U and V as  c m r−2 r−2   (r−2)! (1 − F (u)) gm (F (u)) fU,V (u, v) = × [1 − F (u + v)]γr −1 f (u) f (u + v) , 0 < u, v < ∞, (5.3.1)   0, otherwise. If X ∼ E (λ), then 1 − F (x) = exp (−λx) and (
1 1 − e−(m+1)λx , for m 6= −1, m+1 gm (F (x)) = λx, for m = −1. Upon simplification, we arrive at fU,V (u, v) =

 λ2 cr−2 r−2  gm 1 − e−λu e−γr−1 λu e−γr λv (r − 2)!

(5.3.2)

Thus U and V are independent. Now, let fU (u) be the pd f of U, then fU (u) =

cr−2 (1 − F (u))−1+γr−1 gr−2 m (F (u)) f (u) . (r − 2)!

(5.3.3)

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Using (5.3.2), (5.3.3) and the relation γr−1 = γr + m + 1, we obtain the conditional pd f of V given U = u as h
−1 iγr −1
−1 f (u + v) F (u) , (5.3.4) fV |U (v|u) = F (u + v) F (u) for all v, 0 < v < ∞ and all u. Integrating the expression in (5.3.4) with respect to v from v1 to ∞, we obtain h
−1iγr F V |U (v1 |U = u) = F (u + v1 ) F (u) , (5.3.5) for all v1 and u, 0 < u, v1 < ∞. Since U and V are independent, we obtain from (5.3.5) for all u and v h
−1 iγr F (u + v) F (u) = G (v), where G (v) is clearly a
function of v only. Now, upon taking limit as u → 0, we γ arrive at G (v) = F (v) r . Hence for all u and v, 0 < u, v < ∞ and a fixed γr , we have F (u + v) = F (u)F (v) . (5.3.6) The solution of (5.3.6) is (see Acz´el (1966)) F (x) = e−λx , where λ is arbitrary. Since F (x) = 1 − F (x) is a cd f , λ must be positive, which complete the proof of the theorem. Remarks 5.3.16. (i) If k = 1 and m = 0, then from Theorem 5.3.15 we obtain the result of Rossberg (1972) . If k = 1 and m = −1, then we obtain, from Theorem 5.3.15, the characterization of the exponential distribution of Ahsanul
lah (1978) based on the independence of XU(n) − XU(n−1) and XU(n−1), where XU(k) is defined in Chapter 3. (ii) It is clear that for exponential distribution E (λ), E[X ( j + 1, n, m, k)|X ( j, n, n, k) = x] is a linear function of x. It can be shown that this property characterizes the exponential distribution. Proof of the following theorem is similar to that of Theorem 5.3.15 and hence will be omitted. Theorem 5.3.17. Let X be as in Theorem 5.3.15. Then the following properties are equivalent.

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(a) X ∼ E (λ). (b) For 1 < j ≤ n, the rv0 s γ j (X ( j, n, m, k) − X ( j − 1, n, m, k)) and γ1X (1, n, m, k) are identically distributed. Remarks 5.3.18. (i) For k = 1, m = 0, Theorem 5.3.17 gives a characterization of the exponential distribution based on the identical distribution of (n − j + 1) (X j,n − X j−1,n ) and nX1,n (Ahsanullah, 1977) . (ii) For k = 1, m = −1, Theorem 5.3.17 gives a characterization of the exponential distribution based on the identical distribution of XU( j) −XU( j−1) and X (Ahsanullah, 1978). (iii) For k = 1, m = 0, Theorem 5.3.17 gives a characterization of the exponential distribution based on the identical distribution of (n − j) (X j+1,n − X j,n ) and X. The following two theorems are based on the monotonicity of the hazard rate. Theorem 5.3.19. Let X be a non-negative rv with an absolutely continuous (with respect to Lebesgue measure) and strictly increasing cd f F for all x > 0 and F (x) < 1 for all x > 0. Then the following properties are equivalent. (a) X ∼ E (λ) . (b) X has a monotone hazard rate and for one n , one j (1 < j ≤ n), one m and one k, the rv 0 s {k + (n − j) (m + 1)}(X ( j, n, m, k)− X ( j − 1, n, m, k)) and X are identically distributed. Proof of this theorem is based on the following lemma. Lemma 5.3.20. Let X be as in Theorem 5.3.19 with corresponding pd f f . If the hazard rate is monotone and if Z ∞
−1+γ j−1 j−2 c j−2 F (u) gm (F (u)) H (u, v) du = 0, 0 ( j − 2)! for all v, one j, one m, and one γ j−1, where

 γ j   F u + γvj  , H (u, v) = F (v) −  F (u)

then H (0, v) = 0 for all v > 0.

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Proof. We have H (u, 0) = 0 = H (u, ∞) and     ∂H (u, v) v = q (u, v) r u + − r (u) , ∂u γj "   #γ j

where r (u) is the hazard rate and q (u, v) =

F u+ γv

j

F(u)

.

= 0, for all u and v. (i) If r (x) = constant for all x, then ∂H(u,v) ∂u (ii) If r (x) is strictly monotone increasing in x, then H (u, v) is strictly increasing in u for any fixed v. Thus for all v, if H (u, v) is not identically zero, then it must be negative in an interval containing zero. Let H (u, v) < 0 for u ∈ I = [0, b], where b is a real number. Then for u ∈ I,  γ j   v   F u + γj ∂H (u, v) v   = − f (v) + r u+ ∂v γj F (u)   v > − f (v) + F (v) r u + , for u ∈ I, since H (u, v) < 0, γj     γj v u. − r (v) < 0, for all u ∈ I and all v > > F (v) r u + γj γj −1 Thus for all u ∈ I, H (u, v) decreases to 0 as v → ∞. But for all u ∈ I, H (u, 0) = 0, H (u, v) < 0 and H (u, v) decreases to 0 as v → ∞. Therefore, by continuity of H (u, v) we must have H (u, v) = 0 for all v > 0 and all u ∈ I. Hence H (0, v) = 0 for all v > 0. (iii) If r (x) is strictly monotone decreasing in x, similar argument can be used. Proof of Theorem 5.3.19 can now easily be completed, which we omit it. Remarks 5.3.21. (i) The condition of monotonicity of hazard rate in Theorem 5.3.19 can be replaced by NBU/NWU property. (ii) Kamps and Gather (1997) presented a proof of Theorem 5.3.19 under the equality of the expectations instead of identical distribution. The proof of the following theorem is similar to that of Theorem 5.3.19 and hence will be omitted.

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Theorem 5.3.22. Let X be as in Theorem 5.3.19. Then the following are equivalent. (a) X ∼ E (λ). (b) X has a monotone hazard rate and for one n , one j (1 < j ≤ n), one m and one k (k ≥ 1, m is a real number) the rv0 s {k + (n − j) (m + 1)} (X ( j, n, m, k) − X ( j − 1, n, m, k)) and {k + (n − j − 1) (m + 1)}(X ( j + 1, n, m, k) − X ( j, n, m, k)) are identically distributed.

Chapter 6

Characterizations of Exponential Distribution II 6.1. Characterizations Based on Record Values The problem of characterizing exponential distribution based on record values, as far as we have gathered, started in late sixties by Tata (1969) and followed in seventies by Nagaraja (1977), Sirvastava (1978) and Ahsanullah (1978, 1979). The goal of this chapter is to review characterization results related to the exponential distribution based on record values. As in Chapter 5, we will discuss these results in the chronological order rather than their importance. We will be using the same notation employed in Chapter 5 throughout this chapter whenever needed. We apologize if we missed to report some pertinent results. To refresh the reader’s memory we recall the definition of record values from Chapter 3. Let X1, X2, . . . be a sequence of i.i.d. random variables with cd f F and corresponding pd f f . Set Yn = max {X1 , X2, . . ., Xn} , for n ≥ 1. We say that X j is an upper record value of (Xn)n≥1 if X j > X j−1 . Lower record value is similarly defined. In what follows, unless otherwise stated, by record values we mean upper record values. By definition X1 is a record value. The indices at which the record values occur are given by the n ≥ 0}, where U (0) = 1, and U (n)
=  record value times {U (n), min j| j > U (n − 1) , X j > XU(n−1) . We will denote R (x) = − ln F (x) and
−1 , for F (x) > 0. r (x) is called the hazard rate. We say that r (x) = f (x) F (x)

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F belongs to the class H ∗ if r (x) is either monotone increasing or decreasing. Tata (1969) presented a characterization of the exponential distribution based on the independence of the random variables XU(0) and XU(1) − XU(0) as follows . Theorem 6.1.1. If F is absolutely continuous, then F is exponential if and only if XU(0) and XU(1) − XU(0) are independent. Gupta (1984) pointed out that the following characterizations have appeared in literature for the i.i.d. case. (1). The independence of XU( j+1) − XU( j) and XU( j) characterizes the exponential distribution, Sirvastava 
(1978),  Ahsanullah (1979) and Pfeifer (1982). (2). E XU( j+1) − XU( j) |XU( j) is independent of XU( j) characterizes the exponential distribution, Sirvastava (1978), Ahsanullah (1978) and Nagaraja (1977).   (3). Var (XU( j+1) − XU( j) )|XU( j) is independent of XU( j) characterizes the exponential distribution, Ahsanullah (1981). We will get back to Gupta (1984), but for now we would like to mention that Ahsanullah (1979) presented two characterizations of exponential distribution based on record values generalizing Tata’s (1969) result as well as other characterizations reported in this direction. Here are Ahsanullah’s (1979) results stated in Theorems 6.1.2 and 6.1.4 below. Theorem 6.1.2. Let (Xn)n≥1 be a sequence of i.i.d. random variables having an absolutely continuous (with respect to Lebesgue measure) cd f F such that α = inf{x|F (x) > 0} = 0 and f (x) > 0 for α = 0 < x < ∞. Then for Xn ∼ E (λ), it is necessary and sufficient that X U(n−1) and XU(n) − XU(n−1) are independent. Remark 6.1.3. For n = 1, the above theorem reduces to Theorem 6.1.1. Theorem 6.1.4. Under the assumptions of Theorem 6.1.2, the following two statements are equivalent: (a) X1 ∼ E (λ).
(b) for 0 ≤ m < n − 1, the conditional distributions of XU(n) − XU(n−1)
given XU(m) and XU(m+1) − XU(m) given XU(m) are i.d. and X1 ∈ H ∗.

