Exponential Distribution Theory and Methods

8/2/2019 Exponential Distribution Theory and Methods

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MATHEMATICSRESEARCHDEVELOPMENTS

EXPONENTIAL DISTRIBUTION:

THEORY AND METHODS

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MATHEMATICSRESEARCH

DEVELOPMENTS

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MATHEMATICSRESEARCHDEVELOPMENTS

EXPONENTIAL DISTRIBUTION:

THEORY AND METHODS

M.AHSANULLAH

ANDG.G.HAMEDANI

Nova Science Publishers, Inc.

New York


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Copyright 2010 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or

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Telephone 631-231-7269; Fax 631-231-8175Web 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

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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 ADECLARATION 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.

ISBN978-1-61324-566-8 (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


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To Masuda, Nisar, Tabassum, Faruk,

Angela, Sami, Amil and Julian

MA

To Azam , Azita , Hooman , Peter ,

Holly , Zadan and Azara

GGH


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Contents

Preface ix

1. Introduction 1

1.1 Preliminaries 3

2. Order Statistics 11

2.1 Preliminaries and Definitions 11

2.2 Minimum Variance Linear Unbiased Estimators Based

on Order Statistics

18

2.3 Minimum Variance Linear Unbiased Predictors

(MVLUPs)

24

2.4 Limiting Distributions 27

3. Record Values 31

3.1 Definitions of Record Values and Record Times 31

3.2 The Exact Distribution of Record Values 31

3.3 Moments of Record Values 38

3.4 Estimation of Parameters 44

3.5 Prediction of Record Values 46

3.5 Limiting Distribution of Record Values 48

4. Generalized Order Statistics 51

4.1 Definition 51

4.2 Generalized Order Statistics of Exponential Distribution 52


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vi Contents

5. Characterizations of Exponential Distribution I 65

5.1 Introduction 655.2 Characterizations Based on Order Statistics 66

5.3 Characterizations Based on Generalized Order Statistics 86

6. Characterizations of Exponential Distribution II 99

6.1 Characterizations Based on Record Values 99

6.2 Characterizations Based on Generalized Order Statistics 120

References 121

Index 143


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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-bulbs 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 sixchapters. 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


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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


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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, Char-

acterizations 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


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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 al-

lows 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 aboutthe 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-valuedrandom variables; boldface XXX, YYY, ZZZ, . . . denote vector-valued random variables.

Suppose that X is a real-valued continuous random variable for the basic experi-

ment with cumulative distribution function F and the corresponding probability

density function f. We perform n independent replications of the basic exper-

iment to generate a random sample of size n from X: (X1,X2, . . . ,Xn). Theseare independent random variables, each with the same distribution as that of X.

If Xis are exponential random variables with cumulative distribution function

F(x) = 1 ex, x 0, where > 0 is a parameter, then ni=1Xi is distributedas Gamma with parameters n and . The random variable 2 ni=1Xi has a Chi-square distribution with 2 n degrees of freedom. Consider a series system (a

system which works only if all the components work) with independent com-

ponents with common cumulative distribution function F(x) = 1 ex, x 0,and let T be the life of the system. Then P (T > t) = P (min1inXi > t) =


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2 M. Ahsanullah and G.G. Hamedani

P (X1 > t,X2 > t, . . . ,Xn > t) =ni=1 P(Xi > t) = e

nt, which is an exponentialrandom variable with parameter n.

Let N be a geometric random variable with probability mass functionP (N = k) = p qk1, k = 1,2, . . . where p + q = 1. Now if Xi s are indepen-dent and identically distributed with cumulative distribution function F(x) =1 ex , x 0 and ifV = Ni=1 Xi is the geometrically compounded randomvariable, then pV

d= Xi

d= means equal in distribution

. To see this, let L (t)

be the Laplace transform ofV, then

L (t) = EEetV|N =

k=11 +t

k

pqk

1 = 1 + tp

1.

Thus, p Vd= Xi.

Suppose the random variable X has cumulative distribution function F(x) =1 ex, x 0, and Y = [X], the integral part of X, then Y has the geometricdistribution with probability mass function P (Y = k) = pqk, k = 0,1, . . . andp = 1 e,

P (Y = y) = P (y X< y + 1) = F(y + 1) F(y)= ey e(y+1) =

1 e

ey.

Let Xk,n denote the kth smallest of(X1,X2, . . . ,Xn). Note that Xk,n is a func-

tion 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 ex, x 0, then 1 F1,n (x) = P (X1,n x) = enx and Fn,n (x) = P (Xn,n x) =

1 exn .

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


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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 largerthan that of the first one. We can define the lower records similarly by consider-

ing 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 or-

der statistics. In chapter 4, we have presented generalized order statistics of

exponential distribution.

The problem of characterizing a distribution is an important problem whichhas attracted the attention of many researchers in recent years. Consequently,

various characterization results have been reported in the literature. These char-

acterizations 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 defini-

tions, which most of the readers may very well be familiar with them. The read-ers 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 experi-ment means random experiment.


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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;S

b=

{1,2, . . . ,6

}.

Note that the set {even,odd} is also an acceptable sample space for E ofExample 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 arecalled 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 exclusiveevents (subsets) ofS ( or elements ofP(S) ), then

P(A1 A2 ) = P (A1) + P (A2) + .

Definition 1.1.7. A random variable (rv for short) is a real-valued functiondefined on S, a sample space for an experiment E.

We denote rv s with capital letters X,Y,Z, . . . (as mentioned before) andtheir values with lower case letters x,y,z, . . .. Range of a rv X is the set of all

possible values ofX and is denoted by R(X) .


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Definition 1.1.8. A rv X is called

(i) discrete ifR(X) is countable;

(ii) continuous ifR(X) is an interval and P (X = x) = 0, for all x R(X) ;(iii) mixed ifX 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 ofcd f F X :

(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 discontinuitypoints of a distributionfunction

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 > 0, F(x + ) F(x ) > 0. The set of all such pointsis called the support of F and is denoted by Supp F.

We will restrict our attention, throughout this book, to continuous rv s, inparticular exponential rv.

Definition 1.1.12. Let X be a continuous rv with cd f F X.

Then the proba-

bility density function (pd f) ofX (or pd f corresponding to cd f F X) is denotedby fX and is defined by

fX(t) =

ddt

FX (t), if derivative exists,

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 fX(t) = 0 at those points.


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Properties of pd f f X :

(i) fX(t) 0 for all t R;(ii)

R fX (t)dt = 1.

Definition 1.1.14. The rv X has an exponential distribution with location

parameter ( < < ) and scale parameter (> 0) if its cd f is given by

FX (t) =

0, t< ,

1 e(t), t,

where = 1

.

Graph ofFX for = 0 and different values of

It is clear that ddt

FX (t) exists everywhere except at t = , so the correspondingpd f ofFX is given by

fX(t) =

e(t), t> ,0, otherwise,

Figure 1.1. Graph o f FX f o r = 0 and di f f erent values o f .


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Introduction 7

Graph of fX for = 0 and different values of

We use the notation X

E(,) for such a rv. The rv X

E(0,) will be de-

noted by XE(). We use the notation XE(1) for the standard exponentialrandom variable.

Figure 1.2. Graph o f f X f o r = 0 and di f f erent values o f .

We observe that the condition P (X> s + t|X> s) = P (X> t) is equivalentto 1 F(s + t) = (1F(s))(1F(t)). Now, ifX is a non-negative and non-degenerate rv satisfying this condition, then cd f of X will be F(x) = 1 ex,

x 0. To see this, note that condition 1 F(s + t) = (1 F(s))(1 F(t)) willlead to the condition

1 F(nx) = (1 F(x))n , for all n 1 and all x 0,that is, 1 F(x) = 1 F(x

n)n. The solution of this last equation with bound-

ary conditions F(0) = 0 and F() = 1 is F(x) = 1 ex.

The hazard rate (f(x)/(1 F(x))) is constant for E(,). In fact E(,)is the only family of continuous distributions with constant hazard rate. It can

easily be shown that the constant () hazard rate of a continuous cd f F togetherwith boundary conditions F(0) = 0 and F() = 1 imply that F(x) = 1 ex.


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The linear exponential distribution with increasing hazard rate has pd f of

the form

f(x) = (+x) e(x+x2/2)

, ,> 0, x 0,and the corresponding cd f is F(x) = 1 e(x+x2/2) , , > 0, x 0. Thehazard rate is + x. If = 0, then it is the exponential with c d f F (x) =1 ex.

IfX E(), 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 F1 (p). For X E(), we haveF1 (p) = ln(1p) . The first, second and fourth quartiles are 1 ln

43

, ln2 and

ln 4

respectively.

Definition 1.1.15. Let X be a continuous rv with pd f f X, then the rth mo-

ment ofX about the origin is defined by

r = E[Xr] =R

xrfX(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).

