Order Statistics: Theory & Methods. N. Balakrishnan Department of Mathematics and Statistics McMaster University Hamilton, Ontario, Canada. C. R.

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1 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

2 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 Approximations Characterizations Asymptotics Best linear unbiased estimation and prediction Inference under censoring Results for some specific distributions Outliers and robust inference Goodness-of-fit tests Related statistics Generalizations 20 References 21 Ch. 2. Order Statistics: A Historical Perspective 25 H. Leon Harter and N. Balakrishnan 1. Introduction Distribution theory and properties Measures of central tendency and dispersion 27

3 x Table of contents 4. Regression coefficients Treatment of outliers and robust estimation Maximum likelihood estimators Best linear unbiased estimators Recurrence relations and identities Bounds and approximations Distribution-free tolerance procedures Prediction Statistical quality control and range Multiple comparisons and studentized range Ranking and selection procedures Extreme values Plotting positions on probability paper Simulation methods Ordered characteristic roots Goodness-of-fit tests Characterizations Moving order statistics and applications Order statistics under non-standard conditions Multivariate order statistics and concomitants Records 47 References 48 Ch. 3. Computer Simulation of Order Statistics 65 Pandu R. Tadikamalla and N. Balakrishnan 1. Introduction Direct generation of order statistics Generation of uniform (0, 1) ordered samples 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 The Lorenz order Order statistics and record values Lorenz ordering of order statistics Lorenz ordering of record values Remarks 86 References 86

4 Ch. 5. Stochastic Ordering of Order Statistics 89 Philip J. Boland, Moshe Shaked and J. George Shanthikumar 1. Introduction Stochastic orderings Stochastic order for order statistics from one sample 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 Distribution bounds Moment and support bounds Moment bounds for restricted families 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 Notations Recurrence relations for single moments Recurrence relations for product moments Relations between moments of order statistics from two related populations Normal and half normal distributions Cauchy distribution Logistic and related distributions Gamma and related distributions Exponential and related distributions Power function and related distributions Pareto and related distributions Rayleigh distribution Linear-exponential distribution Lomax distribution Log-logistic and related distributions Burr and truncated Burr distributions Doubly truncated parabolic and skewed distributions Mixture of two exponential distributions Doubly truncated Laplace distribution A class of probability distributions 216 Acknowledgement 221 References 222

5 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 Some basic tools Characterizations based on order statistics 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 Characterizations of exponential distributions based on normalized spacings Related characterizations of other continuous distributions Characterizations of uniform distributions Characterizations of specific continuous distributions 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 Characterizations by sequences of moments and complete function sequences Characterizations of exponential distributions Related characterizations in classes of distributions Characterizations based on a single identity 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 The classical models 317

6 3. Applications and statistical inference Deviations from the classical models 329 Acknowledgements 330 References 331 Ch. 12. Order Statistics: Asymptotics in Applications 335 Pranab Kumar Sen 1. Introduction Some basic results in order statistics Some basic asymptotics in order statistics Robust estimation and order statistics: asymptotics in applications Trimmed LSE and regression quantiles Asymptotics for concomitants of order statistics Concomitant i-functionals and nonparametric regression Applications of order statistics in some reliability problems TTT asymptotics and tests for aging properties Concluding remarks 370 References 371 Ch. 13. Zero-One Laws for Large Order Statistics 375 R. J. Tomkins and Hong Wang 1. Introduction Zero-One laws for the upper-case probability Zero-one laws for the lower-case probability 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 Preliminaries The structure of Cook's D, Normal-Theory properties Modified versions of D/ Summary 400 References 401

7 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 Relations for single moments Relations for product moments Results for the multiple-outlier model (with a slippage of/) observations) Generalization to the truncated Pareto distribution Robustness of the MLE and BLUE Robustness of the censored BLUE 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 A semiparametric bootstrap approximation to A) A saddlepoint approximation to the bootstrap distribution Numerical implementation Simulation results 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 Smirnov's lemma Normal approximation Saddlepoint approximation Bootstrap approximation 479 References 482

8 PART VIII. RELATED STATISTICS Ch. 18. Concomitants of Order Statistics 487 H. A. David and H. N. Nagaraja 1. Introduction and summary Finite-sample distribution theory and moments Asymptotic theory Estimation and tests of hypotheses The rank of Y [rn] Selection through an associated variable Functions of concomitants 506 References 510 Ch. 19. A Record of Records 515 Valery B. Nevzorov and N. Balakrishnan 1. Introduction Classical records Definitions Representations of record times and record values using sums of independent terms Distributions and probability structure of record times Moments of records times and numbers of records Limit theorems for record times Inter-Record times Distributions and probability structure of record values in sequences of continuous random variables Limit theorems for record values from continuous distributions Record values from discrete distributions Weak records Bounds and approximations for moments of record values Recurrence relations for moments of record values Joint distributions of record times and record values Generalizations of the classical record model k lh record times k lh inter-record times k' h record values for the continuous case k lh record values for the discrete case Weak k' h record values ^-records Records in sequences of dependent random variables Random record models Nonstationary record models Multivariate records Relations between records and other probabilistic and statistical problems Nonclassical characterizations based on records Processes associated with records Diverse results 560

9 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 Weighted empirical processes based on observations "Bridge-type" two-time parameter empirical processes Weighted empirical processes based on ranks Weighted empirical processes based on sequential ranks "Bridge-type" empirical processes of sequential ranks Contiguous alternatives 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 Deviations between the general and uniform quantile processes and their sequential versions Weighted sequential quantile processes in supremum and /.^-metrics A summary of the classical Bahadur-Kiefer process theory via strong invariance principles An extension of the classical Bahadur-Kiefer process theory via strong invariance principles 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