Hardbound. Major theoretical advances were made in this area of research, and in the course of these developments order statistics has also found important applications in many diverse areas. These include life-testing and reliability, robustness studies, statistical quality control, filtering theory, signal processing, image processing, and radar target detection.
Theoretical researchers working on theoretical and methodological advancements on order statistics and applied statisticians and engineers developing new and innovative applications of order statistics have been successfully brought together to create this handbook. For the convenience of readers, the subject matter has been divided into two volumes. This volume focuses on theory and methods, and volume 17 deals primarily with applications. Each volume has been divided into parts, each part specializing in one aspect of order statistics. The articles in this volume have been classified into
Author(s): Balakrishnan N., Rao C.R. (eds.)
Edition: 1
Publisher: Elsevier Science
Year: 1998
Language: English
Pages: 752
Tags: Математика;Теория вероятностей и математическая статистика;Математическая статистика;
Front cover......Page 1
Series......Page 3
Title page......Page 5
Date-line......Page 6
Preface......Page 7
Table of contents......Page 11
Contributors......Page 19
PART I. INTRODUCTION AND BASIC PROPERTIES......Page 21
1. Introduction......Page 23
2. Marginal distributions of order statistics......Page 24
3. Joint distributions of order statistics......Page 25
5. Moments and product moments......Page 27
6. Recurrence relations and identities......Page 28
8. Approximations......Page 30
9. Characterizations......Page 31
11. Best linear unbiased estimation and prediction......Page 32
12. Inference under censoring......Page 34
13. Results for some specific distributions......Page 36
14. Outliers and robust inference......Page 37
16. Related statistics......Page 38
17. Generalizations......Page 40
References......Page 41
1. Introduction......Page 45
2. Distribution theory and properties......Page 46
3. Measures of central tendency and dispersion......Page 47
4. Regression coefficients......Page 48
6. Maximum likelihood estimators......Page 49
7. Best linear unbiased estimators......Page 50
9. Bounds and approximations......Page 52
10. Distribution-free tolerance procedures......Page 53
11. Prediction......Page 55
13. Multiple comparisons and studentized range......Page 56
14. Ranking and selection procedures......Page 58
15. Extreme values......Page 59
16. Plotting positions on probability paper......Page 60
18. Ordered characteristic roots......Page 61
19. Goodness-of-fit tests......Page 63
21. Moving order statistics and applications......Page 65
22. Order statistics under non-standard conditions......Page 66
24. Records......Page 67
References......Page 68
3. Generation of uniform (0,1) ordered samples......Page 85
4. Generation of progressive Type-II censored order statistics......Page 88
5. Miscellaneous topics......Page 89
References......Page 90
PART II. ORDERINGS AND BOUNDS......Page 93
2. The Lorenz order......Page 95
3. Order statistics and record values......Page 97
4. Lorenz ordering of order statistics......Page 98
5. Lorenz ordering of record values......Page 101
References......Page 106
1. Introduction......Page 109
2. Stochastic orderings......Page 110
3. Stochastic order for order statistics from one sample......Page 114
4. Stochastic order for order statistics from two samples......Page 119
References......Page 122
1. Introduction......Page 125
2. Distribution bounds......Page 126
3. Moment and support bounds......Page 132
4. Moment bounds for restricted families......Page 143
5. Quantile bounds for restricted families......Page 155
References......Page 162
PART III. RELATIONS AND IDENTITIES......Page 167
1. Introduction......Page 169
2. Notations......Page 173
3. Recurrence relations for single moments......Page 174
4. Recurrence relations for product moments......Page 181
5. Relations between moments of order statistics from two related populations......Page 191
6. Normal and half normal distributions......Page 192
7. Cauchy distribution......Page 195
8. Logistic and related distributions......Page 197
9. Gamma and related distributions......Page 204
10. Exponential and related distributions......Page 208
11. Power function and related distributions......Page 210
12. Pareto and related distributions......Page 213
13. Rayleigh distribution......Page 219
14. Linear-exponential distribution......Page 220
15. Lomax distribution......Page 223
16. Log-logistic and related distributions......Page 224
17. Burr and truncated Burr distributions......Page 229
18. Doubly truncated parabolic and skewed distributions......Page 231
20. Doubly truncated Laplace distribution......Page 232
21. A class of probability distributions......Page 236
Acknowledgement......Page 241
References......Page 242
PART IV. CHARACTERIZATIONS......Page 249
1. Introduction......Page 251
2. Some basic tools......Page 252
3. Characterizations based on order statistics......Page 256
4. Characterizations involving record values and monotonic stochastic processes......Page 269
Acknowledgment......Page 273
References......Page 274
1. Introduction......Page 277
2. Characterizations of exponential distributions based on normalized spacings......Page 279
3. Related characterizations of other continuous distributions......Page 288
4. Characterizations of uniform distributions......Page 290
5. Characterizations of specific continuous distributions......Page 292
6. Characterizations of geometric and other discrete distributions......Page 300
References......Page 305
1. Introduction......Page 311
2. Characterizations by sequences of moments and complete function sequences......Page 313
3. Characterizations of exponential distributions......Page 316