Remarks 1.1.16.

(a)

0 = 1,

1 = E [X] is expected value or mean of X . 2

X =

2 21 is

variance ofX and X is standard deviation of X.(b) The rth moment ofX about X =

1 is defined by

r = E[(XX)r] =R

(xX)rfX(x) dx, r= 1,2, . . . ,

provided the right hand side (RHS) exists. Note that 2 = 2

X.

(c) It is easy to show that from rs one can calculate r s and vice versa.

In fact if the moments about any real number a are known, then moments about


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Introduction 9

any other real number b can be calculated from those about a. Moments about

zero, r s , are the most common moments used.

Example 1.1.17. Let X E(). Find all the moments of X which exist.

Solution:

r =

0

xrexdx =(r+ 1)

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

Definition 1.1.18. Let X be a continuous rv with pd f f X. The MGF (Mo-

ment Generating Function) of X denoted by MX (t) is defined by

MX (t) = E

etX

=

R

etx fX(x) dx,

for those t s for which the RHS exists.

Properties ofMGF :

(i) MX(0) = 1;

(ii) M(r)

X (0) =

r, r = 1,2, . . ., where M(r)

X (0) is the rth derivative of theMGF evaluated at 0.

Example 1.1.19. For X E(,), the MGF is

MX (t) =

etxe(x)dx = et

0e(t)xdx = et( t)1 , if t< ,

from which we obtain

1 = M(1)

X (0) = +1,

2 = M(2)

X (0) = 2 +

2

+

2

2.

So, X = +1 ,

2X =

2 + 2 +22

( + 1 )2 = 12 and X = 1 .

For X E(), MX(t) = ( t)1, i f t< and

M(r)X (t) = (r!) ( t)(r+1) , for r= 1,2, . . . ,


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Then

r =r!r

= r

r1, which is a recurrence relation for the moments of E().

The nth

cumulant of a rv X is defined by Kn =dn

dtn ln MX(t) | t=0. Here ln is used for natural logarithm. For X E(), MX(t) = ( t)1, t< andKn = (n)/

n.

Remarks 1.1.20. If X1,X2, . . . ,Xn form an independent sample from anexponential distribution with parameter , then

(i) method of moments estimator of is = 1X

, where X = 1n

ni=1 Xi;

(ii) maximum likelihood estimator of is also = 1X

;

(iii) entropy of is 1 ln .For E(,), the maximum likelihood estimators of and

are given by = X1,n and = 1/XX1,n

respec-

tively, where, as mentioned before, X1,n = min{X1,X2, . . . ,Xn}. The entropy ofE(,) denoted by EEEX is

EEEX =

( ln f(x)) f (x) dx =

lne(x)

e(x)dx = 1 ln,

which does not depend on location parameter . It is the same as the entropy of

the exponential distribution E().


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Chapter 2

Order Statistics

2.1. Preliminaries and Definitions

Let X1,X2, . . . ,Xn be n independent and identically distributed (i.i.d.) rvs with

common c d f F and pd f f . Let X1,n X2,n Xn,n denote the orderstatistics corresponding to 1,X2, . . . ,Xn. We call Xk,n, 1

k

n, the kth or-

der statistic based on a sample X1,X2, . . . ,Xn. The joint pd f of order statisticsX1,n,X2,n, . . . ,Xn,n has the form

f1,2,...,n:n (x1,x2, . . . ,xn)

=

n!nk=1 f(xk) , < x1 < x2 < < xn < ,0, otherwise.

(2.1.1)

Let f k:n denote the pd f ofXk,n. From (2.1.1) we have

fk:n (x)

=

. . .

f1,2,...,n:n (x1, . . . ,xk1,x,xk+1, . . . ,xn) dx1 dxk1dxk+1 dxn

= n!f(x)

. . .

k1j=1

f(xj)n

j=k+1

f(xj)dx1 dxk1dxk+1 dxn, (2.1.2)

where the integration is over the domain

< x1 < < xk1 < xk+1 < < xn < .


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The symmetry ofk1j=1 f (xj) with respect to x1, . . . ,xk1 and that ofnj=k+1 f

(xj) with respect to xk+1, . . . ,xn help us to evaluate the integral on the RHS of

(2.1.2) as follows:

. . .

k1j=1

f(xj)n

j=k+1

f(xj)dx1 dxk1dxk+1 dxn

=1

(k1)!k1j=1

x

f(xj) dxj1

(n k)!n

j=k+1

x

f(xj) dxj

= (F(x))k1 (1

F(x))nk/(k

1)! (n

k)!. (2.1.3)

Combining (2.1.2) and (2.1.3), we arrive at

fk:n (x) =n!

(k1)! (n k)! (F(x))k1 (1F(x))nkf(x) . (2.1.4)

Clearly, equality (2.1.3) immediately follows from the corresponding formulafor cd fs 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)

=n!

r+1j=1 (k(j)k(j 1) 1)!

r+1

j=1

(F(xj)

F(xj

1))

k(j)k(j1)1 rj=1

f(xj) ,

if x1 < x2 < < xr,

and = 0, otherwise.In particular, if r= 2, 1 i < j n, and x1 < x2, then

fi,j:n (x1,x2) =n!

(i 1)! (j i 1)! (n j)! (F(x1))i1 [F(x2)F(x1)]ji1 [1 F(x2)]nj f(x1) f(x2) .


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Order Statistics 13

The conditional pd f ofXj,n given Xi,n = x1 is

fj|i,n

(x2|x1)

=(n i)!

(j i 1)! (n j)!

F(x2)F(x1)1 F(x1)

ji11 F(x2)1 F(x1)

njf(x2)

1 F(x1).

Thus, Xj,n given Xi,n = x1 is the (j i)th order statistic in a sample of n i fromtruncated distribution with cd f F c (x2|x1) = F(x2)F(x1)1F(x1) . For F(x) = 1 ex,x 0, we will have Fc (x2|x1) = 1 e(x2x1),x2 x1.

If X

E(1) and Z1,n

Z2,n

Zn,n are the n order statistics corre-

sponding to a sample of size n from X, then it can be shown that the joint pd fofZ1,n, Z2,n, . . . ,Zn,n is

f1,2,...,n (z1,z2, . . . ,zn) =

n!e(

ni=1zi), 0 z1 z2 zn


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and

Cov (Zk,n,Zs,n) =

k

i=1Var Win i + 1 =k

i=11

(n i + 1)2 , k s.

Furthermore, letting ki,n =EXki,n

,k 1, n 1, then we have the following

theorems (see, Joshi, (1978)).

Theorem 2.1.1. k1,n =knk11,n , k 1, n 1.

Proof.

k1,n =

0

xknenxdx = xkenx|0 +

0kxk1enxdx =

k

nk11,n .

Theorem 2.1.2. ki,n = ki1,n1 +

knk1i,n , k 1, 2 i n.

Proof. For k 1 and 2 i n,

k

1

i,n =

n!

(i 1)! (n i)!

0xk

1 1 exi1 e(ni+1)xdx.

Integrating by parts, we obtain

k1i,n =n!

(i 1)! (n i)!k[

0(n i + 1)xk1 exi1 e(ni+1)xdx

0(i 1)xk1 exi2 e(ni+2)xdx]

=n!

(i 1)! (n i)!k[n

0 xk1 exi1 e(ni+1)xdx

(i 1)

0xk

1 exi2 e(ni+1)xdx]

=n

kki,n

n

kki1,n1.

Thus,

ki,n = ki

1,n

1 +

k

n

k1i,n .


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Order Statistics 15

Theorem 2.1.3. Leti,j,n = E[Xi,nXj,n], 1 i < j n, then

i,i+1,n = 2i,n +1

n ii,n, 1 i n 1,and

i,j,n = i1,j,n +1

n j + 1i,j1,n, 1 i < j n, j i 2.

Proof.

i,n = EXi,nX0i+1,n=

n!

(i 1)! (n i + 1)!

0

xi

xi

1 exii1 exi e(ni)xi+1 dxi+1dxi=

n!

(i 1)! (n i + 1)!

0xi

1 exii1 exiIxi dxi,where

Ixi =

xi e(n

i)xi+1

dxi+1 = xi+1e(n

i)xi+1

|

xi + (n i)

xi xi+1e(n

i)xi+1

dxi+1.

Thus,

i,n =n! (n i)

(i 1)! (n i + 1)!

0

xi

xixi+1

1 exii1 exi e(ni)xi+1 dxi+1dxi n!

(i 1)! (n i + 1)!

0x2i

1 exi

i1

e(ni+1)xi dxi

= (n i)i,i+1,n (n i)2i,n.

Upon simplification, we obtain

i,i+1,n = 2i,n +

1

n ii,n, 1 i n 1.

