Order Statistics: Theory & Methods Edited by N. Balakrishnan Department of Mathematics and Statistics McMaster University Hamilton, Ontario, Canada C. R. Rao Center for Multivariate Analysis Department of Statistics, The Pennsylvania State University University Park, PA, USA 1998 ELSEVIER AMSTERDAM LAUSANNE NEW YORK OXFORD SHANNON SINGAPORE TOKYO
Preface v Contributors xvii PART I. INTRODUCTION AND BASIC PROPERTIES Ch. 1. Order Statistics: An Introduction 3 N. Balakrishnan and C. R. Rao 1. Introduction 3 2. Marginal distributions of order statistics 4 3. Joint distributions of order statistics 5 4. Properties 7 5. Moments and product moments 7 6. Recurrence relations and identities 8 7. Bounds 10 8. Approximations 10 9. Characterizations 11 10. Asymptotics 12 11. Best linear unbiased estimation and prediction 12 12. Inference under censoring 14 13. Results for some specific distributions 16 14. Outliers and robust inference 17 15. Goodness-of-fit tests 18 16. Related statistics 18 17. Generalizations 20 References 21 Ch. 2. Order Statistics: A Historical Perspective 25 H. Leon Harter and N. Balakrishnan 1. Introduction 25 2. Distribution theory and properties 26 3. Measures of central tendency and dispersion 27
x Table of contents 4. Regression coefficients 28 5. Treatment of outliers and robust estimation 29 6. Maximum likelihood estimators 29 7. Best linear unbiased estimators 30 8. Recurrence relations and identities 32 9. Bounds and approximations 32 10. Distribution-free tolerance procedures 33 11. Prediction 35 12. Statistical quality control and range 36 13. Multiple comparisons and studentized range 36 14. Ranking and selection procedures 38 15. Extreme values 39 16. Plotting positions on probability paper 40 17. Simulation methods 41 18. Ordered characteristic roots 41 19. Goodness-of-fit tests 43 20. Characterizations 45 21. Moving order statistics and applications 45 22. Order statistics under non-standard conditions 46 23. Multivariate order statistics and concomitants 47 24. Records 47 References 48 Ch. 3. Computer Simulation of Order Statistics 65 Pandu R. Tadikamalla and N. Balakrishnan 1. Introduction 65 2. Direct generation of order statistics 65 3. Generation of uniform (0, 1) ordered samples 65 4. Generation of progressive Type-II censored order statistics 6J 5. Miscellaneous topics 69 References 70 PART II. ORDERINGS AND BOUNDS Ch. 4. Lorenz Ordering of Order Statistics and Record Values 75 Barry C. Arnold and Jose A. Villasenor 1. Introduction 75 2. The Lorenz order 75 3. Order statistics and record values 77 4. Lorenz ordering of order statistics 78 5. Lorenz ordering of record values 81 6. Remarks 86 References 86
Ch. 5. Stochastic Ordering of Order Statistics 89 Philip J. Boland, Moshe Shaked and J. George Shanthikumar 1. Introduction 89 2. Stochastic orderings 90 3. Stochastic order for order statistics from one sample 94 4. Stochastic order for order statistics from two samples 99 Acknowledgement 102 References 102 Ch. 6. Bounds for Expectations of L-Estimates 105 Tomasz Rychlik 1. Introduction 105 2. Distribution bounds 106 3. Moment and support bounds 112 4. Moment bounds for restricted families 123 5. Quantile bounds for restricted families 135 References 142 PART III. RELATIONS AND IDENTITIES Ch. 7. Recurrence Relations and Identities for Moments of Order Statistics 149 N. Balakrishnan and K. S. Sultan 1. Introduction 149 2. Notations 153 3. Recurrence relations for single moments 154 4. Recurrence relations for product moments 161 5. Relations between moments of order statistics from two related populations 171 6. Normal and half normal distributions 172 7. Cauchy distribution 175 8. Logistic and related distributions 177 9. Gamma and related distributions 184 10. Exponential and related distributions 188 11. Power function and related distributions 190 12. Pareto and related distributions 193 13. Rayleigh distribution 199 14. Linear-exponential distribution 200 15. Lomax distribution 203 16. Log-logistic and related distributions 204 17. Burr and truncated Burr distributions 209 18. Doubly truncated parabolic and skewed distributions 211 19. Mixture of two exponential distributions 212 20. Doubly truncated Laplace distribution 212 21. A class of probability distributions 216 Acknowledgement 221 References 222