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Nayak (1981) also presented a generalization of Tata’s result by showing that for an absolutely continuous distribution the independence of XU(r) and XU(n) − XU(s) for some 0 ≤ r < s < n is already sufficient to characterize the exponential distribution. This result was independently proved by Dallas (1981) whose method of proof can be used to characterize other distributions. Ahsanullah (1981), which was referred to in (3) above, as well as Pfeifer (1982) employed the independence of certain functions of record values (or record increments) to established further characterizations of the exponential distribution. The similarity between characterizations of exponential distribution based on order statistics and based on record values motivated Gupta (1984) to investigate the relationship between record values and order statistics. The same problem was taken up by Deheuvels (1984) independently. Gupta showed that the conditional distributions of Xi+1,i+1 − Xi,i+1 given Xi,i+1 and XU(i+1) − XU(i) given XU(i) are the same. Gupta also investigated a relation between conditional distributions of X j+1,n given X j,n and XU( j) given XU( j−1) . For different set of conditions see Deheuvels (1984). The following two results are due to Gupta (1984). He proved a general theorem from which all the results mentioned in (1) − (3) above follow as special cases.
r   Theorem 6.1.5. E XU( j+1) − XU( j) |XU( j) = y = c (independent of y) for fixed j, r ≥ 1 if and only if F is exponential. Remarks 6.1.6. (a) Gupta (1984) pointed out that the proof of Theorem 6.1.5 requires only the continuity of F. (b) Theorem 6.1.5 was extended by Rao and Shanbhag (1986), where they obtained the same
r characterization if, in Gupta’s
condition, the expression XU( j+1) − XU( j) is replaced by G XU( j+1) − XU( j) , where G is a monotone function satisfying certain conditions.   Theorem 6.1.7. E XU( j+1) − XU( j) = E [X1 ], for one fixed j, characterizes the exponential distribution in the class of NBU or NWU distributions. Remark 6.1.8. Gupta (1984) pointed given in the  out that the assertion  above theorem cannot be replaced by ‘E XU( j+1) − XU( j) for a fixed j determines the distribution’. He provided a counter-example in his Remark 2 of the paper.

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Deheuvels (1984) pointed out, as did Gupta (1984), that a great deal of what has been achieved in the characterization of exponential distribution based on order statistics can be expressed equivalently based on record values and vice versa. Then he presented the following result, Theorem 6.1.9 below. For a fixed j ≥ 1, Deheuvels (1984) defines the jth record times by o n ( j) ( j) ( j) n1 = j, nk = min m > nk−1 |Xm− j+1,m > Xn( j) − j+1,n( j) , k = 2, 3, . . ., k−1

k−1

and the jth record values by ( j)

Rk = Xn( j)− j+1,n( j) . k

k

Theorem 6.1.9. Let (Xn )n≥1 be a sequence of i.i.d. random variables with ( j)

( j)

a continuous cd f F. For a fixed j ≥ 1, let R1 , R2 , . . . be the corresponding ( j) sequence of jth record values defined above. Then, if k ≥ 1 is a given integer, R k ( j) ( j) and Rk+1 − Rk are independent if and only if F (x) = 1 − exp {−b (x − B)} , x ≥ B, for some finite constants b(b > 0) and B. Remark 6.1.10. As pointed out by Deheuvels (1984), the preceding result gives a simple proof of a theorem given in Ahsanullah (1978) for the case j = 1. Another extension, again due to Deheuvels (1984), is the following theorem. Theorem 6.1.11. With the assumptions of Theorem 6.1.9, if it is assumed further that F (x) is absolutely continuous with respect to Lebesgue measure and that g (x, y) > 0 is a Borel-measurable function such that Z ∞Z ∞ −∞ −∞

g (x, y) dF (x) dF (y) < ∞,

  ( j) ( j) then, if for some j and k ≥ 1, and g Rk , Rk+1 are independent, there exists a constant c such that, for all z , ( j) Rk

Z ∞

g (z, x)dF (x) = cF (z) .

z

Iwi´nska (1985) presented a characterization of the exponential distribution by a distributional property of the difference of two arbitrary record values. She

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also characterized the exponential distribution based on the conditional distributions of the difference of two, not necessarily consecutive, record values. Then in 1986, Iwi´nska established a characterization of the exponential distribution based on the expectation of spacings between two, not necessarily consecutive, record values.  Nagaraja (1988) lets Yj, j ≥ 1 be the sequence of (upper) record values from a sequence of i.i.d. random variables (X j ) j≥1 having a continuous cd f F, for which E [Ym+1 ] is finite. He pointed out

that this (expectation) condition holds if E [X1] and  m are finite (see, Nagaraja, 1978). Nagaraja then estabE X1+ ln X1+ lished the following result. Theorem 6.1.12. E [Ym+1 |Ym] and E [Ym|Ym+1] are both linear in the conditioning random variable for some m if and only if F is an exponential (type) cd f . (A dual result holds for lower record values ). Using concepts of NBU (NWU) and IHR (DHR) Ahsanullah presented certain characterizations of the exponential distribution based on the record values, see Theorems 6.1.13, 6.1.15 and 6.1.16 below. Theorem 6.1.13. Let (X j ) j≥1 be a sequence of i.i.d. non-negative random variables with an absolutely continuous (with respect to Lebesgue measure) cd f F and corresponding pd f f . If F is in the class of NBU or NWU and for some m, m ≥ 2, XU(m) ∼ XU(m−1) + U, where U is independent of XU(m) and XU(m−1) and U ∼ X1 , then X ∼ E (λ), for some λ > 0. Remarks 6.1.14. (a) Theorem 6.1.12 can be used to obtain the following known results pertinent to two parameter exponential distribution ( F (x) = e−λ(x−µ))   E XU(m) = µ + mλ−1 ,   Var XU(m) = mλ−2 , and

  Cov XU(m) , XU(n) = mλ−2 , m < n.

104

M. Ahsanullah and G.G. Hamedani (b) Using Theorem 6.1.13, it can be shown that for the exponential distribu-

tion XU(m) ∼ U1 +U2 + · · · +Um , where U1 ,U2, . . . are i.i.d. exponential. This property is also a characteristic property of the exponential distribution. The following theorem is a generalization of Ahsanullah’s (1987) result. Theorem 6.1.15. Let (X j ) j≥1 be a sequence of i.i.d. non-negative random variables with absolutely continuous (with respect to Lebesgue measure) cd f F and the corresponding pd f f . If F is in the class ofIHR or DHR, and for some m, n with 1 ≤ m < n, E[XU(n) − XU(m) ] = E XU(n−m) , then X1 ∼ E (λ) , for some λ > 0. Theorem 6.1.16. Let (X j ) j≥1 be a sequence of i.i.d. non-negative random variables with an absolutely continuous (with respect to Lebesgue measure) cd f F and corresponding pd f f . Let m be a geometric random variable with parameter p, independent of X 0j s . Then the following two properties are equivalent: (i) X 0j s are exponentially distributed. (ii) p ∑mj=1 X j ∼ XU(n) − XU(n−1), for some fixed n, n ≥ 1, F is in the class of NBU or NWU and E [X1 ] < ∞. Haung and Li (1993) investigated some extensions of various results given in Ahsanullah (1978,1979), Dallas (1981), Gupta (1984) characterizing the exponential distribution based on record values. The following three theorems are due to Haung and Li (1993). Theorem 6.1.17. Assume F has pd f f . Let G be a non-decreasing function such that for every x > 0, G has a point of increase in (0, x) . Assume for some fixed integer j ≥ 1,

    E G XU( j) − XU( j−1) |XU( j) = x = E G XU(0) |XU( j) = x , for all x > 0. Then X1 has an exponential distribution.

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In the following theorem, Haung and Li (1993) define XU(−1) = 0 for convenience. Theorem 6.1.18. Let F and G be as in Theorem 6.1.17. Also, assume f is continuous, F (x) > 0 and G (0) = 0. If for some fixed integers 1 ≤ i ≤ j, 
 E G XU(i) − XU(i−1) |XU( j) = x  
= E G XU(i−1) − XU(i−2) |XU( j) = x , for all x > 0, then X1 has an exponential distribution. The following theorem is a generalization of Gupta’s (1984) and Rao and Shanbhag’s (1986) results where Haung and Li (1993) will consider the difference of any two adjacent record values after XU( j) instead of the difference of XU( j+1) and XU( j) . Theorem 6.1.19. Assume that F has pd f f and F (x) > 0 for x > 0. Let G be a non-decreasing function having non-lattice support on x > 0 with G (0) = 0 and E [G (X1 )] < ∞. If, for some fixed non-negative integers j and k ,
  E G XU( j+k+1) − XU( j+k) |XU( j) = x = c, for every x > 0, where c > 0 is a constant, and if for some ξ > 0, c<

Z ∞ 0

e−ξx dG (x) < ∞,

then c = E [G (X1)] and X1 is exponentially distributed. Remark 6.1.20. a sequence of populations and sequences of   Considering (n) (stemming from the nth population), Witte (1990, random variables Xi i≥1

1993) characterized the exponential distribution based on the equidistribution (m) of XU(n) − XU(n−1) and X1 . Their results are very interesting, but they are not in the same directions as the ones reported in this section so far. Franco and Ruiz (1996) studied characterization of continuous distributions by conditional expectation of record values including exponential distribution.

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Grudzie´n and Szynal (1996) characterized the exponential distribution in terms of record statistics with random index. In their (1997) paper, they mentioned that Lin and Too (1989) characterized the exponential distribution via moments of record values. From Lin and Too’s result it follows that the exponential distribution can be characterized by an equality involving two moments of record values. Grudzie´n and Szynal (1997) characterized the exponential distribution by moments of the kth record values extending the above mentioned result due to Lin and Too. n and Szynal (1997), for a fixed integer k ≥ 1, define the sequence   Grudzie´ (k) Yn of the kth record values as follows: n≥1

(k)

Yn = XUk (n),Lk (n)+k−1,

n = 1, 2, . . .,

where the sequence (Uk (n))n≥1 of the kth record times is given by Uk (1) = 1,  Uk (n + 1) = min j| j > Uk (n), X j, j+k−1 > XUk (n),Lk (n)+k−1 , n = 1, 2, . . .  
(1) Note that for k = 1 the sequence Yn is the sequence XU(n) n≥1 of n≥1

record values defined in the beginning of this chapter. Here are their results stated in Theorems 6.1.21 and 6.1.22 below. i h Theorem 6.1.21. Assume E |min{X1 , X2, . . ., Xk }|2p < ∞ for a fixed integer k ≥ 1 and some p > 1. Suppose that N is a positive integer-valued random variable independent of (Xn)n≥1. Then F ∼ E (1) if and only if  h  i (k) 2 (k) − 2k−1 E NYN+1 + k−2 E [N (N + 1)] = 0, E YN   provided that E N 2 < ∞. i h Theorem 6.1.22. Assume E |min{X1 , X2, . . ., Xk }|2p < ∞ for a fixed integer k ≥ 1 and some p > 1. Then F ∼ E (1) if and only if    2 h i 2 (k) 2 (k) − E Y2 + 2 = 0, E Y1 k k

Characterizations of Exponential Distribution II 107      i h (k) 2 (k) proving that each set E Y1 , k ≥ 1, characterizes the expo, E Y2 nential distribution. For k = 1, Theorem 6.1.22 reduces to the following corollary.   Corollary 6.1.23. Assume E |X1|2p < ∞ for some p > 1. Then F ∼ E (1) if and only if E

h

XL(n)


2 i

−

  2n E XL(n+1) + n (n + 1) = 0. (n − 1)!