For j > i + 1,

i,j1,n = EXi,nX0j1,n n!(i 1)! (j i 1)! (n i + 1)!


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0

xi

xi

1 exii1 (exi exj )ji1exi e(nj+1)xj dxjdxi=

n!

(i 1)! (n i 1)! (n j)!

0xi 1 exii1 exiJxi dxi,

where

Jxi =

xi

(exi exj )ji1e(nj+1)xj dxj

= (n j + 1)

xi

xj(exi exj )ji1e(nj+1)xj dxj

(j i 1)

xi(exi exj )ji2e(nj+2)xj dxj.

Thus,

i,j1,n =n!

(i 1)! (j i 1)! (n j)!

0xi

1 exii1 exi[(n j + 1)

xi

xj (exi exj )ji1e(nj+1)xj dxj

(j i 1)

xi(exi exj )ji2e(nj+2)xj dxj]dxi

= (n j + 1)i,j,n (n j + 1)i1,j,n.Upon simplification, we arrive at

i,j,n = i1,j,n +1

n j + 1i,j1,n, 1 i < j n, j i 2.

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

it can be shown that

Xi,nd= lnUni+1,n or equivalently Xi,n d= ln (1 Ui,n) .

Let

f1,...,r1,r+1,...,n|r(x1, . . . ,xr1,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


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Order Statistics 17

v, where fr:n ,as usual, denotes the pd f ofXr,n. The standard procedure gives us

the required pd f:

f1,...,r1,r+1,...,n|r(x1, . . . ,xr1,xr+1, . . . ,xn|v)= f1,2,...,n:n (x1, . . . ,xr1,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,...,r1,r+1,...,n|r(x1, . . . ,xr1,xr+1, . . . ,xn|v)

= (r1)!r1

j=1f(xj)

F(v) (n j)!n

j=r+1f(xj)

1 F(v) ,x1 < < xr1 < xr+1 < < xn, (2.1.7)

and equal zero otherwise.

Finally, we would like to present Fishers Information, I, for the order statis-

tics from E(). Fishers Information for a continuous random variable X withpd f f(x,) and parameter , under certain regularity conditions, is given by

I = E2

2ln(f(X,))

.

The exponential distribution E() satisfies the regularity conditions and theFishers Information for order statistics from this distribution are as follows:

For X1,n,

I1 = E 22 lnnenX = 12 .For X2,n,

I2 = E2

2ln{n(n 1)(1 eX)e(n1)X}

= E 12 + X2eX(1 eX)2


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=

0

1

2+

x2ex

(1

ex)2

n(n 1)(1 ex)e(n1)xdx

=1

2+

0n(n 1) x

2

1 ex enxdx

=1

2+

2n(n 1)2

k=0

1

(n + k)3.

For Xr,n, r> 2,

Ir =

E 2

2ln n!(r1)!(n r)!(1eX)r1e(nr+1)X

= E

1

2+

(r1)X2eX(1 eX)2

=

0

1

2+

(r1)X2eX(1 eX)2

n!

(r1)!(n r)!(1 ex)r1e(nr+1)xdx

=1

2+

n!

(r

2)!(n

r)!

0

x2(1 ex)r3e(nr+2)xdx

=1

2+

n(n r+ 1)(r2)

(n 1)!(r3)!(n r+ 1)!

0

x2(1ex)r3e(nr+2)xdx

=1

2+

n(n r+ 1)(r2)2 E[X

2r2,n]

=1

2

1 +

n(n r+ 1)(r2)

r3

k=0

1

(n k)2 + (r3

k=0

1

n k)2

.

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

F(x) , < < , > 0.


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Order Statistics 19

Further, assume that

E[Xr,n] = +r, r= 1,2, . . . ,n,Var[Xr,n] = vrr

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

Cov (Xr,n,Xs,n) = Cov (Xs,n,Xr,n) = vrs2, 1 r< s n.

Let

XXX = (X1,n,X2,n, . . . ,Xn,n) .

We can write

E[XXX] = 111 +++, (2.2.1)

where

111 = (1,1, . . . ,1) , = (1,2, . . . ,n)

,

and

Var(XXX) = 2,

where is a matrix with elements vrs,1 r s n.Then the MVLUEs of the location and scale parameters and are

=1

1(111 111)1

X, (2.2.2)

and

=1

111

1(1111)1

X, (2.2.3)

where

=

1

1111111 1112 .

The variances and covariance of these estimators are given by

Var() =2

1

, (2.2.4)

Var() =2 1111 111 , (2.2.5)


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and

Cov (,) =

2

1 111

. (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 = (rs) be an nn matrix with elements, which satisfythe relation

rs = sr = crds, 1 r,s n, for some positive numbers c1,c2, . . . ,cn and d1,d2, . . . ,dn. Then its inverse

1

= (rs

) has elements given as follows:

11 = c2/c1 (c2d1 c1d2) ,nn = dn1/dn (cndn1 cn1dn) ,

k+1k = kk+1 = 1/(ck+1dkckdk+1) ,kk = (ck+1dk1 ck1dk+1)/(ckdk1 ck1dk) (ck+1dkckdk+1),

k= 2,3, . . . ,n 1,and

i j = 0, if |i j| > 1.

Let

f(x) =

1

exp((x)/), < < x < , 0 < < ,

0, otherwise.

We have seen that

E[Xr,n] = +r

j=1

1

n j + 1

Var[Xr,n] = 2

r

j=1

1

(n j + 1)2 , r= 1,2, . . . ,n,

and

Cov (Xr,n,Xs,n) = 2

r

j=11

(n j + 1)2 , 1 r s n.


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Order Statistics 21

One can write that

Cov (Xr,n,Xs,n) =

2

crds, 1 r s n,where

cr =r

j=1

1

(n j + 1)2 , 1 r n,

and

ds = 1, 1 s n.Using Lemma 2.2.1, we obtain

j j = (n j)2 + (n j + 1)2 , j = 1,2, . . . ,n,j+1j = j j+1 = (n j)2 , j = 1,2, . . . ,n 1,

and

i j = 0, if |i j| > 1, i, j = 1,2, . . . ,n.It can easily be shown that

1111 = n2,0,0, . . . ,0 , 1 = (1,1, . . . ,1)

and

= n2 (n 1) .The MVLUEs of the location and scale parameters and are respectively

=nX1,n X

n 1, (2.2.7)

and

=nXX1,n

n 1 , (2.2.8)

where X = nr=1Xr,n

n.

The corresponding variances and covariance of the estimators are

Var[] =

2

n (n 1) , (2.2.9)


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Var[] =2

n 1 , (2.2.10)

andCov (, ) =

2

n (n 1) . (2.2.11)

The remainder of this section will be devoted to MVLUEs based on cen-

sored samples. We consider the case, when some smallest and largest obser-

vations 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

Xr1+1,n Xnr2,n.We will consider here the MVLUEs of the location and scale parameters

based on Xr1+1,n, . . . ,Xnr2,n.Suppose that X has an absolutely continuous cd f F of the form

F((x)/) , < < , > 0.Further, we assume that

E[Xr,n] = +r,

Var[Xr,n] = vrr2, r1 + 1 r n r2,

Cov (Xr,n,Xs,n) = vrs2, r1 + 1 r, s n r2.

Let XXX = (Xr1+1,n, . . . ,Xnr2,n), then we can write

E

XXX

= 111 +++,

with 111 === (1,1, . . . ,1) , === (r1+1, . . . ,nr2), and

VarXXX

= 2,

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 and based on theorder statistics XXX are

= 11(111 111)1X, (2.2.12)


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Order Statistics 23

and

=1

111

1(1111)1X, (2.2.13)where

=

1

1111

111

11112

.

The variances and covariance of these estimators are given as

Var

=2

1

, (2.2.14)

Var = 2

1111 111 , (2.2.15)

and

Cov

,

= 21 111

. (2.2.16)

Now, we consider the exponential distribution with cd f F as

F(x) = 1 exp{(x )/} , < < x < , 0 <


38/158


and

Cov , = r1+1

2

n r2 r1 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.

2.3. Minimum Variance Linear Unbiased Predictors

(MVLUPs)

Suppose that X1,n, X2,n, . . . ,Xr,n are r(r< n) order statistics from a distributionwith location and scale parameters and respectively. Then the best (in thesense of minimum variance) linear predictor Xs,n ofXs,n (r< s n) is given by

Xs,n = +s+WWWsVVV

1 (XXX 111 ),where and are MVLUEs of and respectively, based on

XXX = (X1,n,X2,n, . . . ,Xr,n) ,

s = E[(Xs,n )/] ,and

WWWs = (W1s,W2s, . . . ,Wrs) ,

where

Wjs = Cov (Xj,n,Xs,n) , j = 1,2, . . . , r.

Here VVV1 is the inverse of the covariance matrix of XXX.