xii Table of contents PART IV. CHARACTERIZATIONS Ch. 8. Recent Approaches to Characterizations Based on Order Statistics and Record Values 231 C. R. Rao and D. N. Shanbhag 1. Introduction 231 2. Some basic tools 232 3. Characterizations based on order statistics 236 4. Characterizations involving record values and monotonic stochastic processes 249 Acknowledgment 253 References 254 Ch. 9. Characterizations of Distributions via Identically Distributed Functions of Order Statistics 257 Ursula Gather, Udo Kamps and Nicole Schweitzer 1. Introduction 257 2. Characterizations of exponential distributions based on normalized spacings 259 3. Related characterizations of other continuous distributions 268 4. Characterizations of uniform distributions 270 5. Characterizations of specific continuous distributions 272 6. Characterizations of geometric and other discrete distributions 280 References 285 Ch. 10. Characterizations of Distributions by Recurrence Relations and Identities for Moments of Order Statistics 291 Udo Kamps 1. Introduction 291 2. Characterizations by sequences of moments and complete function sequences 293 3. Characterizations of exponential distributions 296 4. Related characterizations in classes of distributions 297 5. Characterizations based on a single identity 302 6. Characterizations of normal and other distributions by product moments 305 References 308 PART V. EXTREMES AND ASYMPTOTICS Ch. 11. Univariate Extreme Value Theory and Applications 315 Janos Galambos 1. Introduction 315 2. The classical models 317
3. Applications and statistical inference 324 4. Deviations from the classical models 329 Acknowledgements 330 References 331 Ch. 12. Order Statistics: Asymptotics in Applications 335 Pranab Kumar Sen 1. Introduction 335 2. Some basic results in order statistics 337 3. Some basic asymptotics in order statistics 341 4. Robust estimation and order statistics: asymptotics in applications 343 5. Trimmed LSE and regression quantiles 350 6. Asymptotics for concomitants of order statistics 352 7. Concomitant i-functionals and nonparametric regression 357 8. Applications of order statistics in some reliability problems 361 9. TTT asymptotics and tests for aging properties 365 10. Concluding remarks 370 References 371 Ch. 13. Zero-One Laws for Large Order Statistics 375 R. J. Tomkins and Hong Wang 1. Introduction 375 2. Zero-One laws for the upper-case probability 376 3. Zero-one laws for the lower-case probability 379 4. Zero-One laws for the lower-case probability when ranks vary 382 Acknowledgements 383 References 384 PART VI. ROBUST METHODS Ch. 14. Some Exact Properties Of Cook's D, 387 D. R. Jensen and D. E. Ramirez 1. Introduction 387 2. Preliminaries 388 3. The structure of Cook's D, 390 4. Normal-Theory properties 393 5. Modified versions of D/ 398 6. Summary 400 References 401
xiv Table of contents Ch. 15. Generalized Recurrence Relations for Moments of Order Statistics from Non-Identical Pareto and Truncated Pareto Random Variables with Applications to Robustness 403 Aaron Chi Ids and N. Balakrishnan 1. Introduction 403 2. Relations for single moments 405 3. Relations for product moments 407 4. Results for the multiple-outlier model (with a slippage of/) observations) 412 5. Generalization to the truncated Pareto distribution 413 6. Robustness of the MLE and BLUE 415 7. Robustness of the censored BLUE 416 8. Conclusions 421 Acknowledgements 426 Appendix A 426 Appendix B 432 References 438 PART