ORDER STATISTICS

Third Edition

H. A. DAVID and H. N. NAGARAJA

2003

WILEY-INTERSCIENCE
ISBN: 0-471-38926-9
wiley.com

In the over twenty years since the publication of the Second Edition of Order Statistics, the theory and applications of this dynamic field have changed markedly.  Meeting the challenges and demands of todayís students and research community, authors H. A. David and H. N. Nagaraja return with a completely revised and updated Order Statistics, Third Edition.

Chapters two through nine of this comprehensive volume deal with finite-sample theory, with individual topics grouped under distribution theory (chapters two through six) and statistical inference (chapters seven through nine).  Chapters ten and eleven cover asymptotic theory, representing double the coverage of this subject as in the previous edition.  Altogether new topics covered include:

The authors further explain application procedures for many data-analysis techniques and quality control.  An appendix provides a guide to related tables and computer algorithms.  Extensive exercise sets have been updated since the last edition.  In spite of many eliminations the total number of references has increased from 1,000 to 1,500.

Expanded coverage of short-cut methods, robust estimation, life testing, reliability, L-statistics, and extreme-value theory complete this one-of-a-kind resource.  Students and researchers of order statistics will appreciate this updated and thorough edition.


H. A. DAVID, PhD, is a Distinguished Professor Emeritus in the Department of Statistics at Iowa State University.  He has authored over 100 publications, including the Second Edition of Order Statistics.

H. N. NAGARAJA, PhD, is a Professor in the Departments of Statistics and Internal Medicine at The Ohio State University.  He co-edited, along with Pranab K. Sen and Donald F. Morrison, Statistical Theory and Applications: Papers in Honor of H. A. David, and coauthored, along with B. C. Arnold and N. Balakrishnan, Records.


This webpage: http://www.stat.ohio-state.edu/~hnn/OS3.html  provides corrections and updates from time to time.  For purchase information please contact www.wiley.com.

July 22, 2003
hnn@stat.ohio-state.edu

ORDER STATISTICS, THIRD EDITION

TABLE OF CONTENTS

1
INTRODUCTION
1
1.1
The subject of order statistics 
1
1.2
The scope and limits of this book 
3
1.3
Notation 
4
1.4
Exercises 
7
 
2
BASIC DISTRIBUTION THEORY
9
2.1
Distribution of a single order statistic 
  9
2.2
Joint distribution of two or more order statistics 
11
2.3
Distribution of the range and of other systematic statistics 
13
2.4
Order statistics for a discrete parent 
16
2.5
Conditional distributions, order statistics as a Markov chain, and independence results 
17
2.6
Related statistics 
20
2.7
Exercises 
22
 
3
EXPECTED VALUES AND MOMENTS
33
3.1
Basic formulae 
33
3.2
Special continuous distributions 
40
3.3
The discrete case 
42
3.4
Recurrence relations 
44
3.5
Exercises 
49
 
4
BOUNDS AND APPROXIMATIONS FOR MOMENTS OF ORDER STATISTICS
59
4.1
Introduction 
59
4.2
Distribution-free bounds for the moments of order statistics and of the range 
60
4.3
Bounds and approximations by orthogonal inverse expansion 
70
4.4
Stochastic orderings 
74
4.5
Bounds for the expected values of order statistics in terms of quantiles of the parent distribution 
80
4.6
Approximations to moments in terms of the quantile function and its derivatives 
83
4.7
Exercises 
86
 
5
THE NON-IID CASE
95
5.1
Introduction 
  95
5.2
Order statistics for independent nonidentically distributed variates 
  96
5.3
Order statistics for dependent variates 
  99
5.4
Inequalities and recurrence relations---non-IID cases 
102
5.5
Bounds for linear functions of order statistics and for their expected values 
106
5.6
Exercises 
113
 
6
FURTHER DISTRIBUTION THEORY
121
6.1
Introduction 
121
6.2
Studentization 
122
6.3
Statistics expressible as maxima 
124
6.4
Random division of an interval 
133
6.5
Linear functions of order statistics 
137
6.6
Moving order statistics 
140
6.7
Characterizations 
142
6.8
Concomitants of order statistics 
144
6.9
Exercises 
148
 
7
ORDER STATISTICS IN NONPARAMETRIC INFERENCE
159
7.1
Distribution-free confidence intervals for quantiles 
159
7.2
Distribution-free tolerance intervals 
164
7.3
Distribution-free prediction intervals 
167
7.4
Exercises 
169
 
8
ORDER STATISTICS IN PARAMETRIC INFERENCE
171
8.1
Introduction and basic results 
171
8.2
Information in order statistics 
180
8.3
Bootstrap estimation of quantiles and of moments of order statistics 
183
8.4
Least-squares estimation of location and scale parameters by order statistics 
185
8.5
Estimation of location and scale parameters for censored data 
191
8.6
Life testing, with special emphasis on the exponential distribution 
204
8.7
Prediction of order statistics 
208
8.8
Robust estimation 
211
8.9
Exercises 
223
 
9
SHORT-CUT PROCEDURES
239
9.1
Introduction 
239
9.2
Quick measures of location 
241
9.3
Range and mean range as measures of dispersion 
243
9.4
Other quick measures of dispersion 
248
9.5
Quick estimates in bivariate samples 
250
9.6
The studentized range 
253
9.7
Quick tests 
257
9.8
Ranked-set sampling 
262
9.9
O-statistics and L-moments in data summarization 
268
9.10
Probability plotting and tests of goodness of fit 
270
9.11
Statistical quality control 
274
9.12
Exercises 
277
 
10
ASYMPTOTIC THEORY
283
10.1
Introduction 
283
10.2
Representations for the central sample quantiles 
285
10.3
Asymptotic joint distribution of central quantiles 
288
10.4
Optimal choice of order statistics in large samples 
290
10.5
The asymptotic distribution of the extreme 
296
10.6
The asymptotic joint distribution of extremes 
306
10.7
Extreme-value theory for dependent sequences 
309
10.8
Asymptotic properties of intermediate order statistics 
311
10.9
Asymptotic results for multivariate samples 
313
10.10
Exercises 
315
 
11
ASYMPTOTIC RESULTS FOR FUNCTIONS OF ORDER STATISTICS
323
11.1
Introduction 
323
11.2
Asymptotic distribution of the range, midrange, and spacings 
324
11.3
Limit distribution of the trimmed mean 
329
11.4
Asymptotic normality of linear functions of order statistics 
331
11.5
Optimal asymptotic estimation by order statistics 
335
11.6
Estimators of tail index and extreme quantiles 
341
11.7
Asymptotic theory of concomitants of order statistics 
345
11.8
Exercises 
350
 
12
APPENDIX GUIDE TO TABLES AND ALGORITHMS
355
 
13
REFERENCES
367
 
14
INDEX
451


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