Suppose that for the exponential distribution with cd f

F(x) = 1 exp{(x )/} , < < x < , 0 < < ,all the observations were available. We recall that

E[Xr,n] = +r

j=1

1

n j + 1 ,

Var[Xr,n] = 2

r

j=11

(n j + 1)2, r= 1,2, . . . ,n,


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Order Statistics 25

and

Cov (Xr,n,Xs,n) = 2

r

j=1

1

(n j + 1)2, 1

r

s

n.

To obtain MVLUEs for the case, when r1 + r2 observations are lost, weneed to deal with the covariance matrix of size (n r1 r2) (n r1 r2),elements of which coincide with

Cov (Xr,n,Xs,n) = 2crds, r1 + 1 r s n r2,

where

cr =

r

j=11

(n j + 1)2 ,and

ds = 1.

We can obtain the inverse matrix 1 using Lemma 2.2.1 as

1

=

(n

r1 1)

2

+ 1/c

r1+1 (n

r1 1)

2

. . . 0(n r1 1)2 (n r1 2)2 + (n r1 1)2 . . . 00 (n r1 2)2 . . . 00 0 . . . 0...

... . . . 0

0 0 . . . (r2 + 1)20 0 . . . (r2 + 1)

2

,

where

11 = (n r1 1)2 + 1/cr1+1,nr1r2 nr1r2 = (r2 + 1)

2 ,

j j = (n r1 j)2 + (n r1 j + 1)2 , j = 2,3, . . . ,n r1 r2 1,j+1j = j j+1 = (n r1 j)2 , j = 1,2, . . . ,n r1 r2 1,

and

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


40/158


Note also that we have

=== (r1+1, . . . ,nr2) ,

where

r = E[(Xr,n )/] =r

j=1

1

n j + 1 .

Simple calculations show that

1

= r1+1c

r1+1 (n

r1

1) ,1,1, . . . ,1, r2 + 1 ,

1 ===

2r1+1cr1+1

+ (n r1 r2 1) ,

1

111 = r1+1/cr1+1,

1111

111 = 1/cr1+1,

1111 === r1+1/cr1+1,

1111

1111

=1

cr1+1 r1+1cr1+1 (n r1 1) ,1,1, . . . ,r2 + 1 ,111

1111

1=

1

cr1+1

r1+1cr1+1

,0,0, . . . ,0

,

=

1

1111

111

11112

= (n r1 r2 1)/cr1+1.

Upon simplification, we obtain

= 11111 1111 11111111X

=1

n r2 r1 1

nr2

j=r1+1

Xj,n (n r1)Xr1+1,n + r2Xnr2,n.

Analogously, from (2.2.12) and (2.2.14)(2.2.16) we have the necessary ex-

pressions for , Var

, Var

, and Cov

,

.


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Order Statistics 27

2.4. Limiting Distributions

Let X1,X2, . . . ,Xn be i.i.d. exponentially distributed rvs with cd f F (x) = 1 ex, x 0. Then with a sequence of real numbers an = ln n and bn = 1, we have

P (Xn,n an + bnx) = P (Xn,n ln n +x) =

1e(ln n+x)n

=

1 e

x

n

n eex as n .

Thus the limiting distribution of Xn,n with the constant an = ln n and bn = 1

whenXjs

areE

(1) is type 1 extreme value distribution. The numbersa

n andb

nare known as normalizing constants.

Remark 2.4.1. We know that if Y has type 1 extreme value distribution,

then E[Y] = , the Euler constant. Thus E[Xn,n] ln n , as n . ButE[Xn,n] =

nj=1

1nj+1 , so we have the known result,

nj=1

1nj+1 ln n , as

n .

For the derivation of the limiting distribution of X1,n, we need the following

lemma.

Lemma 2.4.2. Let(Xn)n1 be a sequence of i.i.d. rvs with cd f F. Consider

a sequence (en)n1 of real numbers. Then for any , 0 < , the followingtwo statements are equivalent:

(a) limn (nF(en)) = ;

(b) limnP (X1,n > en) = e.

Since limnn1 e xn = ex, ifXjs are i.i.d.E(1), thenlim

nP

X1,n >x

n

= ee

x

.

Thus, the limiting distribution of X1,n with constants cn = 0 and dn = 1/n

when Xjs are i.i.d. E(1) is type 3 extreme value distribution. Here again thenumbers 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.


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Theorem 2.4.3. Let X1,X2, . . . ,Xn be n i.i.d. rvs with cd f F and Xnk+1,n

be their(n k+ 1)th order statistic. If for some stabilizing constants a n and bn(an + bnx as n ), F

n

(an + bnx) T(x) as n , for all x, for somedistribution T (x), then

P (Xnk+1,n an + bnx) k1

j=0

T(x) ( ln T(x))j /j! as n ,

for any fixed k and all x.

Proof. Let us consider a sequence (cn)n

1 such that as n

, cn

c. Then

limn 1 cnn n = ec. Take cn (x) = n (1 F(an + bnx)). Now, for every fixedx,

P (Xnk+1,n an + bnx) =n

j=nk+1

n

j

(F(an + bnx))

j (1F(an + bnx))nj

=k1

j=0

n

j

(cn (x)/n)

j (1 cn (x)/n)nj .

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. SinceFn (an + bnx) T(x), as n , we have

n ln [1 (1 F(an + bnx))] T(x) , as n .

Thus, for sufficiently large n, we have

n ln [1 (1 F(an + bnx))] = n (1 F(an + bnx))= cn (x) T(x) , as n ,

from which we obtain limn cn (x) = T(x) uniformly in x. Now, using Pois-son approximation to binomial, we arrive at


j=0

T(x) ( ln T(x))/j!, as n , for all x.


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Order Statistics 29

For the special case ofi.i.d.E(1) rvs with an = ln n and bn = 1, we haveFn (an + bnx) eex , as n , for all x 0. We will then have


j=0

ejx

j!ee

x

, as n , for all x 0.

The asymptotic distribution of Xk,n for fixed k as n is given by thefollowing 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. rvs with cd f F and Xk,n

be their kth order statistic. If for some stabilizing constants a n and bn(an + bnx 0 as n ) , Fn (an + bnx) G (x), as n , for all x, for somedistribution G (x), then

P (Xk,n > an + bnx) k1

j=0

G (x)

ln G (x)jj!

, as n , for any fixed kand all x.

Again, for the special case ofi.i.d. E(1) rvs with an = 0 and bn = 1/n,we have F

n(an + bnx) ex as n . But, in this case Fn

0 + 1

nx

= ex forall n, and hence we will have

P (Xk,n > an + bnx) =k1

j=0

G (x)

ln G (x)jj!

=k1

j=0

exxj

j!, for all x and all n.


44/158


45/158

Chapter 3

Record Values

3.1. Definitions of Record Values and Record Times

Suppose that (Xn)n1 is a sequence of i.i.d. rvs with c d f F . Let Yn =

max (min){Xj|1 j n} for n 1. We say Xj is an upper (lower) recordvalue of

{Xn

|n

1

}, if Xj > ( 1. By definition X1 is an upper

as well as a lower record value. One can transform the upper records tolower records by replacing the original sequence of (Xn)n1 by (Xn)n1 or(ifP (Xn > 0) = 1 for all n) by (1/Xn)n1; the lower record values of this se-quence will correspond to the upper record values of the original sequence.

The indices at which the upper record values occur are

given by the record times {U(n) ,n 1}, where U(n) =min

j|j > U(n 1) ,Xj > XU(n1),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 distributionof U(n), n 1 does not depend on F. We will denote L (n) as the indiceswhere 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 ofXU(n)


46/158


for n 1, then we haveF

1(x) = PXU(1) x = F(x) ,

F2 (x) = PXU(2) x

=

x

y

j=1

(F(u))j1 dF(u) dF(y)

=

x

y

dF(u)

1 F(u)dF(y) =x

R (y) dF(y) ,

(3.2.1)

where R (x) = ln (1 F(x)), 0 < F(x) < 1.IfF has a pd f f, then the pd f ofXU(2) is

f2 (x) = R (x) f(x) . (3.2.2)

The cd f

F3 (x) = PXU(3) x

=

x

y

j=0

(F(u))jR (u) dF(u) dF(y)

= x

y

R (u)

1 F(u)dF(u) dF(y) =

x

(R (u))2

2!dF(u) . (3.2.3)

The pd f f 3 ofXU(3) is

f3 (x) =(R (x))2

2!f(x) . (3.2.4)

It can similarly be shown that the cd f F n ofXU(n) is

Fn (x) = x

(R (u))n1

(n 1)!dF(u) ,

< x < . (3.2.5)

This can be expressed as

Fn (x) =

R(x)

un1

(n 1)! eudu, < x < ,

and

Fn (x) = 1

Fn (x) = F(x)n1

j=0(R (x))j

j!= eR(x)

n1

j=0(R (x))j

j!.


