Order Statistics

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Order Statistics Third Edition

WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: David J. Balding, Noel A. C. Cressie, Nicholas I. Fisher, Iain M. Johnstone, J. B. Kadane, Louise M. Ryan, David W. Scott, Adrian F. M. Smith, JozefL. Teugels Editors Emeriti: Vic Barnett, J. Stuart Hunter, David G. Kendall A complete list of the titles in this series appears at the end of this volume.

Order Statistics Third Edition

H. A. DAVID Iowa State University Department of Statistics Ames, I A

H. N. NAGARAJA The Ohio State University Department of Statistics Columbus, OH


Copyright © 2003 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, e-mail: [email protected] Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representation or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data: David, H. A. (Herbert Aron), 1925Order statistics — 3rd ed. / H.A. David, H.N. Nagaraja. p. cm. Includes bibliographical references and index. ISBN 0-471-38926-9 (cloth) 1. Order statistics. I. Nagaraja, H. N. (Haikady Navada), 1954-. II. Title. QA278.7.D38 2003 519.5—dc21 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1


To Ruth—HAD To my mother, Susheela—HNN

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Contents PREFACE 1 INTRODUCTION 1.1 The subject of order statistics 1.2 The scope and limits of this book 1.3 Notation 1.4 Exercises

xi 1 1 3 4 7

2 BASIC DISTRIBUTION THEORY 2.1 Distribution of a single order statistic 2.2 Joint distribution of two or more order statistics 2.3 Distribution of the range and of other systematic statistics 2.4 Order statistics for a discrete parent 2.5 Conditional distributions, order statistics as a Markov chain, and independence results 2.6 Related statistics 2.7 Exercises

9 9 11

17 20 22

3 EXPECTED VALUES AND MOMENTS 3.1 Basic formulae 3.2 Special continuous distributions

33 33 40

13 16




3.3 3.4 3.5

The discrete case Recurrence relations Exercises

4 BOUNDS AND APPROXIMATIONS FOR MOMENTS OF ORDER STATISTICS 4.1 Introduction 4.2 Distribution-free bounds for the moments of order statistics and of the range 4.3 Bounds and approximations by orthogonal inverse expansion 4.4 Stochastic orderings 4.5 Bounds for the expected values of order statistics in terms of quantiles of the parent distribution 4.6 Approximations to moments in terms of the quantile function and its derivatives 4.7 Exercises 5

THE NON-IID CASE 5.7 Introduction 5.2 Order statistics for independent nonidentically distributed variates 5.3 Order statistics for dependent variates 5.4 Inequalities and recurrence relations—non-IID cases 5.5 Bounds for linear functions of order statistics and for their expected values 5.6 Exercises

6 FURTHER DISTRIBUTION THEORY 6.1 Introduction 6.2 Studentization 6.3 Statistics expressible as maxima 6.4 Random division of an interval 6.5 Linear functions of order statistics 6.6 Moving order statistics

42 44 49

59 59 60 70 74 80 83 86 95 95 96 99 102 106 113 121 121 122 124 133 137 140


6.7 6.8 6.9

Characterizations Concomitants of order statistics Exercises

7 ORDER STATISTICS IN NONPARAMETRIC INFERENCE 7.1 Distribution-free confidence intervals for quantiles 7.2 Distribution-free tolerance intervals 7.3 Distribution-free prediction intervals 7.4 Exercises


142 144 148

159 159 164 167 169

8 ORDER STATISTICS IN PARAMETRIC INFERENCE 8.1 Introduction and basic results 8.2 Information in order statistics 8.3 Bootstrap estimation of quantiles and of moments of order statistics 8.4 Least-squares estimation of location and scale parameters by order statistics 8.5 Estimation of location and scale parameters for censored data 8.6 Life testing, with special emphasis on the exponential distribution 8.7 Prediction of order statistics 8.8 Robust estimation 8.9 Exercises

171 171 180

204 208 211 223

9 SHORT-CUT PROCEDURES 9.1 Introduction 9.2 Quick measures of location 9.3 Range and mean range as measures of dispersion 9.4 Other quick measures of dispersion 9.5 Quick estimates in bivariate samples 9.6 The studentized range 9.7 Quick tests 9.8 Ranked-set sampling

239 239 241 243 248 250 253 257 262

183 185 191



9.9 9.10 9.11 9.12

O-statistics and L-moments in data summarization Probability plotting and tests of goodness of fit Statistical quality control Exercises

10 ASYMPTOTIC THEORY 10.1 Introduction 10.2 Representations for the central sample quantiles 10.3 Asymptotic joint distribution of central quantiles 10.4 Optimal choice of order statistics in large samples 10.5 The asymptotic distribution of the extreme 10.6 The asymptotic joint distribution of extremes 10.7 Extreme-value theory for dependent sequences 10.8 Asymptotic properties of intermediate order statistics 10.9 Asymptotic results for multivariate samples 10.10 Exercises 11 ASYMPTOTIC RESULTS FOR FUNCTIONS OF ORDER STATISTICS 11.1 Introduction 11.2 Asymptotic distribution of the range, midrange, and spacings 11.3 Limit distribution of the trimmed mean 11.4 Asymptotic normality of linear functions of order statistics 11.5 Optimal asymptotic estimation by order statistics 11.6 Estimators of tail index and extreme quantiles 11.7 Asymptotic theory of concomitants of order statistics 11.8 Exercises