Remark 6.1.24. Theorems 6.1.21 and 6.1.22 as well as Corollary 6.1.23 are special cases of the characterization results for more general distributions than exponential distribution. We reduced the original theorems to the exponential case to be consistent with the theme of this chapter and the book as a whole. The interested readers can see the general results in Grudzie´n and Szynal (1997). Lo´ pez-Bl´aquez and Moreno-Rebollo (1997) presented certain characterizations of distributions based on linear regression of record values. One of these distributions is exponential distribution given in Theorem 6.25 below. Their result on exponential distribution extends Nagaraja’s (1988) result in this direction. Theorem 6.1.25. If F is k times differentiable, and for   certain non-negative integer, i, the conditional expectations E XU(i+k)|XU(i) and E XU(i) |XU(i+k) are both linear, then, except for location and scale parameters, F is the cd f of an exponential distribution. The following results (Theorems 6.1.26, 6.1.27 and 6.1.30 and Corollary 6.1.29) due to Rao and Shanbhag (1998), extend the results in Dallas (1981), Gupta (1984), Rao and Shanbahg (1986, 1994), Witte (1988) and Huang and Li (1993) all of which, except Rao and Shanbhag (1994), were mentioned before in this chapter. Theorem 6.1.26. For some k ≥ 1, XU(k+1) − XU(k) ∼ X1 if and only if X 1 has an exponential distribution.

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As a corollary (see Corollary 6.1.29 below) to their following theorem, Rao and Shanbhag (1998) establish a characterization of the exponential distribution. Theorem 6.1.27. Let k be a positive integer and φ be a nonconstant
real  monotonic left continuous function on R+ such that E φ XU(k+1) − XU(k) < ∞ and Fk (x + ·) = Fk ((y + ·) −) a.e. [|φ (·+) − φ (0+)|] whenever 0 < Fk (x) = Fk (y−) < Fk (y) , where Fk is the cd f of XU(k). Then, for some c 6= φ (0+) ,
  E φ XU(k+1) − XU(k) |XU(k) = c a.s. if and only if the left extremity, l k , of Fk is finite, and either φ is nonarithmetic and conditional distribution of X 1 − lk given that X1 > lk is exponential, or for some λ > 0, φ is arithmetic with span λ and for some β ∈ (0, 1) P {X1 − lk ≥ x + nλ} = βnP {X1 − lk ≥ x} , x > 0; n = 0, 1, . . ., where X1 is a random variable with cd f F. Remark 6.1.28. When F is continuous, Theorem 6.1.27 holds without the left continuity assumption of φ. Corollary 6.1.29 (Rao and Shanbhag, 1998). Let the assumptions in Theorem 6.1.27 be met. Then the following assertion holds: If F is continuous or has its left extremity as one of its continuity
pointsand φ is nonarithmetic, then, for some c 6= φ (0+), E φ XU(k+1) − XU(k) |XU(k) = c a.s. holds if and only if F is exponential, with a shift. Theorem 6.1.30. Let F be continuous and k 2 > k1 ≥ 1 be fixed integers. Then, on some interval of the type (−∞, a) , with a > the left extremity of the distribution of X U(k1 ) , the conditional distribution of X U(k2) − XU(k1) given XU(k1 ) = x is independent of x for almost all x if and only if F is exponential, with a shift.

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109

For the next characterization of exponential distribution we need a couple of new notation. Define two sequences of random variables (ξn )n≥1 and (ηn )n≥1 by ( 1 if Xn+1 < ∑ni=1 Xi , n = 1, 2, . . ., ξn = 0 otherwise and ηn =

(

1 if XL(n)+1 < XL(n) , 0 if XL(n)+1 ≥ XL(n) ,

n = 1, 2, . . . .

The following two Theorems are due to Bairamov (2000). Theorem 6.1.31. Let X1 be a non-negative random variable having continuous cd f F satisfying inf {x|F (x) > 0} = 0. Then the following statements are equivalent: (a) X1 has an exponential distribution. (b) For some n > 1 E [ξn ] = E [ηn ] and F is either NBU or NWU. Theorem 6.1.32. Let F be absolutely continuous satisfying inf {x|F (x) > 0} = 0. Then the following properties are equivalent: (a) X1 has an exponential distribution. (b) For some n > 1 n

∑ Xi ∼ XL(n) ,

i=1

and F is either NBU or NWU. The following two characterizations of the exponential distribution, due to Lee (2001), are given in terms of conditional expectations of record values improving similar characterizations mentioned before. Theorem 6.1.33. If F is absolutely continuous with F (x) < 1 for all x, then   E XU(n+1) − XU(n)|XU(m) = y = c, c > 0, n ≥ m + 1, if and only if F is exponential.

110

M. Ahsanullah and G.G. Hamedani Theorem 6.1.34. If F is absolutely continuous with F (x) < 1 for all x, Then   E XU(n+2) − XU(n)|XU(m) = y = 2c, c > 0, n ≥ m + 1,

if and only if F is exponential. Lee et al. (2002) reported two more characterizations of the exponential distribution similar to Theorems 6.1.33 and 6.1.34. The new theorems and the above two theorems can be combined in the following differently worded statement. Theorem 6.1.35. Let F be an absolutely continuous cd f with F (x) < 1 for all x. Then F is exponential if and only if for some n and m , m ≤ n − 1 and some integer i, 1 ≤ i ≤ 4   E XU(n+i) − XU(n)|XU(m) = y = ic, c > 0. The following eight characterizations of the exponential distribution have appeared in Ahsanullah’s book “Record values-Theory and Applications” ; University Press of America Inc. (2004). We present them here for the sake of completeness. The first theorem below is a generalization of Theorem 6.1.2. Theorem 6.1.36. Let (Xn)n≥1 be i.i.d. rv0 s with an absolutely continuous cd f F with F (0) = 0 and F (x) < 1 for all x > 0. Then for X1 ∼ E (λ) , it is necessary and sufficient that XU(n) − XU(m) (0 < m < n) and XU(m) are independent. Proof. The necessary condition is easy to establish. To prove the sufficiency condition, we need the following lemma. Lemma 6.1.37. Let F be an absolutely continuous cd f and F (x) > 0,
−1 = exp {−q (u, v)} and h (u, v) = for all x > 0. Suppose that F (u + v) F (v) r ∂ ∂ q (u, v) 6= 0 for {q (u, v)} × exp {−q (u, v)} ∂u q (u, v), for r ≥ 0, h (u, v) 6= 0, ∂u any positive u and v. If h (u, v) is independent of v, then q (u, v) is a function of u only.

Characterizations of Exponential Distribution II

111

Proof of Lemma. Let g (u) = h (u, v) = (q (u, v))r exp {−q (u, v)} ∞

=

∂ q (u, v) ∂u

(−1) j ∂ ∑ Γ ( j + 1) {q (u, v)}r+1 ∂u q (u, v) j=0 ∞

=

1 ∂ (−1) j ∑ Γ ( j + 1) (r + j + 1) ∂u q (u, v) . j=0

Hence ∞

(−1) j 1 ∑ Γ ( j + 1) {q (u, v)}r+ j+1 (r + j + 1) = c + j=0

Z

g (u)du =: g1 (u), say.

(6.1.1) Here g1 (u) is a function of u only and c is independent of u but may depend on v. Now, letting u → 0+ , we see that q (u, v) → 0 and hence from (6.1.1), we have c as independent of v. Therefore, 0=

∂ g1 (v) = ∂v

∞

(−1) j ∂ ∑ Γ ( j + 1) {q (u, v)}r+ j ∂u q (u, v) j=0  −1 ∂ ∂ . q (u, v) = g (u) q (u, v) ∂v ∂u

Now, we have g (u) = h (u, v) 6= 0 and ∂ ∂v q (u, v)= 0.

∂ ∂u q (u, v)

6= 0, then we must have

Now, we prove the sufficiency condition of Theorem 6.1.36. The joint pd f of Zn,m = XU(n) − XU(m) and XU(m) is f (z, u) =

Rm−1 (x) r (x) {R (z + x) − R (x)}n−m−1 f (z + x) , Γ (m) Γ (n − m) for 0 < z < ∞, 0 < x < ∞,

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M. Ahsanullah and G.G. Hamedani
where R (x) , r (x) and f (x) are − ln F (x) , f (x) /F (x) and pd f corresponding to cd f F , respectively. The conditional pd f of Zn,m given XU(m) = x is
f z|XU(m) = x =

1 f (z + x) , {R (z + x) − R (x)}n−m−1 Γ (n − m) F (x) for 0 < z < ∞, 0 < x < ∞. (6.1.2)

Since Zn,m and XU(m) are independent, we will have for all z > 0, {R (z + x) − R (x)}n−m−1

f (z + x) , F (x)

(6.1.3)

as independent of x. F(z+x) Now, let R (z + x) − R (x) = − ln F (x) = q (z, x), say. Writing (6.1.3) in terms of q (z, x), we get {q (z, x)}n−m−1 exp {−q (z, x)}

∂ q (z, x), ∂z

(6.1.4)

as independent of x. Hence, by Lemma 6.1.37, we have 
−1 − ln F (z + x) F (x) = q (z + x) = c (z) ,

(6.1.5)

where c (z) is a function of z only. Thus
−1 F (z + x) F (x) = c1 (z),

(6.1.6)

where c1 (z) is a function of z only. The relation (6.1.6) is true for all z ≥ 0 and any arbitrary fixed positive number x. The continuous solution of (6.1.6) with the boundary condition F (0) = 1 and F (∞) = 0 is F (x) = exp {−λx} , for x ≥ 0 and any arbitrary positive real number λ.