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Record Values 33

The pd f f n ofXU(n) is

fn (x) = (R (x))

n

1

(n 1)! f(x) , < x < . (3.2.6)

Note that Fn (x)Fn1 (x) = F(x)f(x) fn (x), and for E(), Fn (x)Fn1 (x) =n1 xn1

(n) ex.

A rv X is said to be symmetric about zero if X and X have the samedistribution function. If f is their pd f, then f(x) = f(x) for all x. Tworvs X and Y with cd fs F and G are said to be mutually symmetric if F(x) =

1 G (x) for all x, or equivalently if their corresponding pd fs f and g exist,then f (x) = g (x) for all x. If a sequence ofi.i.d. rvs are symmetric aboutzero, 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)n1 ofi.i.d. rvs, 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)

=n1j=1

r(xj) f(xn) , < x1 < x2 < < xn1 < xn < , (3.2.7)

where, as before,

r(x) =d

dxR (x) =

f(x)

1 F(x) , 0 < F(x) < 1.

The joint pd f ofXU(i) and XU(j) is

fi j (xi,xj) =(R (xi))

i1

(i 1)! r(xi)[R (xj)R (xi)]ji1

(j i 1)! f (xj ) ,

for < xi < xj < . (3.2.8)In particular, for i = 1 and j = n we have

f1n (x1,xn) = r(x1)[R (x

n)

R (x1

)]n2

(n 2)! f(xn) , for < x1 < xn < .


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

f(xj|xi) = fi j (xi,xj)fi (xi)

=[R (xj)R (xi)]ji1

(j i 1)! f(xj)

1 F(xi) , for < xi < xj < . (3.2.9)

For j = i + 1

f(xi+1|xi) = f(xi+1)1

F(xi)

, for < xi < xi+1 0, 1 k< m, the joint conditional pd f of XU(i+k) and XU(i+m) givenXU(i) is

f(i+k)(i+m)x,y|XU(i) = z

=

1

(m k) 1

(k)[R (y)R (x)]mk1 [R (x) R (z)]k1 f (y) r(x)

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),0 < F(u) < 1 and h (u) = d

duH(u), then

PXL(n) x

=

x

(H(u))n1

(n 1)! dF(u) , (3.2.11)

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

f(n) (x) =(H(x))n1

(n 1)! f(x) . (3.2.12)

The joint pd f ofXL(1), XL(2), . . . ,XL(m) can be written as

f(1)(2)...(m) (x1,x2, . . . ,xm)

= m1j=1 h (xj) f (xm) , < xm < xm1 < < x1


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Record Values 35

The joint pd f ofXL(i) and XL(j) is

f(i)(j) (x,y) = (H(x))i

1

(i 1)! [H(y)H(x)]j

i

1

(j i 1)! h (x) f(y) ,j > i and < y < x 0) are parameters.The corresponding cd f F and the hazard rate rof the rv X with pd f (3.2.15)

are respectively

F(x) = 1 exp1 (x) , x ,and

r(x) = f(x)/(1 F(x)) = 1. (3.2.16)Again, as before, we will denote the ex-

ponential distribution with pd f (3.2.15) withE(,), the exponential distribution ( = 0, = 1/) with E(), and the

standard exponential distribution with E(1) .For E(,),the joint pd f ofXU(m) and XU(n), m < n is

fm,n (x,y)

=

n

(m) (x )m1(yx)nm1(nm) exp

1 (y) , x < y


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It can be shown that XU(m)d= XU(m1) +U, (m > 1) where U is independent

of XU(m) and XU(m1) and is identically distributed as X1 if and only if X1 E(). For E(1)with n 1,

PXU(n+1) > wXU(n)

=

0

wx

xn1

(n)eydydx

=

0

xn1

(n)ewxdx = wn.

The conditional pd f ofXU(n) given XU(m) = x is

f (y|x) = mn (yx)nm1(nm) exp1 (yx) , x < y < ,0, otherwise.

(3.2.18)

Thus, P

XU(n)XU(m)

= y|XU(m) = x

does not depend on x. It can be shown

that if = 0, then XU(n)XU(m) is identically distributed as XU(nm), m < n.We take = 0 and = 1 and let Tn =

nj=1XU(j). Since

Tn = XU(n)

XU(n

1) + 2XU(n1)XU(n2)+ + (n 1)XU(2)XU(1)+ nXU(1)

=n

j=1

jWj,

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

n (t) =n

j=1

1

1

jt. (3.2.19)

Inverting (3.2.19), we obtain the pd f f Tn ofTn as

fTn (u) =n

j=1

1

(j) (1)

nj

(n j + 1) eu/jjn2. (3.2.20)

Theorem 3.2.1. Let (Xn)n1 be a sequence of i.i.d. rvs with the standard

exponential distribution. Suppose j =XU(j)

XU(j+1)

, j = 1,2, . . . ,m

1, then s are

independent.


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Record Values 37

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

f(x1,x2, . . . ,xm) = m1j=1

xjexm , 0 < x1 < x2 < < xm


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3.3. Moments of Record Values

Without any loss of generality we will consider in this section the standard ex-ponential distribution E(1), with pd f f (x) = exp(x), x > 0, for which wealso have f (x) = 1 F(x). We already know that XU(n) ,in this setting, can bewritten as the sum of n i.d. rvs V1,V2,..,Vn with common distribution E(1).Further, we have also seen that

EXU(n)

= n,

Var

XU(n)

= n,

and

CovXU(n),XU(m)

= m, m < n. (3.3.1)

For m < n,

EX

p

U(n)X

q

U(m)

=

0

u0

1

(m) 1(n m) u

qexvm+p1 (u v)nm1 dvdu.

Substituting tu = v and simplifying we get

EX

p

U(n)X

q

U(m)

=

0

0

1

(m) 1(n m) u

n+p+q1extm+p1 (1 t)nm1 dtdu

=(m +p)(n +p + q)

(m)(n +p).

Using (3.2.19), it can be shown that for Tn = nj=1XU(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, . . .

EXk+1U(n) = EXk+1U(n1)+ (k+ 1)EXkU(n) , (3.3.2)


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Record Values 39

and consequently, for0 m n 1 we can write

EXk+1U(n) = EXk+1U(m)+ (k+ 1) nj=m+1EXkU(j) , (3.3.3)with E

Xk+1

U(0)

= 0 and E

X0

U(n)

= 1.

Proof. For n 1 and k= 0,1, . . . , we have

EXk

U(n) =1

(n)

0

xr(R (x))n1 f(x) dx

=1

(n)

0

xk(R (x))n1 (1 F(x))dx, (f(x) = 1 F(x)) .

Upon integration by parts, treating xk for integration and the rest of the integrand

for differentiation, we obtain

E

XkU(n)

= 1(k+ 1)(n)

0

xk1 (R (x))n1 f (x) dx (n 1)

0xk+1 (R (x))n2 f(x) dx

=1

k+ 1

0

xk+11

(n)(R (x))n1 f(x) dx

0

xk+11

(n 1) (R (x))n2

f(x) dx

=

1

k+ 1

EXk+1

U(n)

E

Xk+1

U(n1)

,

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 wayto 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 relationsfor the product moments of record values.

Theorem 3.3.3. For m 1 and p, q = 0,1,2, . . .

EXpU(m)Xq+1U(m+1) = EXp+q+1U(m) + (q + 1)EXpU(m)XqU(m+1) , (3.3.4)


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and for1 m n 2, p,q = 0,1,2, . . .

EXpU(m)Xq+1U(n) = EXpU(m)Xq+1U(n1)+ (q + 1)EXpU(m)XqU(n) , (3.3.5)Proof. Let us consider 1 m < n and p, q = 0,1,2, . . .

EX

p

U(m)Xq

U(n)

=

1

(m)(n m)

0xp (R (x))m1 r(x)I(x) dx, (3.3.6)

where

I(x) =

x

yq [R (y)

R (x)]nm1 f(y) dy

=

xyq [R (y) R (x)]nm1 (1 F(y))dy, since f(y) = 1 F(y) .

Upon performing integration by parts, treating yq for integration and the rest of

the integrand for differentiation, we obtain, when n = m + 1, that

I(x) =1

q + 1

x

yq+1f (y) dy xq+1 (1 F(x)),

and when n m + 2, that

I(x) =1

q + 1

x

yq+1 {R (y)R (x)}nm1 f(y) dy

(n m1)

xyq+1 {R (y)R (x)}nm2 f(y) dy

.

Substituting the above expression of I(x) in equation (3.3.6) and simplifying,

we obtain, when n = m + 1 that

EX

p

U(m)X

q

U(m+1)

=

1

q + 1

EX

p

U(m)X

q+1U(m+1)

E

X

p+q+1U(m)

,


EX

p

U(m)Xq

U(n)

=

1

q + 1

E

X

p

U(m)Xq+1

U(n)

E

X

p

U(m)Xq+1

U(n1)

.