268 2 70 274 277 283 283 285 288 290 296 306 309 311 313 315

323 323 324 329 331 335 341 345 350





Preface to Third Edition

Since the publication in 1981 of the second edition of this book both theory and applications of order statistics have greatly expanded. In this edition Chapters 2-9 deal with finite-sample theory, with division into distribution theory (Chapters 2-6) and statistical inference (Chapters 7-9). Asymptotic theory is treated in Chapters 10 and 11, representing a doubling in coverage. In the spirit of previous editions we present in detail an up-to-date account of what we regard as the basic results of the subject of order statistics. Many special topics are also taken up, but for these we may merely provide an introduction if other more extensive accounts exist. The number of references has increased from 1000 in the second edition to around 1500, and this in spite of the elimination of a good many references cited earlier. Even so, we had to omit a larger proportion of relevant references than before, giving some preference to papers not previously mentioned in review articles. In addition to an increased emphasis on asymptotic theory and on order statistics in other than random samples (Chapter 5), the following sections are entirely or largely new: 2.6. Related statistics; 4.4. Stochastic orderings; 6.6. Moving order statistics; 6.7. Characterizations; 7.3. Distribution-free prediction intervals; 8.2. Information in order statistics; 8.3. Bootstrap estimation; 9.6. Studentizedrange;9.8. Ranked-set sampling; and 9.9. O-statistics and L-moments in data summarization. Section 6.6 includes a major application to median and order-statistic filters and Section 9.6 to bioequivalence testing. XI



Order Statistics continues to be both textbook and guide to the research literature. The reader interested in a particular section is urged at least to skim the exercises for that section and where relevant to look at the corresponding appendix section for related statistical tables and algorithms. We are grateful to Stephen Quigley, Editor, Wiley Series in Probability and Statistics, for encouraging us to prepare this edition. Encouragement was also provided by N. Balakrishnan, to whom we owe a special debt for his careful reading of most of the book. This has resulted in many corrections and clarifications as well as several suggestions. Chapter 10 benefited from a careful perusal by Barry Arnold. D. Dharmappa in Bangalore prepared a preliminary version of the manuscript in BTgX with speed and accuracy. The typing of references and index was ably done by Jeanette LaGrange of the Iowa State Statistics Department. We also acknowledge with appreciation the general support of our respective departments. H. A. DAVID H. N. NAGARAJA Ames, Iowa Columbus, Ohio January 2003



Preface to Second Edition In the ten years since the first edition of this book there has been much activity relevant to the study of order statistics. This is reflected by an appreciable increase in the size of this volume. Nevertheless it has been possible to retain the outlook and the essential structure of the earlier account. The principal changes are as follows. Sections have been added on order statistics for independent nonidentically distributed variates, on linear functions of order statistics (in finite samples), on concomitants of order statistics, and on testing for outliers from a regression model. In view of major developments the section on robust estimation has been greatly expanded. Important progress in the asymptotic theory has resulted in the complete rewriting, with the help of Malay Ghosh, of the sections on the asymptotic joint distribution of quantiles and on the asymptotic distribution of linear functions of order statistics. Many other changes and additions have also been made. Thus the number of references has risen from 700 to 1000, in spite of some deletions of entries in the first edition. Many possible references were deemed either insufficiently central to our presentation or adequately covered in other books. By way of comparison it may be noted that the first (and so far only) published volume of Harter's (1978b) annotated bibliography on order statistics contains 937 entries covering the work prior to 1950. I am indebted to P. G. Hall, P. C. Joshi, Gordon Simons, and especially Richard Savage for pointing out errors in the first edition. The present treatment of asymptotic theory has benefitted from contributions by Ishay Weissman as well as Malay Ghosh. All the new material in this book has been read critically and constructively by H. N. Nagaraja. It is a pleasure to thank also Janice Peters for cheerfully given extensive secretarial help. In addition, I am grateful to the U.S. Army Research Office for longstanding support. H. A. DAVID Ames, Iowa July 1980



Preface Order statistics make their appearance in many areas of statistical theory and practice. Recent years have seen a particularly rapid growth, as attested by the references at the end of this book. There is a growing recognition that the large body of theory, techniques, and applications involving order statistics deserves study on its own, rather than as a mere appendage to other fields, such as nonparametric methods. Some may decry this increased specialization, and indeed it is entirely appropriate that the most basic aspects of the subject be incorporated in general textbooks and courses, both theoretical and applied. On the other hand, there has been a clear trend in many universities toward the establishment of courses of lectures dealing more extensively with order statistics. I first gave a short course in 1955 at the University of Melbourne and have since then periodically offered longer courses at the Virginia Polytechnic Institute and especially at the University of North Carolina, where much of the present material has been tried out. In this book an attempt is made to present the subject of order statistics in a manner combining features of a textbook and of a guide through the research literature. The writing is at an intermediate level, presupposing on the reader's part the usual basic background in statistical theory and applications. Some portions of the book, are, however, quite elementary, whereas others, particularly in Chapters 4 and 9, are rather more advanced. Exercises supplement the text and, in the manner of M. G. Kendall's books, usually lead the reader to the original sources. A special word is needed to explain the relation of this book to the only other existing general account, also prepared in the Department of Biostatistics, University of North Carolina, namely, the multiauthored Contributions to Order Statistics, edited by A. E. Sarhan and B. G. Greenberg, which appeared in this Wiley series in 1962. The present monograph is not meant to replace that earlier one, which is almost twice as long. In particular, the extensive set of tables in Contributions will long retain their usefulness. The present work contains only a few tables needed to clarify the text but provides, as an appendix, an annotated guide to the massive output of tables scattered over numerous journals and books; such tables are essential for the ready use of many of the methods described. Contributions was not designed as a textbook and is, of course, no longer quite up to date. However, on a number of topics well developed by 1962 more extensive coverage will be found there than here. Duplication of all but the most fundamental material has been kept to a minimum. In other respects also the size of this book has been kept down by deferring wherever feasible to available specialized monographs. Thus plans for the treatment of the role of order statistics in simultaneous inference have largely been abandoned in view of R. G. Miller's very readable account in 1966.