ATLANTIS STUDIES IN PROBABILITY AND STATISTICS

VOLUME 3

SERIES EDITORS: CHRIS P. TSOKOS

Atlantis Studies in Probability and Statistics

Series Editors:

Chris P. TsokosUniversity of South Florida Tampa, Tampa, USA

(ISSN: 1879-6893)

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An Introduction to Order Statistics

Mohammad AhsanullahRider University,

Department of Management Sciences,2083 Lawrenceville Road,

Lawrenceville, NJ 08648, USA

Valery B. NevzorovSt. Petersburg State University,

Department of Mathematics and Mechanics,198904 St. Petersburg, Russia

Mohammad ShakilMiami Dade College (Hialeah Campus),

Department of Mathematics, 1800 West 49th Street,Miami, FL 33012, USA

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Atlantis Press

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This book is published under the Creative Commons Attribution-Non-commercial license, meaningthat copying, distribution, transmitting and adapting the book is permitted, provided that this is donefor non-commercial purposes and that the book is attributed.This book, or any parts thereof, may not be reproduced for commercial purposes in any form or byany means, electronic or mechanical, including photocopying, recording or any information storageand retrieval system known or to be invented, without prior permission from the Publisher.

Atlantis Studies in Probability and Statistics

Volume 1: Bayesian Theory and Methods with Applications - Vladimir P. Savchuk, C.P. TsokosVolume 2: Stochastic Differential Games. Theory and Applications - K.M. Ramachandran, C.P.Tsokos

ISBNsPrint: 978-94-91216-82-4E-Book: 978-94-91216-83-1ISSN: 1879-6893

c© 2013 ATLANTIS PRESS

To my wife, Masuda

M. Ahsanullah

To my wife, Ludmilla

Valery B. Nevzorov

To my parents & my wife, Nausheen

M. Shakil

Preface

Dear Reader, imagine that you are a participant of Olympic Games, say, you are one of

n competitors in high-jumping. Before the start the future results of participants can be

regarded as some independent random variables X1, X2, . . . , Xn. The competition will range

all attempts of sportsmen and their final results can be considered as the observed values

of the so-called order statistics X1,n � X2,n � · · · � Xn,n. Hence to predict the result of

the winner you must know the distribution of the extreme order statistic Xn,n. The future

destinations of the silver and bronze prizewinners are determined as Xn−1,n and Xn−2,n

correspondingly. If you are a sprinter then the future results of the gold, silver and bronze

medaled sportsmen are associated with minimal order statistics X1,n, X2,n, and X3,n . These

are the simplest examples of the “sport” applications of order statistics. Other examples

of the applicability of order statistics (and especially of extreme order statistics) can be

suggested by meteorologists, hydrologists, business analysts. The knowledge of the theory

of order statistics is useful for specialists in the actuarial science and the reliability theory.

Some attempts to present a systematic exposition of the theory of order statistics and ex-

tremes began essentially from the publication of the David’s (1970) (the second issue of it

appeared in 1981). We can mention also the following books, where the theory of order

statistics and their different applications were presented: Galambos (1978, 1987), Arnold,

Balakrishnan and Nagaraja (1992, 1998), Kamps (1995), Nevzorov (2000), Ahsanullah

and Nevzorov (2001, 2005), David and Nagaraja (2003), Ahsanullah and Kirmani (2008).

Almost all of these books are rather theoretical. We suggest here (see also Ahsanullah

and Nevzorov (2005)) another way to study this theory. Together with the corresponding

theoretical results, which are presented as 21 chapters, we suggest our readers to solve a

lot of exercises. From one side it allows to understand better the main ideas and results

of the theory. From other side the reader can determine his/her level of permeation to this

material. Solutions of these exercises are given in the end of the corresponding chapters.

vii

viii An Introduction to Order Statistics

The aim of the book is to present various properties of the order statistics and inference

based on them. The book is written on a lower technical level and requires elementary

knowledge of algebra and statistics. The first chapter describes some basic definitions and

properties of order statistics. Chapters 2 to 4 present sample quantiles, representation of

order statistics as functions of independent and identically distributed random variables,

conditional distributions and order statistics of discrete distributions. Chapters 5 to 11 give

the moment properties and asymptotic behavior of middle, intermediate and extreme order

statistics. Chapters 12 to 15 discuss estimation of parameters and their properties. Chap-

ters 16 to 20 deal with order statistics from extended samples, record values, characteriza-

tions, order statistics from F-alpha distribution and generalized order statistics. Chapter 21

contains several interesting problems with hints to solve them.

Summer research grant and sabbatical leave from Rider university enabled the first author

to complete his part of the work. The work of the second author was supported by the grant

RFFI 10-01-00314 and the grant of Science School 4472.2010. The third author is grateful

to Miami Dade College for all the supports including STEM grants.

Contents

Preface vii

1. Basic definitions 1

2. Distributions of order statistics 15

3. Sample quantiles and ranges 23

4. Representations for order statistics 37

5. Conditional distributions of order statistics 51

6. Order statistics for discrete distributions 61

7. Moments of order statistics: general relations 75

8. Moments of uniform and exponential order statistics 83

9. Moment relations for order statistics: normal distribution 95

10. Asymptotic behavior of the middle and intermediate order statistics 105

11. Asymptotic behavior of the extreme order statistics 115

12. Some properties of estimators based on order statistics 131

13. Minimum variance linear unbiased estimators 139

14. Minimum variance linear unbiased estimators and predictors based on

censored samples 153

15. Estimation of parameters based on fixed number of sample quantiles 163

ix

x An Introduction to Order Statistics

16. Order statistics from extended samples 169

17. Order statistics and record values 175

18. Characterizations of distributions based on properties of order statistics 189

19. Order statistics and record values based on Fα distributions 201

20. Generalized order statistics 217

21. Compliments and problems 227

Bibliography 231

Index 243