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


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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

EX

p

U(m)Xq+1

U(n)

= E

X

p+q+1U(m)

+ (q + 1)

n

j=m+1

X

p

U(m)Xq

U(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

EXU(m)XU(n)

= E

X2U(m)

+ (n m)EXU(m) . (3.3.8)

Similarly, by setting p = 0 in (3.3.3), we obtain

EXU(n)

= E

XU(m)

+ (n m) , n > m. (3.3.9)

With the help of (3.3.8) and (3.3.9), we get for n m + 1

CovXU(m),XU(n) = EXU(m)XU(n)EXU(m)EXU(n)= E

X2U(m)

+ (n m)EXU(m)EXU(m)2 (n m)EXU(m)

= VarXU(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

EXpU(m)Xq+1U(m+1) =q+1

j=0 (q + 1)(j)

EXp+q+1jU(m) ,and for1 m n 2

EX

p

U(m)Xq+1

U(n)

=

q+1

j=0

(q + 1)(j)EX

p

U(m)Xq+1j

U(n1),

where

(q + 1)(0) = 1 and (q + 1)(j) = (q + 1) q (q + 1 j + 1) , for j 1.


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Remark 3.3.7. The recurrence relations (3.3.4) and (3.3.5) can be used ina 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, . . . ,

EX

p+1U(m1)X

q

U(m)

= E

X

p+q+1U(m)

(p + 1)E

X

p

U(m)Xq

U(m+1)

, (3.3.10)

and for2 m n 2 and p, q = 0,1,2, . . . ,

E

X

p+1U(m1)X

q

U(n1)

= E

X

p+1U(m)X

q

U(n1)

(p + 1)E

X

p

U(m)Xq

U(m+1)

.

(3.3.11)

Proof. For 2 m n and p, q = 0,1,2, . . . ,

EX

p

U(m)X

q

U(n)

=

0

0

xpyqfmn (x,y)dxdy

=1

(m 1)! (n m1)!

0yqf(y)J(y) dy, (3.3.12)

where

J(y) =

y0

xp { ln(1 F(x))}m1

{ ln (1 F(x)) + ln (1 F(y))}nm1 f (x)1 F(x) dx

=

0xp { ln (1 F(x))}m1 { ln(1 F(x)) + ln (1 F(y))}nm1 dx,

since f(x) = 1 F(x). Upon integration by parts, treating xp

for integrationand the rest of the integrand for differentiation, we obtain, for n = m + 1, that

J(y) =1

p + 1

yp+1 { ln(1 F(y))}m+1

(m 1)

y0

xp+1 {ln (1 F(x))}m2 f(x)1 F(x)dx,


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Record Values 43


J(y) =1

p + 1 [(n m 1)y

0x

p+

1

{ ln (1 F(x))}m

1 f (x)

1 F(x){ ln (1 F(y)) + ln (1 F(x))}nm2 dx

(m 1)y

0xp+1 { ln(1 F(x))}m2 f(x)

1 F(x){ ln (1 F(y)) + ln (1 F(x))}m2 dx].

Now, substituting the above expression of J(y) in equation (3.3.12) and simpli-

fying, we obtain, for n = m + 1, that

EX

p

U(m)Xq

U(n)

=

1

p + 1

E

X

p+q+1U(m)

E

X

p+1U(m1)X

q

U(m)

,

and for n m + 2 that

EX

p

U(m)Xq

U(n)

=

1

p + 1

EX

p+1U(m)X

q

U(n)

E

X

p+1U(m1)X

q

U(n1)

.

The recurrence relations (3.3.10) and (3.3.11) follow readily when the abovetwo 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, . . .

EX

p+1U(m1)X

q

U(n1)

= EX

p+q+1U(n1)

(p + 1)

n1

j=m

EX

p

U(j)Xq

U(n)

.

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

EX

p+1U(m1)X

q

U(m)

=

p+1

j=0

(1)j (p + 1)(j)EX

p+q+1jU(m+j)

,

and for2 m n2 that

EXp+1U(m1)XqU(n1) = p+1j=0 (1)j (p + 1)(j)EXp+1jU(mj)XqU(n+1j) ,


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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 and .

(a) Minimum Variance Linear Unbiased Estimator (MVLUE)

Suppose XU(1),XU(2), . . . ,XU(m) are the m record values from an i.i.d. se-quence of rvs with common c d f E (,). Let Yi = 1

XU(i)

, i =

1,2, . . . ,m. Then

E[Yi] = i = Var[Yi] , i = 1,2, . . . ,m,

and

Cov (Yi,Yj) = min

{i, j

}.

Let

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

,

then

E[XXX] = LLL +,

Var[XXX] = 2VVV,

where

LLL = (1,1, . . . ,1) , = (1,2, . . . ,m) ,VVV = (Vi j) , Vi j = min{i, j}, i, j = 1,2, . . . ,m.

The inverse VVV1 =

Vi j

can be expressed as

Vi j =

2 if i = j = 1,2, . . . ,m 1,1 if i = j = m,

1 if

|i

j

|= 1, i, j = 1,2, . . . ,m,

0, otherwise.


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Record Values 45

The MLVLUEs , of and respectively are

= V

VV

1 LLLLLLVVV1XXX///, = LLLVVV1

LLL

LLLVVV1XXX///,where

=LVVV1L

VVV1

LLLVVV12 ,and

Var[] = 2LLLVVV1///,

Var[] = 2LLLVVV1LLL///,Cov (, ) = 2LLLVVV1///.

It can be shown that

LLLVVV1 = (1,0,0, . . . ,0), VVV1 = (0,0, . . . ,0,1),

VVV1 = m and = m 1.

Upon simplification we get

=

mXU(1)XU(m)/(m 1) ,

=XU(m)XU(1)

/(m 1) , (3.4.1)

with

Var[] = m2/(m 1) , Var[] = 2/(m 1) andCov (, ) = 2/(m 1) . (3.4.2)

(b) Best Linear Invariant Estimator (BLIE)

The best linear invariant (in the sense of minimum mean squared error and

invariance with respect to the location parameter ) estimators, BLIEs,

,

of

and are

= E121 +E12 ,


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and

= /(1 +E12) ,

where and are MVLUEs of and andVar[] Cov (, )

Cov (, ) Var[]

= 2

E11 E12

E21 E22

.

The mean squared errors of these estimators are

MSE[

] = 2

E11 E212 (1 +E22)1

,

and

MSE[] = 2E22 (1 +E22)1 .We have

E[( ) ()] = 2E12 (1 +E22)1 .Using the values ofE11, E12 and E22 from (3.4.2), we obtain

= (m + 1)XU(1)XU(m)/m, = XU(m)XU(1)/m,Var[] = 2 m2 + m 1/m,

and

Var[] = 2 (m 1)/m2.3.5. Prediction of Record Values

We will predict the sth upper value based on the first m record values for s > m.Let

WWW = (W1,W2, . . . ,Wm) ,

where

2Wi = CovXU(i),XU(s)

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

and

= 1EXU(s) .


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Record Values 47

The best linear unbiased predictor of XU(s) is XU(s), where

XU(s) = + +WWWVVV

1

(XXX LLL ), and are MVLUEs of and respectively. It can be shown that WWWVVV1(XXXLLL ) = 0, and hence

XU(s) =

(s 1)XU(m) + (m s)XU(1)/(m 1) , (3.5.1)

E[ XU(s)] = + s,

Var[ XU(s)] = 2 m + s2 2s/(m1) ,

MSE[ XU(s)] = E[( XU(s)XU(s))2] = 2 (s m) (s 1)/(m1) .Let XU(s) be the best linear invariant predictor ofXU(s). Then it can be shown

that XU(s) = XU(s)C12 (1 +E22)1 , (3.5.2)where

C122 = Cov,LLL WWWVVV1LLL +WWWVVV1

and

2E22 = Var[].

Upon simplification, we get

XU(s) = m sm

XU(1) +s

mXU(m),

EXU(s) = +ms + m sm ,Var

XU(s) = 2 m2 + ms2 s2/m,MSE

XU(s) = MSE[ XU(s)] + (s m)2m (m1)

2 =s (s m)

m2.

It is well-known that the best (unrestricted) least square predictor of XU(s) is

XU(s) = EXU(s)|XU(1), . . . ,XU(m) = XU(m) + (s m). (3.5.3)


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But XU(s)depends on the unknown parameter . If we substitute the minimum

variance unbiased estimate for , then XU(s)becomes equal to XU(s). Now

E[ XU(s)] = + s = EXU(s)

,Var[ XU(s)] = m

2

and

MSE[ XU(s)] = E[(XU(s)XU(s))2] = (s m)2.

