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H. A. DAVID 写的Order statistics,3ed
PREFACE xi
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
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
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
8.7 Prediction of order statistics
8.8 Robust estimation
8.9 Exercises
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
9.9 ?-statistics and L-moments in data summarization 268
9.10 Probability plotting and tests of goodness of fit 2 70
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
APPENDIX: GUIDE TO TABLES AND ALGORITHMS 355
REFERENCES 367
INDEX 451
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