4. Related characterizations in classes of distributions......Page 317
5. Characterizations based on a single identity......Page 322
6. Characterizations of normal and other distributions by product moments......Page 325
References......Page 328
PART V. EXTREMES AND ASYMPTOTICS......Page 333
1. Introduction......Page 335
2. The classical models......Page 337
3. Applications and statistical inference......Page 344
4. Deviations from the classical models......Page 349
Acknowledgements......Page 350
References......Page 351
1. Introduction......Page 355
2. Some basic results in order statistics......Page 357
3. Some basic asymptotics in order statistics......Page 361
4. Robust estimation and order statistics: asymptotics in applications......Page 363
5. Trimmed LSE and regression quantiles......Page 370
6. Asymptotics for concomitants of order statistics......Page 372
7. Concomitant $L$-functionals and nonparametric regression......Page 377
8. Applications of order statistics in some reliability problems......Page 381
9. TTT asymptotics and tests for aging properties......Page 385
10. Concluding remarks......Page 390
References......Page 391
1. Introduction......Page 395
2. Zero-One laws for the upper-case probability......Page 396
3. Zero-one laws for the lower-case probability......Page 399
4. Zero-One laws for the lower-case probability when ranks vary......Page 402
Acknowledgements......Page 403
References......Page 404
PART VI. ROBUST METHODS......Page 405
1. Introduction......Page 407
2. Preliminaries......Page 408
3. The structure of Cook's $D_I$......Page 410
4. Normal-Theory properties......Page 413
5. Modified versions of $D_I$......Page 418
6. Summary......Page 420
References......Page 421
1. Introduction......Page 423
2. Relations for single moments......Page 425
3. Relations for product moments......Page 427
4. Results for the multiple-outlier model (with a slippage of $p$ observations)......Page 432
5. Generalization to the truncated Pareto distribution......Page 433
6. Robustness of the MLE and BLUE......Page 435
7. Robustness of the censored BLUE......Page 436
8. Conclusions......Page 441
Appendix A......Page 446
Appendix B......Page 452
References......Page 458
PART VII. RESAMPLING METHODS......Page 459
1. Introduction......Page 461
2. A semiparametric bootstrap approximation to $X_j$......Page 464
3. A saddlepoint approximation to the bootstrap distribution......Page 468
4. Numerical implementation......Page 471
5. Simulation results......Page 473
6. Example: The British coal mining data......Page 478
Acknowledgements......Page 480
References......Page 481
1. Introduction and definitions......Page 483
2. Smirnov's lemma......Page 487
3. Normal approximation......Page 488
4. Saddlepoint approximation......Page 495
5. Bootstrap approximation......Page 499
References......Page 502
PART VIII. RELATED STATISTICS......Page 505
1. Introduction and summary......Page 507
2. Finite-sample distribution theory and moments......Page 508
3. Asymptotic theory......Page 513
4. Estimation and tests of hypotheses......Page 516
5. The rank of $Y_{[rn]}$......Page 521
6. Selection through an associated variable......Page 524
7. Functions of concomitants......Page 526
References......Page 530
2. Classical records......Page 535
3. Definitions......Page 537
4. Representations of record times and record values using sums of independent terms......Page 538
5. Distributions and probability structure of record times......Page 540
6. Moments of records times and numbers of records......Page 543
8. Inter-Record times......Page 545
9. Distributions and probability structure of record values in sequences of continuous random variables......Page 547
11. Record values from discrete distributions......Page 548
12. Weak records......Page 549
13. Bounds and approximations for moments of record values......Page 550
14. Recurrence relations for moments of record values......Page 551
15. Joint distributions of record times and record values......Page 552
17. $k$th record times......Page 554
18. $k$th inter-record times......Page 557
19. $k$th record values for the continuous case......Page 558
21. Weak $k$th record values......Page 560
22. $k_n$-records......Page 561
23. Records in sequences of dependent random variables......Page 563
25. Nonstationary record models......Page 565
27. Relations between records and other probabilistic and statistical problems......Page 578
28. Nonclassical characterizations based on records......Page 579
30. Diverse results......Page 580
References......Page 581
PART IX. RELATED PROCESSES......Page 591
1. Introduction......Page 593
2. Weighted empirical processes based on observations......Page 599
3. "Bridge-type" two-time parameter empirical processes......Page 610
4. Weighted empirical processes based on ranks......Page 619
5. Weighted empirical processes based on sequential ranks......Page 624
6. "Bridge-type" empirical processes of sequential ranks......Page 629
7. Contiguous alternatives......Page 634
8. Weighted multi-time parameter empirical processes......Page 641
References......Page 648
1. Introduction: Basic notions, definitions and some preliminary results......Page 651
2. Deviations between the general and uniform quantile processes and their sequential versions......Page 669
3. Weighted sequential quantile processes in supremum and $L_p$-metrics......Page 674
4. A summary of the classical Bahadur-Kiefer process theory via strong invariance principles......Page 682
5. An extension of the classical Bahadur-Kiefer process theory via strong invariance principles......Page 700
6. An outline of a sequential version of the extended Bahadur-Kiefer process theory via strong invariance principles......Page 703
References......Page 706
Author Index......Page 709
Subject Index......Page 721
Contents of Previous Volumes......Page 735
Back cover......Page 752