We like to mention also that by considering the mean squared errors of

XU(s),

XU(s) and

XU(s), it can be shown that

MSE[ XU(s)] = E[( XU(s)XU(s))2] = (s m)2.

3.6. Limiting Distribution of Record Values

We have seen that for = 0 and = 1, EXU(n)

= n and Var

XU(n)

= n.

Hence

PXU(n)nn x = PXU(n) n +xn=

n+xn0

xn1ex

(n)dx, (3.6.1)

= pn (x) , say.

Let

(x) =12

x

et2/2dt.

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

and values of (x) for comparison.


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Record Values 49

Table 1. Values of pn (x)

n\x 2 1 0 1 25 0.0002 0.1468 0.5575 0.8475 0 .9590

10 0.0046 0.1534 0.5421 0.8486 0 .960115 0.0098 0.1554 0.5343 0.8436 0 .965325 0.0122 0.1568 0.5243 0.8427 0 .9684

45 0.0142 0.1575 0.5198 0.8423 0 .9698 (x) 0.0226 0.1587 0.5000 0.8413 0 .9774

Thus for large values of n, (x) is a good approximation of pn (x).Finally, the entropy of nth upper record value XU(n) is

n + ln(n) ln (n 1)(n) ,

where (n) is the digamma function, (n) = (n)/(n). To see this we ob-

serve that pd f ofXU(n) is given by

fn (x) =n

(n)xn1ex, x 0,

and its entropy is computed as follows

En = E[ lnXU (n)]

=

0

n

(n) ex (ln(n) n ln+x (n 1) lnx) dx

= ln(n)n ln+ n (n 1){ ln+(n)}= n + ln(n) ln(n 1)(n) .


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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 val-

ues and sequential order statistics are special cases of the gos. The rvsX(1,n,m,k),X(2,n,m,k), . . . ,X(n,n,m,k), k > 0, m R, are n gos froman absolutely continuous c d f F with corresponding pd f fif their joint pd f

f1,2,...,n (x1,x2, . . . ,xn) can be written as

f1,2,...,n (x1,x2, . . . ,xn)

= k

n1j=1

j

n1j=1

(1 F(xj))m f(xj)

(1 F(xn))k1 f(xn) ,

F1 (0+) < x1 < < xn < F1 (1) , (4.1.1)

where j = k+ (n j) (m + 1) 1 for all j,1 j n,kis a positive integerand 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.


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Ifk= 1 and m = 0, then X(s,n,m,k) reduces to the ordinary sth order statis-tic and (4.1.1) will be the joint pd f of the n order statistics X1,n X2,n Xn,n. Ifk= 1 and m = 1, then (4.1.1) will be the joint pd f of the first n upperrecord values of the i.i.d. rvs with cd f F and pd f f .

Integrating out x1,x2, . . . ,xs1,xs+1,..,xn from (4.1.1) we obtain the pd ffs,n,m,k ofX(s,n,m,k)

fs,n,m,k(x) =cs

(s 1)! (1 F(x))s1 f(x) gs1m (F(x)), (4.1.2)

where cs = sj=1 j and

gm (x) =

1

(m+1)

1 (1 x)m+1

, m = 1,

ln (1 x) , m = 1, x (0,1) .

Since limm1 1m+1

1 (1x)m+1

= ln (1 x) , we will write gm (x) =1

m+1

1 (1 x)m+1

, for all x (0,1) and for all m with g1 (x) =

limm

1 gm (x) .

4.2. Generalized Order Statistics of Exponential

Distribution

Recall that pd f ofX E(,) , is given by

f(x) = 1 exp

1 (x) , x > , > 0,0, otherwise. (4.2.1)Lemma 4.2.1. Let (Xi)i1 be a sequence of i.i.d. rv

s from E(,), then

1X(1,n,m,k) E(1,) and X (s,n,m,k) d= + sj=1 Wjj , whereWj E(0,1) = E(1) for all j s.Proof. From (4.1.2), pd f f s,n,m,k ofX(s,n,m,k), in this case, is

fs,n,m,k(x) =c

s(s 1)! 1e1(x) sgs1m 1 e1(x) . (4.2.2)


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(c) From (4.2.6) it follows that s {X(s,n,m,k)X(s 1,n,m,k)} E(0,). 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 = s {X(s,n,m,k)X(s 1,n,m,k)} .

(d) For k= 1 and m = 1, we obtain XU(s) = +sj=1 Wj.

For XE(,) , we have from (4.2.5),E[X(s,n,m,k)] = +sj=1 1j andthe recurrence relation

E[X(s,n,m,k)]E[X(s 1,n,m,k)] = s.

Let

D (1,n,m,k) = 1X(1,n,m,k),

D (s,n,m,k) = sX(s,n,m,k)X(s 1,n,m,k), 2 s n,

then for X E() all the D (j,n,m,k), j = 1,2, . . . ,n are i.i.d. E() . Thus, wehave 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 ofupper 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 fr,s,n,m,k(x,y) is(see p. 68 of Kamps (1995))

fr,s,n,m,k(x,y) =cs

(r1)! (s r1)! (1 F(x))m

f(x)

gr1m (F(x)) [h (F(y))h (F(x))]sr1 (1F(y))s f(y) , (4.2.7)

where

h (x) = 1m+1 (1 x)m+1 , m

= 1,

ln (1 x) , m = 1.


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Generalized Order Statistics 55

Theorem 4.2.3. For E(1) and s > 1

E(X(s,n,m,k))p+1 = E(X(s 1,n,m,k))p+1+ p + 1s E[(X(s,n,m,k))p] ,and consequently for s > r

E

(X(s,n,m,k))p+1

=E

(X(r,n,m,k))p+1

+s

j=r+1

p + 1

jE[(X(j,n,m,k))p] .

Proof. We have

E[(X(s,n,m,k))p] =

0xp

cs

(s 1)! esxgs1m

1 exdx

=

0

scs(p + 1) (s 1)!x

p+1esxgs1m

1 exdx

0

cs (s 1)(p + 1) (s 1)!x

p+1esxgs2m

1 exe(m+1)dx=

s

(p + 1) E(X(s,n,m,k))p+1

E(X(s 1,n,m,k))p+1

,from which the result follows.

For k= 1 and m = 1, Theorem 4.2.3 coincides with Theorem 3.3.1. Fork= 1 and m = 0, we obtain

E

(Xs,n)p+1

= E

(Xs1,n)p

+1

+p + 1

n s + 1E[(Xs,n)p]

and consequently

E

(Xs,n)p+1

= E

(Xs1,n)p+1

+

s

j=r+1

p + 1

n j + 1E[(Xj,n)p] .

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

E[Xs,n] = E[X1,n] +s

j=2

1

n

j + 1,


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that is,

E[Xs,n] =s

j=1

1

n j + 1.

Theorem 4.2.4. For E(1) ,1 r< s n and p,q = 0,1,2, . . . we have

E

(X(r,n,m,k))p (X(s,n,m,k))q+1

= E

(X(r,n,m,k))p (X(s 1,n,m,k))q+1

+

q + 1

s E[(X(r,n,m,k))p

(X(s,n,m,k))q

].

Proof. We have

E[(X(r,n,m,k))p (X(s,n,m,k))q]

=

0

cs

(r1)! (s r1)! e(m+1)xgr1m

1 exI(x) dx, (4.2.8)

where

I(x) =

xyq

1

m + 1

1 e(m+1)y

1

m + 1

1 e(m+1)x

sr1esydy

=s

q + 1

x

yq+1

1

m + 1

1 e(m+1)y

1

m + 1

1 e(m+1)x

sr1esydy

1q + 1

x

yq+1

1

m + 1 1 e(m+1)y

1

m + 1 1 e(m+1)x

sr2

es1ydy.

Upon substituting for I(x) in (4.2.8), we obtain

E[(X(r,n,m,k))p (X(s,n,m,k))q]

=s

q + 1E

(X(r,n,m,k))p (X(s,n,m,k))q+1

E

(X(r,n,m,k))p (X(s 1,n,m,k))q+1

.


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Thus,

E(X(r,n,m,k))p (X(s,n,m,k))q+1= E

(X(r,n,m,k))p (X(s 1,n,m,k))q+1

+

q + 1

sE[(X(r,n,m,k))p (X(s,n,m,k))q].

For k= 1 and m = 1, Theorem 4.2.4 coincides with Theorem 3.3.3. Fork= 1 and m = 0, we obtain from Theorem 4.2.4

EXpr,nXq+1s,n = EXpr,nXq+1s1,n+ qn s + 1EXpr,nXqs1,n .Estimation of and

Minimum Variance Linear Unbiased Estimators (MVLUEs)

Lemma 4.2.5. Let

and be the MVLUEs of and respectively, based

on n gos X(1,n,m,k),X(2,n,m,k), . . . ,X(n,n,m,k)from an absolutely contin-

uous cd f F with pd f f. Then

= X(1,n,m,k) ( /1)

and

=

1

n 1

n

j=1

(j j+1)X(j,n,m,k)1X(1,n,m,k), with n+1 = 0,

Var

= n2/(n 1)21,Var = 2/(n 1) ,Cov

,

= 2/(n 1)1.