VII. RESAMPLING METHODS Ch. 16. A Semiparametric Bootstrap for Simulating Extreme Order Statistics 441 Robert L. Strawderman and Daniel Zelterman 1. Introduction 441 2. A semiparametric bootstrap approximation to A) 444 3. A saddlepoint approximation to the bootstrap distribution 448 4. Numerical implementation 451 5. Simulation results 453 6. Example: The British coal mining data 458 Acknowledgements 460 References 461 Ch. 17. Approximations to Distributions of Sample Quantiles 463 Chunsheng Ma and John Robinson 1. Introduction and definitions 463 2. Smirnov's lemma 467 3. Normal approximation 468 4. Saddlepoint approximation 475 5. Bootstrap approximation 479 References 482
PART VIII. RELATED STATISTICS Ch. 18. Concomitants of Order Statistics 487 H. A. David and H. N. Nagaraja 1. Introduction and summary 487 2. Finite-sample distribution theory and moments 488 3. Asymptotic theory 493 4. Estimation and tests of hypotheses 496 5. The rank of Y [rn] 501 6. Selection through an associated variable 504 7. Functions of concomitants 506 References 510 Ch. 19. A Record of Records 515 Valery B. Nevzorov and N. Balakrishnan 1. Introduction 515 2. Classical records 515 3. Definitions 517 4. Representations of record times and record values using sums of independent terms 518 5. Distributions and probability structure of record times 520 6. Moments of records times and numbers of records 523 7. Limit theorems for record times 525 8. Inter-Record times 525 9. Distributions and probability structure of record values in sequences of continuous random variables 527 10. Limit theorems for record values from continuous distributions 528 11. Record values from discrete distributions 528 12. Weak records 529 13. Bounds and approximations for moments of record values 530 14. Recurrence relations for moments of record values 531 15. Joint distributions of record times and record values 532 16. Generalizations of the classical record model 534 17. k lh record times 534 18. k lh inter-record times 537 19. k' h record values for the continuous case 538 20. k lh record values for the discrete case 540 21. Weak k' h record values 540 22. ^-records 541 23. Records in sequences of dependent random variables 543 24. Random record models 545 25. Nonstationary record models 545 26. Multivariate records 558 27. Relations between records and other probabilistic and statistical problems 558 28. Nonclassical characterizations based on records 559 29. Processes associated with records 560 30. Diverse results 560
Acknowledgement 561 References 561 PART IX. RELATED PROCESSES Ch. 20. Weighted Sequential Empirical Type Processes with Applications to Change-Point Problems 573 Barbara Szyszkowicz 1. Introduction 573 2. Weighted empirical processes based on observations 579 3. "Bridge-type" two-time parameter empirical processes 590 4. Weighted empirical processes based on ranks 599 5. Weighted empirical processes based on sequential ranks 604 6. "Bridge-type" empirical processes of sequential ranks 609 7. Contiguous alternatives 614 8. Weighted multi-time parameter empirical processes 621 Acknowledgement 628 References 628 Ch. 21. Sequential Quantile and Bahadur-Kiefer Processes 631 Miklos Csorgo and Barbara Szyszkowicz 1. Introduction: Basic notions, definitions and some preliminary results 631 2. Deviations between the general and uniform quantile processes and their sequential versions 649 3. Weighted sequential quantile processes in supremum and /.^-metrics 654 4. A summary of the classical Bahadur-Kiefer process theory via strong invariance principles 662 5. An extension of the classical Bahadur-Kiefer process theory via strong invariance principles 680 6. An outline of a sequential version of the extended Bahadur-Kiefer process theory via strong invariance principles 683 Acknowledgement 686 References 686 Author Index 689 Subject Index 701 Contents of Previous Volumes 715