Characterizations of Exponential Distribution II

113

Remark 6.1.38. The assumption of “absolute continuity of cd f F” in Theorem 6.1.36 can be replaced by “continuity of cd f F ”. We have seen that if the sequence (Xn )n≥1 of i.i.d.rv0s are from E (λ), then d

XU(n) =

n

∑ Z j,

j=1

where Z 0j s are i.i.d. ∼ E (λ) . The following theorem gives a characterization of the exponential distribution using the above property. Theorem 6.1.39. Let (Xn)n≥1 be a sequence of i.i.d. rv0 s with an absolutely continuous cd f F and corresponding pd f f and F (0)
= 0, F (x) < 1, for all x > 0. If F NBU or NWU and Z n+1,n = XU(n+1) − XU(n) ∼ X1 , then X1 ∼ E (λ). Proof. The pd f gn of Zn+1,n can be written as (R ∞ (R(u))n−1 0 Γ(n) r (u) f (u + z) du, z ≥ 0, gn (z) = 0, otherwise.

(6.1.7)

Since Zn+1,n ∼ X1, we must have Z ∞ 0

Substituting

(R (u))n−1

r (u) f (u + z) du = f (z), for all z > 0. Γ (n) Z ∞ 0

(R (u))n−1 f (u) du = Γ (n) ,

(6.1.8)

(6.1.9)

we have Z ∞

(R (u))n−1 r (u) f (u + z) du

0

= f (z)

Z ∞

(R (u))n−1 f (u)du, for all z > 0. (6.1.10)

0

Thus

Z ∞ 0

h i
−1 (R (u))n−1 f (u) f (u + z) F (u) − f (z) du = 0, for all z > 0. (6.1.11)

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M. Ahsanullah and G.G. Hamedani

Integrating (6.1.11) with respect to z from z1 to ∞, we arrive at Z ∞ 0

h i
−1 (R (u))n−1 f (u) F (u + z1 ) F (u) −F (z1 ) du = 0, for all z1 > 0. (6.1.12)

If F is NBU, then (6.1.12) is true if
−1 F (u + z1 ) F (u) = F (z1 ), for all z1 > 0.

(6.1.13)

The only continuous solution of (6.1.13) with boundary conditions F (0) = 1 and F (∞) = 0, is F (x) = exp (−λx) , where λ is an arbitrary real positive number. Similarly, if F is NWU then (6.1.12) is true if (6.1.13) is satisfied and hence X1 ∼ E (λ). The following theorem is proved under the assumption of monotone hazard rate. Theorem 6.1.40. Let (Xn)n≥1 be a sequence of i.i.d. rv0 s with an absolutely continuous cd f F with corresponding pd f f and F (0) = 0. If Zn+1,n and Zn,n−1 are i.d. and F belong to H ∗ , then X1 ∼ E (λ). Proof. We have P (Zn+1,n > z) =

(R

∞ (R(u))n−1 r(u) F (u + z) du, 0 Γ(n)

0,

for all z > 0, otherwise.

Since Zn+1,n and Zn,n−1 are i.d., using the above equation we have Z ∞

(R (u))n r (u)F (u + z) du

0

=n

Z ∞

(R (u))n−1 r (u) F (u + z) du, for all z > 0. (6.1.14)

0

Substituting the identity n

Z ∞ 0

(R (u))n−1 r (u)F (u + z) du =

Z ∞ 0

(R (u))n f (u + z) du

Characterizations of Exponential Distribution II

115

in (6.1.14), we get, on simplification, that   Z ∞ r (u + z) n−1 (R (u)) r (u) F (u + z) 1 − du = 0, for all z > 0. (6.1.15) r (u) 0 Thus if F ∈ H ∗ , then (6.1.15) is true if for almost all u and any z > 0, r (u + z) = r (u) .

(6.1.16)

That is the constant hazard rate. The relation (6.1.16) is a well-known characteristic property of the exponential distribution. Hence we have X1 ∼ E (λ) . Theorem 6.1.41. Let (Xn )n≥1 be a sequence of non-negative i.i.d. rv0 s with an absolutely continuous cd f F with the corresponding
pd f f . If F ∈ H ∗ and for some fixed n, m, 1 ≤ m < n < ∞, Zn,m = XU(n) − XU(m) ∼ XU(n−m) , then X1 ∼ E (λ). Proof. The pd f 0 s f1 of XU(n−m) and f2 of Zn,m can be written as f1 (x) =

1 (R (x))n−m−1 f (x) , for 0 < x < ∞, Γ (n − m)

(6.1.17)

and f 2 (x) =

Z ∞ (R (u))m−1 [R (x + u) − R (x)]n−m−1 0

Γ (n) Γ (n − m) for 0 < x < ∞.

r (u) f (x + u) du

(6.1.18)

Integrating (6.1.17) and (6.1.18) with respect to x from 0 to x0 , we get F1 (x0 ) = 1 − g1 (x0) , where

n−m

g1 (x0 ) =

∑

j=1

(6.1.19)

(R (x0 )) j−1 −R(x0 ) , e Γ ( j)

and F2 (x0 ) = 1 − g2 (x0 , u),

(6.1.20)

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M. Ahsanullah and G.G. Hamedani

where [R (x0 + u) − R (u)] j−1 exp {−R (x0 + u) − R (u)} . ∑ Γ ( j) j=1

n−m

g2 (x0 , u) =

Now equating (6.1.19) and (6.1.20), we get Z ∞ (R (u))m−1 0

Γ (m)

f (u)[g2 (x0 , u) − g1 (x0 )] du = 0, for all x0 ≥ 0.

(6.1.21)

Thus, if F ∈ H ∗ , then (6.1.21) is true if r (x0 + u) = r (u) , for almost all u and any fixed x0 ≥ 0.

(6.1.22)

Hence X1 ∼ E (λ), where λ is an arbitrary positive number. The following theorem uses the property of homocedasticity but not IFR, DFR, NBU or NWU property. Theorem 6.1.42. Let (Xn)n≥1 be a sequence of i.i.d.rv0 swith an absolutely continuous cd f F such that inf{x|F (x) >0} = 0 and E Xn2 < ∞. Then X1 has exponential distribution if and only if Var Zn+1,n |XU(n) = x = b for all x, where b is a positive constant independent of x. Proof. We only need to prove the “only if” condition. Note that  Z  2 |XU(n) = x = E Zn+1,n

∞ 0


−1 z2 F (x) dF (z + x) =2

Z ∞ 0


−1 z F (x) dF (z + x) dz, (6.1.23)

and   Z E Zn+1,n |XU(n) = x =

∞ 0


−1 z F (x) dF (z + x) =

Z ∞ 0


F (x) F (z + x) dz. (6.1.24)

Characterizations of Exponential Distribution II

117

R

Substituting G (x) = 0∞ zF (z + x) dz and denoting G(r) (x) as the rth derivative of G (x) , we have on simplification G(1) (x) =

Z ∞

F (z + x) dz,

G(2) (x) = F (x)

and

G(3) (x) = − f (x) ,

0

where f is pd f corresponding to F. Expressing (6.1.23) and (6.1.24) in terms of G(r) (x), we have   Var Zn+1,n |XU(n) = x  −1   −1 2 = 2G (x) G(2) (x) − G(1) (x) G(2) (x) = b, for all x > 0. (6.1.25) Differentiating (6.1.25) with respect to x and simplifying, we obtain   −3  2 (3) (2) (1) (2) − G (x) − G (x) G (x) = 0. (6.1.26) 2G (x) G (x) Since G(3) (x) 6= 0 for all x > 0, we must have 2 G(1) (x) − G (x) G(2) (x) = 0,

 i.e.

  −1  d (1) G (x) G (x) = 0, for all x > 0. dx

(6.1.27)

(6.1.28)

The solution of (6.1.28) is G (x) = ae−cx ,

x > 0,

(6.1.29)

where a and c are arbitrary constants. Hence F (x) = G(2) (x) = ac2e−cx ,

x > 0.

Since F is a cd f with F (0) = 0, it follows that F (x) = e−λx ,

x > 0 and

λ>0

is an arbitrary constant.

118

M. Ahsanullah and G.G. Hamedani

The following theorem gives a characterization of the exponential distribution using the hazard rate. Theorem 6.1.43. Let (Xn)n≥1 be a sequence of i.i.d. rv0 s with an absolutely continuous cd f F and corresponding pd f f such that inf {x|F (x) = 0} = 0, F (x) < 1 for all x > 0 and F ∈ H ∗ . Then X1 ∼ E (λ) if and only if for some fixed n, n ≥ 1, the hazard rates r 1 of Zn+1,n and r of X1 are equal. Proof. We only need to prove the “only if” condition. Suppose r1 = r. The joint pd f of XU(n+1) and XU(n) is given by fn+1,n (x, y) =

(

1 n−1 r (x) Γ(n) (R (x))

f (y) , 0 < x < y < ∞,

0,

otherwise.

Substituting Zn+1,n = XU(n+1) − XU(n) and Vn = XU(n), we have the joint pd f of Zn+1,n and Vn as ( 1 (R (v))n−1 r (v) f (z + v) , for 0 < z, v < ∞, (6.1.30) f1∗ (z, v) = Γ(n) 0, otherwise. By (6.1.30), we can write R∞

(R (v))n−1 r (v) f (z + v) dv

0

(R (v))n−1 r (v)F (z + v) dv

r1 (z) = R 0∞

, for all z ≥ 0.

Since r1 (z) = r (z) for all z, the RHS of (6.1.31) must be equal to z ≥ 0, from which, after simplification, we obtain

(6.1.31) f (z) F(z)

for all

Z ∞

(R (v))n−1 r (v) F (z)F (z + v) [r (z + v) − r (z)]dv = 0, for all z ≥ 0.

0

(6.1.32) Since F ∈ H ∗ , for (6.1.32) to be true, we must have r (z + v) = r (v) , for all z ≥ 0 and almost all v, v ≥ 0. Hence, X1 ∼ E (λ) .