Proof. It is not hard to show that

E[X(s,n,m,k)] = +s,

Var[X(s,n,m,k)] = 2Vs, for 1 s n,


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where s = sj=1

1j, Vs =

sj=1

12j. Let

XXX = (X(1,n,m,k),X(2,n,m,k), . . . ,X(n,n,m,k)) ,

then

E[XXX] = 1 +,

Cov (X(j,n,m,k),X(i,n,m,k)) = 2Vi, 1 i < j n,Var[XXX] = 2VVV,

where, as in Chapter 3, 111is an n 1vector of units, === (1,2, . . . ,n), VVV ===(Vi j) and Vi j = Vi for 1 j n.

Let === VVV1 =

Vi j, then

Vii = 2i + 2i+1, i = 1,2, . . . ,kn, n+1 = 0,

Vi+1i = Vii+1 = i+1,Vi j = 0, for |i j|> 1.

The MVLUEs

and respectively are (see, David, (1981))

= VVV1(111111)VVV1XXX///,

= 111VVV1(111111)VVV1XXX///,

where

= (111VVV1111)))(VVV1)))

(111VVV1)))2.

We also have

Var

= 2VVV1, Var

= 2111VVV1111///,

and

Cov

,

= 2111VVV1///.


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It can easily be shown that

111V

VV

1

= (1,0,0, . . . ,0) ,VVV1 = (1 2,2 3, . . . ,n1 n,n) ,

VVV1 = n,111VVV1111 = 21, 111VVV1 = 1 and = (n 1)21.

Now,

111VVV1(111111)VVV1XXX/// = 1

(111VVV1111111VVV1111)VVV1XXX

=1

(21VVV1XXX1111VVV1XXX) =

1

n 1 (VVV1XXX1X(1,n,m,k)).Hence

=

1

n 1

n

j=1

(j j+1)X(j,n,m,k)1X(1,n,m,k).

We can write = 11 111 + ccc, where

ccc =

0,1

2,

1

2+

1

3,

1

2+

1

3+

1

4, . . . ,

n

j=2

1

j

.

Thus

= cccVVV1(111 111)VVV1XXX///

1.

We have

cccVVV1111 = 0,cccVVV1 = n 1 and hence = X(1,n,m,k)

1.

Ifk= 1 and m = 0, then j = n j + 1 and and coincide with MVLUEs

given by order statistics (see, Arnold et al., (1992), p. 176). If k = 1 andm = 0, then j = 1 and

and

coincide with MVLUEs given by Ahsanullah,

((1980), p.466).


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The variances and covariance of

and are

Var = 2 (VVV1) = n2/(n 1)21,Var

=2

(VVV1111) = 2/(n 1) ,

Cov

,

=

2

(VVV1111) = 2/(n 1)1.

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 )

and of and are

=

E12

1 +E22

and

=

/(1 +E22) ,

where

and are MVLUEs of and and Var[] Cov,

Cov

,

Var[

]

= 2 E11 E12E21 E22 .The mean squared errors of these estimators are

MSE

= 2

E11 E212 (1 +E22)1

and

MSE

= 2E22 (1 +E22)1 .

Substituting the values of E12 and E22 in the above equations, we have, on sim-

plification, that

=

+

1

n1

and

=

n 1n

,

MSE = n + 1n 2

21 andMSE = 1n2.


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Prediction ofX(s,n, m, k)

We shall assume that s > n. Let = (1,2, . . . ,n) where

j = Cov(X(s,n,m,k),X(j,n,m,k)), j = 1,2, . . . ,n and =1E[X(x,n,m,k)]. The best linear unbiased predictor (BLUP)X(s,n,m,k) ofX(s,n,m,k) is X(x,n,m,k) =

+ +VVV1(XXX 111 ),

where

and are the MVLUEs of and respectively. But = s and

= (V1,V2, . . . ,Vn) . It can be shown that VVV1 = (0,0, . . . ,0,1) and hence

X(s,n,m,k) =

+s+X(n,n,m,k)

n

= X(n,n,m,k) + (s n). (4.2.9)

Ifk= 1 and m = 0, then j = n j + 1 and

X(s,n,m,k) coincides with theBLUP based on the order statistics (see, Arnold et al. (1992), p. 181). If k= 1and m = 1, then j = 1 and X(s,n,m,k) coincides with the BLUP based onrecord values (see, Ahsanullah (1980), p. 467).

We have

E[ X(x,n,m,k)] = + (s n),

Var[ X(x,n,m,k)] = 2Vn + (s n)2 2

n 1 + 2 (s n)Cov(X(n,n,m,k),

)

= 2

1

21+ + 1

2n

+

1

n 1

1

n+1+ 1

s

2+

2

n 1 1

n+1+

+

1

s1

2+

+

1

n,MSE[ X(x,n,m,k)] = E[( X(x,n,m,k)X(s,n,m,k))2]

= E[(X(n,n,m,k)X(x,n,m,k) + (s n))2]

= 2

Vn +Vs 2Vn + (s n)2 1n 1

= 2

Vs Vn + (s n)2 (n 1)1

.

Ifk= 1 and m = 1, then the BLUP XU(s) of the sth upper record value from


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(4.2.9) is

XU(s) =(x1)XU(n) (s n)XU(1)

(n 1), (4.2.10)

and

E[ XU(n)] = 2

m + s2 2s/(m1) . (4.2.11)Let X(x,n,m,k) be the best linear invariant predictor of X(s,n,m,k). Then

X(s,n,m,k) = X(s,n,m,k) c

12

1 + c22

, (4.2.12)

where

c122 = Cov(

,

1 VVV1111 +VVV1 ) and c222 = Var[ ].It can easily be shown that c12 = (s n)/(n 1) and since c22 = 1/(n 1),we have

c121+c22

= snn

. Thus

X(s,n,m,k) = X(s,n,m,k)s n

n

= X(n,n,m,k)+n 1

n(s n)

(4.2.13)

E[X(s,n,m,k)] = +s + s nn

(4.2.14)

and

Var[X(s,n,m,k)] = 2Vn +n 1

n 2

(sn)

2 1

n 1= 2

s n + n 1

n2(s n)2

. (4.2.15)

The bias term is

E[

X(s,n,m,k)X(s,n,m,k)] = (n s)+ n 1

n(s n)

= 1

n (s n).


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Thus,

MSE[X(s,n,m,k)] = Var[X(s,n,m,k)]+(bias)2= 2

s n + n1

n2(s n)2

+

1

n(s n)

2= 2

s n + 1

n(s n)2

= MSE[ X(s,n,m,k)] 1

n (n 1) (s n)2 .

For k= 1 and m = 0, we obtain

E[ XU(s)] = +

s +

s nn

,

Var[ XU(s)] = 2

n2 + ns s2

n2

.


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79/158

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 earlysixties 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 re-

lated 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 inadvance if we missed to report some of the existing pertinent results.

Let X1and X2 be two i.i.d. random variables with common c d f F (x) and letX(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 condi-

tion for F to be the cd f of an exponential random variable with parameter ,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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showed that X(2) X(1) X1 ( means having the same distribution) charac-terizes the exponential distribution among the class of absolutely continuous

distributions. Seshardi et al. (1969) reported a characterization of the expo-nential distribution based on the identical distribution of an (n1)-dimensionalrandom vector of random variables Vr = Sr/Sn,r= 1,2, . . . ,n 1, where Sr isthe rth partial sum of the random sample, and vector of order statistics of n 1i.i.d. U(0,1) random variables. Csorgo 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,nand Xsr,nr(these rvs will be defined in the next paragraph) . Rossberg (1972)gave a more general result when s = r+ 1, which will be stated in the followingsection. 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 c d 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 Xr1,n) , 2 r n (5.1.1)

from an exponential distribution with parameter , i.e., F(x) = 1 ex

,x 0, > 0, are again independent and identically exponentially distributed. Thus,we have

F E() implies that D1,n,D2,n, . . . ,Dn,n are i.i.d.E(). (5.1.2)

5.2. Characterizations Based on Order Statistics

We start this section with the following result due to Desu (1971).


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Characterizations of Exponential Distribution I 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 ex forx 0, where 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 rela-

tively prime positive integers k1, k2 > 1 . Here is his theorem:

Theorem 5.2.2. Let supp(F) = (0,). Then X1 E() if and only ifniX1,ni E() for1 < n1 < n2 with ln n1/ ln n2 irrational.

Ahsanullah and Rahman (1972)

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