(6.1.33)

Characterizations of Exponential Distribution II

119

The following theorem is due to Basak (1996), which is based on lower k−records. This result generalizes the work of Ahsanullah and Kirmani (1991). Theorem 6.1.44. Let (Xn )n≥1 be a sequence of i.i.d. rv0 s with cd f F such that F (0) = 0 and limx→0+ x = λ, λ > 0. If (L (n, k) − k + 1) XL(n,k) and X1,n, k ≥ 1, are identically distributed, then X ∼ E (λ). F (x)

The following theorem, due to Ahsanullah (2004), replaces the equality of the distributions in Theorem 6.1.44 with the equality of the expectations. For this we need to define a special class of distributions. We say that a cd f F of a rv X with F (0) = 0 and E [X] < ∞, belongs to harmonic new better (worse) than used in expectation ,HNBUE (HNWUE), if for t > 0, Z ∞

t

F (x) dx ≤ (≥) µe− µ .

t

Theorem 6.1.45. Let (Xn )n≥1 be a sequence of i.i.d. rv0 s with a cd f F with F (0) = 0. If F is HNBUE or HNWUE, then   E (U (n, k) − k + 1)XU(n,k) = E [X1,k ] if and only if F is exponential. Iwi´nska (2005) presented characterizations of the exponential distribution based on the distributional properties and the expected values of the record values. It is assumed throughout the paper that the random variables are continuous and non-negative and their cd f F has the property that lim x→0+ F (x) /x exists and is finite. It is also assumed that the index of the record values has a geometric distribution. Bairamov et al. (2005) characterized the exponential distribution in terms of the regression of a function of a record value with its adjacent record values as covariate. Some of the characterizations mentioned before in this chapter are of similar nature. Yanev et al. (2007) extended Bairamov et al.’s (2005) results to truncated exponential distributions with support (lF , ∞) where, as in Chapter 5, lF = inf{x|F (x) > 0} . Clearly, in case lF = 0, their results reduce to Bairamov et al.’s (2005) for the exponential distribution. For more details of truncated exponential distributions we refer the reader to Yanev et al. (2007).

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6.2. Characterizations Based on Generalized Order Statistics So far, in Section 6.1, we have been considering characterizations of the exponential distribution based on record values. As we mentioned in Chapter 5, the concept of generalized order statistics was introduced by Kamps (1995) as a unified approach to order statistics and record values. We presented various characterizations of the exponential distribution based on generalized order statistics in Section 5.3. These characterizations can be included in this section as well, however we will not repeat them here and instead refer the reader to Section 5.3 one more time, for a review of the characterizations of the exponential distribution in terms of generalized order statistics.

References Abramowitz, M. and Stegan, I. (1972). Handbook of Mathematical Functions. Dover, New York, NY. Aczel, J. (1966). Lectures on Functional Equations and Their Applications. Academic Press, New York, NY. Ahsanullah, M. (1975). A characterization of the exponential distribution. In: G. P. Patil, S. Kotz and J.O rd (eds.), Statistical Distributions in Scientific Work, Vol. 3, 71–88. Dordrecht–Holland, D. Reidel Publishing Company. Ahsanullah, M. (1976). On a characterization of the exponential distribution by order statistics. J. Appl. Prob., 13, 818–822. Ahsanullah, M. (1977). A Characteristic Property of the Exponential Distribution. Ann. of Statist., 5, 580–582. Ahsanullah, M. (1978a). A characterization of the exponential distribution by spacing. J. Appl. Prob., 15, 650–653. Ahsanullah, M. (1978b). On characterizations of exponential distribution by spacings. Ann. Inst. Stat. Math., 30, A, 163–166. Ahsanullah, M. (1978c). Record Values and the Exponential Distribution. Ann. Inst. Statist. Math., 30, A, 429–433. Ahsanullah, M. (1979). Characterization of the Exponential Distribution by Record Values. Sankhy¯a, Ser. B, 41, 116–121.

122

References

Ahsanullah, M. (1980). Linear Prediction of Record Values for the Two Parameter Exponential Distribution. Ann. Inst. Stat. Math. 32, A, 363–368. Ahsanullah, M. (1981a). Record Values of the Exponentially Distributed Random Variables. Statistiche Hefte, 2, 121–127. Ahsanullah, M. (1981b). On a Characterization of the Exponential Distribution by Weak Homoscedasticity of Record Values. Biom. J., 23, 715–717. Ahsanullah, M. (1981c). On characterizations of the exponential distribution by spacings. Statische Hefte, 22, 316–320. Ahsanullah, M. (1982). Characterizations of the Exponential Distribution by Some Properties of Record Values. Statistiche Hefte, 23, 326–332. M.Ahsanullah (1984). A characterization of the exponential distribution by higher order gap. Metrika, 31, 323–326. Ahsanullah, M. (1987a). Two Characterizations of the Exponential Distribution. Comm. Statist. Theory-Methods, 16(2), 375–381. Ahsanullah, M. (1987b). Record Statistics and the Exponential Distribution. Pak. J. Statist., 3, A, 17–40. Ahsanullah,M. (1988a). On a conjecture of Kakosian, Klebanov and Melamed. Statistiche Hefte, 29, 151–157. Ahsanullah, M. (1988b). Introduction to Record Statistics, Ginn Press, Needham Heights, MA. Ahsanullah, M. (1988c). Characteristic properties of order statistics based on a random sample size from an exponential distribution. Statistica Neerlandica, 42, 193–197. Ahsanullah, M. (1991). Some Characteristic Properties of the Record Values from the Exponential Distribution. Sankhy¯a, Ser. B, 53, 403–408. Ahsanullah, M. (1992). Record Values of Independent and Identically Distributed Continuous Random Variables. Pak. J. Statist., 8(2), A, 9–34.

References

123

Ahsanullah, M. (1994). Records of Univariate Distributions. Pak. J. Statist., 9(3), 49–72. Ahsanullah, M. (1995). Record Statistics. Nova Science Publishers Inc., New York, NY. Ahsanullah, M. (2000). Generalized order statistics from exponential distribution. J. Statist. Plan. and Inf., 25, 85–91. Ahsanullah, M. (2003). Some characteristic properties of the generalized order statistics from exponential distribution. J. Statist. Research,Vol. 37, 2, 159–166. Ahsanullah, M. (2004). Record Values–Theory and Applications. University Press of America, Lanham, MD. Ahsanullah, M. (2006). The generalized order statistics from exponential distribution. Pak.J. Statist., Vol. 22, 2, 121–128. Ahsanullah, M. and Aliev, F. (2008). Some characterizations of exponential distribution by record values. Journal of Statistical Research , Vol. 2, No. 1, 11–16. Ahsanullah, M. and Kirmani, S.N.U.A. (1991). Characterizations of the Exponential Distribution by Lower Record Values. Comm. Statist. TheoryMethods, 20(4), 1293–1299. Ahsanullah, M. and Nevzorov, V.B. (2001a). Ordered Random Variables. Nova Science Publishers Inc, New York, NY. Ahsanullah, M. and Nevzorov, V.B. (2001b). Distribution between uniform and exponential. In: M. Ahsanullah, J. Kenyon and S.K. Sarkar (eds.), Applied Statistical Science IV, 9–20. Ahsanullah, M. and Wesolowski, J. (1998). Linearity of Best Non–adjacent Record Values. Sankhy¯a, Ser. B, 231–237. Arnold, B.C. (1971). Two characterizations of the exponential distribution. Technical Report, Iowa State University.

124

References

Arnold, B.C. (1980). Two characterizations of the geometric distribution. J. Appl. Prob. 17, 570–573. Arnold, B.C. and Balakrishnan, N. (1989). Relations, Bounds and Approximations for Order Statistics. Lecture Notes in Statistics, No. 53, Springer– Verlag, New York, NY. Arnold, B.C., Balakrishnan, N. and Nagaraja, H. N. (1998). Records. John Wiley & Sons Inc., New York. NY. Arnold,B.C. and Ghosh, M. (1976). A characterization of geometric distributions by distributional properties of order statistics. Scan. Actuarial J., 232–234. Azlarov, T. A. and Volodin, N. A. (1986). Characterization Problems Associated with the Exponential Distribution. Springer-Verlag, New York, NY. Bairamov, I.G. and Ahsanullah, M. (2000). Distributional Relations Between Order Statistics and the Sample itself and Characterizations of Exponential Distribution. J. Appl. Statist. Sci., Vol. 10(1), 1–16. Bairamov, I.G., Ahsanullah,M. and Pakes, A.G. (2005). A characterization of continuous distributions via regression on pairs of record values. Australian and New Zealand J. Statistics, Vol. 474, 243–247. Bairamov, M., Ahsanullah, M. and Akhundov, I. (2002). A Residual Life Function of a System Having Parallel or Series Structure. J. of Statistical Theory & Applications, 119–132. Bairamov, I. and Ozkal, T. (2007). On characterizations of distributions through the properties of conditional expectations of order statistics. Commun. in Statist. Theory-Methods, 36, 1319–1326. Balakrishnan, N. and Ahsanullah, M. (1993a). Relations for Single and Product Moments of Record Values from Exponential Distribution. J. Appl. Statist. Sci., 2(1), 73–88.

References

125

Balakrishnan, N. and Ahsanullah, M. (1993b). Relations for Single and Product Moments of Record Values from Lomax Distribution. Sankhy¯a, B, 56, 140–146.. Balakrishnan, N. and Balasubramanian, K. (1995). A Characterization of Geometric Distribution Based on Record Values. J. Appl. Statist. Sci., 2, 277–282. Barton, D.E. and Mallows, C.L. (1965). Some Aspects of the Random Sequence. Ann. Math. Statist. 36, 236–260. Basak, P. (1996). Lower Record Values and Characterization of Exponential Distribution. Calcutta Statistical Association Bulletin, 46, 1–7. Basu, A.P. (1965). On characterizing the exponential distribution by order statistics. Ann. Inst. Statist. Math. 17, 03–96. Beg, M.I. and Ahsanullah, M. (2006). On characterizating distributions by conditional expectations of functions of generalized order statistics. J. Appl. Statist. Sci., Vol. 15(2), 229–244. Bieniek, M. and Szynal, D. (2003). Characterizations of Distributions via Linearity of Regression of Generalized Order Statistics. Metrika, 58, 259– 271. Blom, G. Thorburn, D. and Vessey, T. (1990). The Distribution of the Record Position and its Applications. Amer. Statistician, 44, 152–153. Bruss, F.T. (1988). Invariant Record Processes and Applications to Best Choice Modeling. Stoch. Proc. Appl. 17, 331–338. Bell, C.B. and Sarma, Y.R.K. (1980). A characterization of the exponential distribution based on order statistics. Metrika, 27, 263–269. Chan, L.K. (1967). On a characterization of distributions by expected values of extreme order statistics. Amer. Math. Monthly, 74, 950–951. Chandler, K.N. (1952). The Distribution and Frequency of Record Values. J. Roy. Statist. Soc., B, 14, 220–228.

126

References

Cinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, New Jersey. Cramer, E., Kamps, U. and Keseling, C. (2004). Characterizations via Linearity of Ordered Random Variables: A Unifying Approach. Commun. Statist. Theory-Methods, Vol. 33, No. 12, 2885–2911. Crawford, G.B. (1966). Characterizations of geometric and exponential distributions. Ann. Math. Statist., 37, 1790–1794. Csorgo, M. Seshadi, V. and Yalovsky, M. (1975). Applications of characterizations in the area of goodness of fit. In: G.P. Patil et al. (eds.), Statististical distributions in Scientific Work, Vol. 2, Reidel Dordrecht, 79–90. Dallas, A.C. (1981). Record Values and the Exponential Distribution. J. Appl. Prob., 18, 959–951. Dallas, A.C. (1981). A Characterization Using Conditional Variance. Metrika, 28, 151–153. Davies, P.L. and Shabbhag, D.N.(1987). A generalization of a theorem by Deny with applications in characterization theory. Quart. J. Math. Oxford, 38(2), 13–34. David, H.A. (1981). Order Statistics. Second Edition, John Wiley & Sons, Inc., New York, NY. David, H.A. and Nagaraja, H.N. (2003). Order Statistics. Third Edition, Wiley & Sons, Hoboken, New Jersey. DeHaan, L. and Resnick, S.I. (1973). Almost Sure Limit Points of Record Values. J. Appl. Prob., 10, 528–542. Deheuvels, P. (1983). The Complete Characterization of the Upper and Lower Class of Record and Inter-Record Times of i.i.d. Sequence. Zeit. Wahrscheinlichkeits theorie Verw. Geb., 62, 1–6. Deheuvels, P. (1984). The Characterization of Distributions by Order Statistics and Record Values – A Unified Approach. J. Appl. Prob., 21, 326–334.

References

127

Deken, J.G. (1978). Record Values, Scheduled Maxima Sequences. J. Appl. Prob., 15, 491–496. Dembi´nska, A. and Wesolowski, J. (1998). Linearity of Regression for Non– Adjacent Order Statistics. Metrika, 48, 215–222. Dembi´nska, A. and Wesolowski, J. (2000). Linearity of Regression for Non– Adjacent Record values. J. of Statistical Planning and Inference , 90, 195– 205. Dembi´nska, A. and Wesolowski, J. (2003). Constancy of Regression for size two Record Spacings. Pak. J. Statist., 19(1), 143–149. Dembi´nska, A. and Wesolowski, J. (2000). Linearity of Prediction for Non– adjacent Record Values. J. Statist. Plan. Inf., 90, 195–205. Desu, M.M. (1971). A characterization of the exponential distribution by order statistics. Ann. Math. Statist., 42, 837–838. Dufur, R. (1982). Tests d’ajustement pour des echantillon tonques ou censures.Ph.D. Thesis, Universite de Montreal. Dufur, R., Maag, V.R. and van Eeden, C. (1984). Correcting a proof of a characterization of the exponential distribution. J. Roy. Statist. Soc., B, 46, 238–241. Dunsmore, J.R. (1983). The Future Occurrence of Records. Ann. Inst. Stat. Math., 35, 267–277. Embrechts, P. and Omey, E. (1983). On Subordinated Distributions and Random Record Processes. Ma. P. Cam. Ph. 93, 339–353. Feller, W. (1957). An Introduction to Probability Theory and Its Applications, Vol. I, 2nd Edition. Wiley & Sons, New York, NY. Feller, W. (1966). An Introduction to Probability Theory and its Applications . Vol. II, Wiley & Sons, New York, NY. Ferguson, T.S. (1964). A characterization of the negative exponential distribution. Ann. Math. Statist., 35, 1199–1207.

128

References

Ferguson, T.S. (1967). On characterizing distributions by properties of order statistics. Sankhy¯a, Ser. A, 29, 265–277. Fisz, M. (1958). Characterizations of some probability distributions. Skand. Aktirarict., 65–67. Fosam, E.B. Rao, C.R. and Shanbhag, D.N. (1993). Comments on some papers involving Cauchy functional equation. Statist. Prob.Lett., 17, 299–302. Foster, F.G. and Stuart, A. (1954). Distribution Free Tests in Time Series Based on the Breaking of Records. J. Roy. Statist. Soc., B, 16, 1–22. Foster, F.G. and Teichroew, D. (1955). A Sampling Experiment on the Powers of the Record Tests for a Trend in a Time Series. J. Roy. Statist. Soc., B, 17, 115–121. Franco, M. and Ruiz, J.M. (1995). On Characterization of Continuous Distributions with Adjacent Order Statistics. Statistics, 26, 275–385. Franco, M. and Ruiz, J.M. (1996). On Characterization of Continuous Distributions by Conditional Expectation of Record Values. Sankhy¯a, 58, Series A, Pt. 1, 135–141. Franco, M. and Ruiz, J.M. (1997). On Characterizations of Distributions by Expected Values of Order Statistics and Record Values with Gap. Metrika, 45, 107–119. Franco, M. and Ruiz, J.M. (2001). Characterization of Discrete distributions based on conditional expectations of Record Values. Statistical Papers, 42, 101–110. Fr´echet, M. (1927). Sur La loi probabilite de l’?cart Maximum. Ann. de la oc. Polonaise de Math., 6, 93–116. Freudenberg, W. and Szynal, D. (1976). Limit Laws for a Random Number of Record Values. Bull. Acad. Polon. Sci. Ser. Math. Astr. Phys., 24, 195–199. Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics . Robert E. Krieger Publishing Co. Malabar, FL.

References

129

Galambos, J. (1975a). Characterizations of probability distributions by properties of order statistics II. In: G.P. Patil et al. (eds.), Statistical Distributions in Scientific Work, Vol. 3, Reidel Dordrecht, 89–101. Galambos, J. (1975b). Characterizations of probability distributions by properties of order statistics I. In: G.P. Patil et al. (eds.), Statistical Distributions in Scientific Work, Vol. 3, Reidel Dordrecht, 71–86. Galambos, J. and Kotz, S. (1978). Characterizations of Probability Distributions. Lecture Notes in Mathematics, No. 675, Springer–Verlag, New York, NY. Galambos, J. and Kotz, S. (1983). Some characterizations of the exponential distribution via properties of geometric distribution. In: P.K. Sen, ed., Essays in honor of Norman L. Johnson, North Holland, Amsterdam, 159– 163. Galambos, J. and Seneta, E. (1975). Record Times. Proc. Amer. Math. Soc., 50, 383–387. Gajek, L. and Gather, U. (1989). Characterizations of the Exponential Distribution by Failure Rate and Moment Properties of Order Statistics. Lecture Notes in Statistics , 51. Extreme Value Theory, Proceeding, 114–124. J.Husler, R.D. Reiss (eds.), Springer–Verlag, Berlin, Germany. Gather, U. (1989). On a characterization of the exponential distribution by properties of order statistics, Statist. Prob. Lett., 7, 93–96. Gather, U., Kamps, U. and Schweitzer, N. (1998). Characterizations of distributions via identically distributed functions of order statistics. In: N.Balakrishnan and C.R. Rao (eds.), Handbook of Statistics , Vol. 16, 257–290. Gaver, D.P. (1976). Random Record Models. J. Appl. Prob., 13, 538–547. Gaver, D.P. and Jacobs, P.A. (1978). Non Homogeneously Paced Random Records and Associated Extremal Processes. J. Appl. Prob., 15, 543– 551.

130

References

Glick, N. (1978). Breaking Records and Breaking Boards. Monthly, 85(1), 2–26.

Amer. Math.

Gnedenko, B. (1943). Sur la Distribution Limite du Terme Maximum d’une Serie Aletoise. Ann. Math., 44, 423–453. Govindrarajulu, Z. (1966). Characterizations of the exponential and power function distributions. Scand. Aktuarietdskr., 49, 132–136. Goldburger, A.S. (1962). Best Linear Unbiased Predictors in the Generalized Linear Regression Model. J. Amer. Statist. Assoc., 57, 369–375. Goldie, C.M. (1989). Records, Permutations and Greatest Convex Minorants. Math. Proc. Camb. Phil. Soc., 106, 169–177. Goldie, C.M. and Resnick, S.I. (1989). Records in Partially Ordered Set. Ann. Prob., 17, 675–689. Goldie, C.M. and Resnick, S.I. (1995). Many Multivariate Records. Stoch. Proc. And Their Appl., 59, 185–216. Goldie, C.M. and Rogers, L.C.G. (1984). The k-Record Processes are i.i.d. Z. Wahr. Verw. Geb., 67, 197–211. Gradshteyn, I.S. and Ryzhik, I.M. (1980). Tables of Integrals, Series, and Products, Corrected and Enlarged Edition. Academic Press, Inc. Grosswald, E. and Kotz, S. (1981). An Integrated Lack of Memory Property of the Exponential Distribution. Ann. Inst. Statist. Math., 33, A, 205–214. Grudzien, Z. and Szynal, D. (1985). On the Expected Values of kth Record Values and Characterizations of Distributions. Probability and Statistical Decision Theory, Vol A. F. Konecny, J. Mogyorodi and W. Wertz, (eds.), Dordrecht – Reidel, 1195–214. Grudzien, Z. and Szynal, D. (1996). Characterizations of distributions by order statistics and record values, A Unified Approach. J. Appl. Prob., 21, 326– 334..

References

131

Grudzien, Z. and Szynal, D. (1997). Characterizations of uniform and exponential distributions via moments of the kth record values randomly indexed. Applications Mathematicae, 24, 307–314. Gulati, S. and Padgett, W.J. (1994a). Nonparametric Quantitle Estimation from Record Breaking Data. Aust. J. of Stat., 36, 211–223. Gulati, S. and Padgett, W.J. (1994b). Smooth Nonparametric Estimation of Distribution and Density Function from Record Breaking Data. Comm. In Statist. Theory-Methods, 23, 1259–1274. Gulati, S. and Padgett, W.J. (1994c). Smooth Nonparametric Estimation of the Hazard and Hazard Rate Functions from Record Breaking Data. J. Statist. Plan. Inf., 42, 331–341. Gupta, R.C. (1973). A characteristic property of the exponential distribution. Sankhy¯a, Ser. B, 35, 365–366. Gupta, R. C. (1984). Relationships between Order Statistic and Record Values and Some characterization Results. J. Appl. Prob., 21, 425–430. Gupta, R.C. and Ahsanullah, M. (2004). Characterization results based on the conditional expectation of a function of non–adjacent order statistic (Record Value). Annals of Statistical Mathematics, Vol. 56(4), 721–732. Gut, A. (1990). Convergence Rates for Record Times and the Associated Covering Processes. Stoch. Processes Appl., 36, 135–152. Hamedani, G,G,, Ahsanullah, M. and Sheng, R. (2008). Characterizations of certain continuous univariate distributions based on the truncated moment of the first order statistic. Aligarh Journal of Statistics, 28, 75–81. Hill, B.M. (1975). A Simple General Approach to Inference about the Tail of a Distribution. Ann. statist., 3, 1163–1174. Holmes, P.T. and Strawderman, W. (1969). A Note on the Waiting Times Between Record Observations. J. Appl. Prob., 6, 711–714. Huang, J.S. (1974). Characterizations of exponential distribution by order statistics. J. Appl. Prob., 11, 605–609.

132

References

Huang, J.S., Arnold, B.C. and Ghosh, M. (1979). On characterizations of the uniform distribution based on identically distributed spacings. Sankhy¯a, Ser B, 41, 109–115. Huang, Wen–Jang and Li, Shun–Hwa (1993). Characterization Results Based on Record Values. Statistica Sinica, 3, 583–599. Iwinska, M. (1985). On a characterization of the exponential distribution by order statistics. In Numerical Methods and Their Applications. Proc. 8th Sess Poznan Circle Zesz Nauk Ser 1. Akad. Ekad. Poznan, 132, 51–54. Iwinska, M. (1986). On the characterizations of the exponential distribution by order statistics and record values. Fasciculi Mathematici, 16, 101–107. Iwinska, M. (2005). On characterization of the exponential distribution by record values with a random index. Fasciculi Mathematici, 36, 33–39.. Jenkinson, A.F. (1955). The Frequency Distribution of the Annual Maximum (or Minimum) Values of Meteorological Elements. Quart. J. Meter. Soc., 87, 158–171. Joshi, P.C. (1978). Recurrence relation between moments of order statistics from exponential and truncated exponential distribution. Sankhy¯a, Ser B, 39, 362–371. Johnson, N.L. and Kotz, S. (1977). Distributions in statistics: Continuous Multivariate Distributions. Wiley & Sons, New York, NY. Kakosyan, A. V., Klebanov, L. B. and Melamed, J. A. (1984). Characterization of Distribution by the Method of Intensively Monotone Operators. Lecture Notes in Math. 1088, Springer Verlag, New York, N.Y. Kaminsky, K.S. (1972). Confidence intervals for the exponential scale parameter using optimally selected order statistics. Technometrics, 14, 371–383. Kaminsky, K.S. and Nelson, P.L. (1975). Best linear unbiased prediction of order statistics in location and scale families. J. Amer. Statist. Assoc., 70, 145–150.

References

133

Kamps, U. (1991). A general recurrence relation for moments of order statistics in a class of probability distributions and characterizations. Metrika, 38, 215–225. Kamps, U. (1982a). Identities for the difference of moments of successive order statistics and record values. Metron, 50, 179–187. Kamps, U. (1992 b). Characterizations of the exponential distributions by equality of moments. Allg. Statist. Archiv., 78, 122–127. Kamps, U. (1995). A Concept of Generalized Order Statistics. Teubner, Stuttgart. Kamps, U. (1998a). Order Statistics. Generalized in Encyclopedia of Statistical Sciences. Update Vol. 3 (S. KotzmRead, C.B. and Banks, D.L., eds.) To appear (Wiley & Sons, New York, NY). Kamps, U. (1998b). Subranges of generalized order statistics from exponential distributions. Fasciculi Mathematici, 28, 63–70. Kamps, U. Cramer, E. (2001). On distribution of generalized order statistics. Statistics, 35, 269–280. Kamps, U, and Gather, U. (1997). Characteristic properties of generalized order statistics from exponential distributions. Applications Mathematicae, 24, 383–391. Karlin, S. (1966). A First Course in Stochastic Processes. Academic Press, New York, NY. Keseling, C. (1999). Conditional Distributions of Generalized Order Statistics and some Characterizations. Metrika , 49, 27–40. Kirmani, S.N.U.A. and Beg, M.I. (1983). On Characterization of Distributions by Expected Records. Sankhy¯a, Ser. A, 48, 463–465. Klebanov, L.B. and Melamed, J. A. (1983). A Method Associated with Characterizations of the exponential distribution. Ann. Inst. Statist. Math., A, 35, 105–114.

134

References

Korwar, R.M. (1984). On Characterizing Distributions for which the Second Record has a Linear Regression on the First. Sankhya, Ser B, 46, 108– 109. Korwar, R.M. (1990). Some Partial Ordering Results on Record Values. Commun. Statist. Theroy-Methods, 19(1), 299–306. Kotz, S. (1974). Characterizations of statistical distributions : a supplement in recent surveys. Internal. Statist. Rev., 42, 39–65. Lau, Ka–sing and Rao, C. R. Integrated Cauchy Functional Equation and Characterization of the Exponential. Sankhy¯a, Ser A, 44, 72–90. Leadbetter, M.R., Lindgreen, G. and Rootzen, H. (1983). Extremes and Related Properties of Random Sequences and Series , Springer–Verlag, New York, N.Y. Lee Min-Young (2001). On a characterization of the exponential distribution by conditional expectations of record values. Commun. Korean Math. Soc., 16, 287–290. Lee, Min–Young, Cheng, S.K and Jung, K.H. (2002). Characterizations of the exponential distribution by order statistics and conditional expectations of random variables. Commun. Korean Math. Soc., 17, 39–65. Leslie, J.R. and van Eeden,C. (1993). On a characterization of the exponential distribution on a Type 2 right censored sample. Ann. Statist., 21, 1640– 1647. Lin, G.D. (1987). On Characterizations of Distributions Via Moments of Record Values. Probab. Th. Rel. Fields, 74, 479–483. Lin, G.d. and Too, Y.H. (1989). Characterizations of uniform and exponential distributions. Statist. Prob. Lett., 7, 357–359.. Lloyd, E.H. (1952). Least Squares Estimation of Location and Scale Parameters Using Order Statistics. Biometrika, 39, 88–95.

References

135

Lopez–Blaquez, F. and Moreno–Reboollo, J.L. (1997). A characterization of distributions based on linear regression of order statistics and random variables. Sankhy¯a, Ser A, 59, 311–323. Malov, S.V. (1997). Sequential τ-ranks. J. Appl. Statist. Sci., 5, 211–224. Mann, N.R. (1969). Optimum Estimators for Linear Functions of Location and Scale Parameters. Ann. Math. Statist., 40, 2149–2155. Marsglia, G. and Tubilla, A. (1975). A Note on the Lack of Memory Property of the Exponential Distribution. Ann. Prob., 3, 352–354. Mellon, B. (1988). The Olympic Record Book. Garland Publishing Inc., New York, NY. Menon, M.V. and Seshardi, V. (1975). A characterization theorem useful in hypothesis testing in contributed papers. 40th session of the Inrtnal. Statist . Inst. Voorburg, 566–590. Mohan, N.R. and Nayak, S.S. (1982). A Characterization Based on the Equidistribution of the First Two Spacings of Record Values. Z. Wahr. Verw. Geb., 60, 219–221. Maag, U., Dufour, R. and van Eeden, C. (1984). Correcting a proof of a characterization of the exponential distribution. J. Roy. Statist. Sco., B, 46, 238–241. Nagaraja, H. N. (1977). On a Characterization Based on Record Values. Austral. J. Statist., 19, 70–73. Nagaraja, H. N. (1978). On the Expected Values of Record Values. Austral. J. Statist., 20, 176–182. Nagaraja, H. N. (1988a) Record Values and Related Statistics – a Review, Commun. Statist. Theory-Methods, 17, 2223–2238. Nagaraja, H.N. (1988b). Some characterizations of continuous distributions based on regression of adjacent order statistics of random variables. Sankhya, Ser A, 50, 70–73.

136

References

Nagaraja, H.N. and Nevzorov, V.B. (1977). On characterizations Based on Recod Values and Order Statistics. J. Statist. Plan. Inf., 61, 271–284. Nagaraja, H. N., Sen, P. and Srivastava, R.C. (1989). Some Characterizations of Geometric Tail Distributions Based on Record Values. Statistical Papers, 30, 147–155. Nayak, S.S. (1981). Characterizations Based on Record Values. J. Indian Statist. Assn., 19, 123–127. Nayak, S.S. (1985). Record Values for and Partial Maxima of a Dependent Sequence. J. Indian Statist. Assn., 23, 109–125. Neuts, M.F. (1967). Waiting Times Between Record Observations. J. Appl. Prob., 4, 206–208. Nevzorov, V.B. (1988). Records. Theo. Prob. Appl., 32, 201–228. Nevzorov, V. B. (1992). A Characterization of Exponential Distributions by Correlation between Records. Mathematical Methods of Statistics , 1, 49– 54. Nevzorov, V.B. (1995). Asymptotic distributions of Records in Non – Stationary Schemes. J. Statist.. Plan. Inf., 45, 261–273. Nevzorova, L.N.,Nevzorov, V. B. and Balakrishnan, N. ( 1997). Characterizations of distributions by extreme and records in Archimedian copula processes. In Advances in the Theory and Practice of Statistics – A volume in Honor of Samuel Kotz (N.L. Johnson and N. Balakrishnan, eds.), 469–478. John Wiley & Sons, New York, NY. Nevzorov, V. B. (2001). Records: Mathematical Theory. Translation of Mathematical Monographs, Vol. 194. American Mathematical Society. Providence, RI. Nevzorov, V.B. and Ahsanullah, M. (2001a). Some Distributions of Induced Records. Biometrical Journal , 42(8), 1069–1081.

References

137

Nevzorov, V.B. and Ahsanullah, M. (2001b). Extremes and Records for Concomitants of Order Statistics and Record Values. J.Appl. Statist. Sci., 10(3), 181–190. Oncei, S.Y., Ahsanullah, M., Gebizlioglu, O.I. and Aliev, F.A. (2001). Characterization of Geometric Distribuion through Normal and Weak Record Values. Stochastc Modelling and Applications, 4(1), 53–58 Pawlas, P. and Szynal, D. (1999). Recurrence Relations for Single and Product Moments of K–th Record Values From Pareto, Generalized Pareo and Burr Distributions. Commun. Statist. Theory-Methods, 28(7), 1699– 1701. Pfeifer, D. (1981). Asymptotic Expansions for the Mean and Variance of Logarithmic Inter–Record Times. Meth. Operat. Res., 39, 113–121. Pfeifer, D. (1982). Characterizations of exponential distributions by independent non–stationary record increments. J. Appl. Prob., 19, 127–135 (Corrections 19, p. 906). Pudeg, A. (1990). Charakterisierung von wahrseinlichkeitsver–teilungen durch ver teilungseigenschaften der ordnungesstatistiken und rekorde. Dissertation Aachen University of Technology. Puri, P.S. and Rubin, H. (1970). A characterization based on the absolute difference of two i.i.d. random variables. Ann. Math. Statist., 41, 2113– 2122. Rahimov, I. and Ahsanullah, M. (2001). M. Records Generated by the total Progeny of Branching Processes. Far East J. Theo. Statist, 5(10), 81–84. Rahimov, I. and Ahsanullah, M. (2003). Records Related to Sequence of Branching Stochastic Process. Pak. J. Statist., 19, 73–98. Ramachandran, B. (1979). On the strong memoryless property of the exponential and geometric laws. Sankhy¯a, Ser A, 41, 244–251. Ramachandran, B. and Lau, R.S. (1991). Functional Equations in Probability Theory. Academic Press, Boston, MA.

138

References

Rao. C.R. (1983). An extension of Deny’s theorem and its application to characterizations of probability distributions . In: P.J. Bickel et al. (eds.), A festschrift for Erich L. Lehman, Wordsworth, Belmont, 348–366. Rao, C.R. and Shanbhag, D.N. (1986). Recent results on characterizations of probability distributions. A unified approach through extensions of Deny’s theorem. Adv. Appl. Prob., 18, 660–678. Rao, C.R. and Shanbhag, D.N. (1994). Choquet Deny Type Functional Equations with Applications to stochastic models. Wiley & Sons, Chichestere. Rao, C.R. and Shanbhag, D.N. (1995a). Characterizations of the exponential distribution by equality of moments. Allg. Statist. Archiv, 76, 122–127. Rao, C.R. and Shanbhag, D.N. (1995b). A conjecture of Dufur on a characterization of the exponential distributions. Center for Multivariate Analysis. Penn State University Technical Report, 95–105. Rao, C.R. and Shanbhag, D.N. (1998). Recent Approaches for Characterizations Based on Order Statistics and Record Values. Handbook of Statistics, (N. Balakrishnan and C.R. Rao, eds.) Vol. 10, 231–257. Raqab, M.Z. (1997). Bounds Based on Greatest Convex Minorants for Moments of Record Values. Statist. Prob. Lett., 36, 35–41. Raqab, M.Z. (2002). Characterizations of Distributions Based on Conditional Expectations of Record Values. Statistics and Decisions, 20, 309–319. Raqab, M.Z. and Ahsanullah, M. (2000). Relations for Marginal and Joint Moment Generating Functions of Record Values From Power Function Distribution. J. Appl. Statist. Sci., Vol. 10(1), 27–36. Raqab, M.Z. and Ahsanullah, M. (2001). Estimation of the location and scale parameters of generalized exponential distribution based on order statistics. Journal of Statistical Computing and Simulation , Vol. 69(2), 109– 124. Raqab, M.Z. and Ahsanullah, M. ( 2003). On Moment Generating Functions of Records from Extreme Value Distributions. Pak. J. Statist., 19(1), 1– 13.

References

139

Raqab, M.Z. and Amin, W.A. (1997). A Note on Reliability Properties of k Record Statistics. Metrika, 46, 245–251. Reidel, M. and Rossberg, H.J. (1994). Characterizations of the exponential distribution function by properties of the difference Xk+s,n − Xk,n of order statistics. Metrika, 41, 1–19. Reiss, R.D. (1989). Approximate Distributions of Order Statistics . Springer– Verlag, New York, NY. Renyi, A. (1962). Theorie des Elements Saillants d’une Suit d’observations Collq. Combinatorial Meth. Prob. Theory , Aarhus University, 104–115. Resnick, S. (1987). Extreme Values, Regular Variation and Point Processes. Springer–Verlag, New York, NY. Resnick, S.I. (1973). Record Values and Maxima. Ann. Probab., 1, 650–662. Roberts, E.M. (1979). Review of Statistics of Extreme Values with Application to Air Quality Data. Part II: Application. J. Air Polution Control Assoc., 29, 733–740. Rossberg, H.J. (1972). Characterization of the exponential and Pareto distributions by means of some properties of the distributions which the differences and quotients of order statistics subject to. Math Operationsforsch Statist., 3, 207–216. Salamingo, F.J. and Whitaker, L.D. (1986). On Estimating Population Characteristics from Record Breaking Observations. I. Parametric Results. Naval Res. Log. Quart., 25, 531–543. Sen, P.K. (1959). On the Moments of the Sample Quantiles. Calcutta Statist. Assoc. Bull., 9, 1–19. Sen, P.K. and Salama, I.A. (eds.). (1992). Order Statistics and Nonparametric Theory and Applications. Elsevier, Amsterdam. Seshardi, V., Csorgo, N.M. and Stephens, M.A. (1969). Tests for the exponential distribution using Kolmogrov–type statistics. J. Roy Statist. Soc., B, 31, 499–509.

140

References

Shah, B.K. (1970). Note on Moments of a Logistic Order Statistics. Ann. Math. Statist., 41, 2151–2152. Shah, S.M. and Jani, P.N. (1988). UMVUE of reliability and its variance in two parameter exponential life testing model. Calcutta. Statist. Assoc. Bull.,37, 209–214. Shimizu, R. (1979). A characterization of exponential distribution involving absolute difference of i.i.d. random variables. Proc. Amer. Math. Soc., 121, 237–243. Shorrock, S. (1972). Record Values and Record Times. J. Appl. Prob., 9, 316–326. Shorrock, S. (1973). Record Values and Inter–Record Times. J. Appl. Prob., 10, 543–555. Siddiqui, M.M. and Biondini, R.W. (1975). The Joint Distribution of Record Values and Inter–Record Times. Ann. Prob., 3, 1012–1013. Singpurwala, N.D. (1972). Extreme Values from a Lognormal Law with Applications to Air Population Problems. Technometrics, 14, 703–711. Smith, R.L. (1986). Extreme Value Theory Based on the r Largest Annual Events. J. Hydrology, 86, 27–43. Smith, R.L. and Weissman, I. (1987). Large Deviations of Tail Estimators based on the Pareto Approximations. J. Appl. Prob., 24, 619–630. Srivastava, R.C. (1978). Two Characterizations of the Geometric Distribution by Record Values. Sankhy¯a, Ser B, 40, 276–278. Stepanov, A. V (1992). Limit Theorems for Weak Records. Theory of Probab. and Appl., 37, 579–574 ( English Translation). Sukhatme, P.V.(1937). Tests of significance of the –population with two degrees of freedom. Ann. Eugenics, 8, 52–56.

References

141

Tallie, C. (1981). A Note on Srivastava’s Characterization of the Exponential Distribution Based on Record Values. In: C. Tallie, G.P. Patil and B. Boldessari (eds.), Statistical Distributions in Scientific Work. ReidelDordrecht., 4, 417–418. Tanis, E.A. (1964). Linear forms in the order statistics from an exponential distribution. Ann. Math. Statist., 35, 270–276. Tata, M.N. (1969). On Outstanding Values in a Sequence of Random Variables. Z. Wehrscheinlichkeitsth., 12, 9–20. Teugels, J.L. (1984). On Successive Record Values in a Sequence of Independent and Identically Distributed Random Variables. Statistical Extremes and Applications. (Tiago de Oliveira, ed.). Reidel-Dordrecht, 639–650. Tiago de Oliveira, J. (1958). Extremal Distributions. Rev. Fac. Cienc. Univ. Lisboa, A, 7, 215–227. Tippett, L.H.C. (1925). On the Extreme Individuals and the Range of Samples Taken from a Normal Population. Biometrika, 17, 364–387. Vervaat, W. (1973). Limit Theorems for Records from Discrete Distributions. Stoch. Proc. Appl., 1, 317–334. Weissman, I. (1978). Estimations of Parameters and Large Quantiles Based on the k Largest Observations. J. Amer. Statist. Assoc., 73, 812–815. Wesolowski, J, and Ahsanullah, M. (2000). Linearity of Regresion for Non– adjacent weak Records. Statistica Sinica, 11, 30–52. Westcott, M. (1977). The Random Record Model. Proc. Roy. Soc. Lon., A, 356,529–547. Westcott, M. (1979). On the Tail Behavior of Record Time Distributions in a Random Record Process. Ann. Prob., 7, 868–237. White, H.J. (1988). Some characterizations of distributions based on the integrated Cauchy functional equations. Sankhy¯a, Ser A, 50, 59–63.

Index B  Best Linear Invariant Estimator, 45, 60

C  Cauchy Functional Equation, 134 Characterization, 121, 122, 124, 125, 126, 128, 131, 132, 133, 135, 136, 139, 141, 142 Conditional Distribution, 133

D 

G  Generalized Order Statistics, 51, 52, 53, 55, 57, 59, 61, 63, 125, 133 Geometric, 136, 137, 140, 142

H  Hazard Rate, 131

L  Limiting Distribution, 27, 48

Discrete, 128, 141

M  E 

Exponential Distribution, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 122, 124, 126, 130, 135, 136, 142 Extreme Value Distribution, 138

Minimum Variance Linear Unbiased Predictor, 24 Moments, 9, 38, 124, 125, 134, 137, 139, 140

 

          144

Index

O  Order Statistics, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 66, 124, 126, 127, 128, 129, 133, 134, 136, 137, 138, 139, 140

P  Prediction, 46, 61, 122, 127

R  Record Times, 31, 126, 129, 131, 137, 140, 142 Record Values, 31, 33, 35, 37, 38, 39, 41, 43, 45, 46, 47, 48, 49, 99, 121, 122, 123, 124, 125, 126, 127, 128, 131, 132, 134, 135, 136, 137, 138, 139, 140, 141, 142 Recurrence Relations, 137