Approximate Distributions of Order Statistics

This book is designed as a unified and mathematically rigorous treatment of some recent developments of the asymptotic distribution theory of order statistics (including the extreme order statistics) that are relevant for statistical theory and its applications. Particular emphasis is placed on results concern­ ing the accuracy oflimit theorems, on higher order approximations, and other approximations in quite a general sense. Contrary to the classical limit theorems that primarily concern the weak convergence of distribution functions, our main results will be formulated in terms of the variational and the Hellinger distance. These results will form the proper springboard for the investigation of parametric approximations of nonparametric models of joint distributions of order statistics. The approxi­ mating models include normal as well as extreme value models. Several applications will show the usefulness of this approach. Other recent developments in statistics like nonparametric curve estima­ tion and the bootstrap method will be studied as far as order statistics are concerned. 1n connection with this, graphical methods will, to some extent, be explored.

0 Introduction.- 0.1. Weak and Strong Convergence.- 0.2. Approximations.- 0.3. The Role of Order Statistics in Nonparametric Statistics.- 0.4. Central and Extreme Order Statistics.- 0.5. The Restriction to Independent and Identically Distributed Random Variables.- 0.6. Graphical Methods.- 0.7. A Guide to the Contents.- 0.8. Notation and Conventions.- I Exact Distributions and Basic Tools.- 1 Distribution Functions, Densities, and Representations.- 1.1. Introduction to Basic Concepts.- 1.2. The Quantile Transformation.- 1.3. Single Order Statistics, Extremes.- 1.4. Joint Distribution of Several Order Statistics.- 1.5. Extensions to Continuous and Discontinuous Distribution Functions.- 1.6. Spacings, Representations, Generalized Pareto Distribution Functions.- 1.7. Moments, Modes, and Medians.- 1.8. Conditional Distributions of Order Statistics.- P.1. Problems and Supplements.- Bibliographical Notes.- 2 Multivariate Order Statistics.- 2.1. Introduction.- 2.2. Distribution Functions and Densities.- P.2. Problems and Supplements.- Bibliographical Notes.- 3 Inequalities and the Concept of Expansions.- 3.1. Inequalities for Distributions of Order Statistics.- 3.2. Expansions of Finite Length.- 3.3. Distances of Measures: Convergence and Inequalities.- P.3. Problems and Supplements.- Bibliographical Notes.- II Asymptotic Theory.- 4 Approximations to Distributions of Central Order Statistics.- 4.1. Asymptotic Normality of Central Sequences.- 4.2. Expansions: A Single Central Order Statistic.- 4.3. Asymptotic Independence from the Underlying Distribution Function.- 4.4. The Approximate Multivariate Normal Distribution.- 4.5. Asymptotic Normality and Expansions of Joint Distributions.- 4.6. Expansions of Distribution Functions of Order Statistics.- 4.7. Local Limit Theorems and Moderate Deviations.- P.4. Problems and Supplements.- Bibliographical Notes.- 5 Approximations to Distributions of Extremes.- 5.1. Asymptotic Distributions of Extreme Sequences.- 5.2. Hellinger Distance between Exact and Approximate Distributions of Sample Maxima.- 5.3. The Structure of Asymptotic Joint Distributions of Extremes.- 5.4. Expansions of Distributions of Extremes of Generalized Pareto Random Variables.- 5.5. Variational Distance between Exact and Approximate Joint Distributions of Extremes.- 5.6. Variational Distance between Empirical and Poisson Processes.- P.5. Problems and Supplements.- Bibliographical Notes.- 6 Other Important Approximations.- 6.1. Approximations of Moments and Quantiles.- 6.2. Functions of Order Statistics.- 6.3. Bahadur Approximation.- 6.4. Bootstrap Distribution Function of a Quantile.- P.6. Problems and Supplements.- Bibliographical Notes.- 7 Approximations in the Multivariate Case.- 7.1. Asymptotic Normality of Central Order Statistics.- 7.2. Multivariate Extremes.- P.7. Problems and Supplements.- Bibliographical Notes.- III Statistical Models and Procedures.- 8 Evaluating the Quantile and Density Quantile Function.- 8.1. Sample Quantiles.- 8.2. Kernel Type Estimators of Quantiles.- 8.3. Asymptotic Performance of Quantile Estimators.- 8.4. Bootstrap via Smooth Sample Quantile Function.- P.8. Problems and Supplements.- Bibliographical Notes.- 9 Extreme Value Models.- 9.1. Some Basic Concepts of Statistical Theory.- 9.2. Efficient Estimation in Extreme Value Models.- 9.3. Semiparametric Models for Sample Maxima.- 9.4. Parametric Models Belonging to Upper Extremes.- 9.5. Inference Based on Upper Extremes.- 9.6. Comparison of Different Approaches.- 9.7. Estimating the Quantile Function Near the Endpoints.- P.9. Problems and Supplements.- Bibliographical Notes.- 10 Approximate Sufficiency of Sparse Order Statistics.- 10.1. Comparison of Statistical Models via Markov Kernels.- 10.2. Approximate Sufficiency over a Neighborhood of a Fixed Distribution.- 10.3. Approximate Sufficiency over a Neighborhood of a Family of Distributions.- 10.4. Local Comparison of a Nonparametric Model and a Normal Model.- P. 10. Problems and Supplements.- Bibliographical Notes.- Appendix 1. The Generalized Inverse.- Appendix 2. Two Technical Lemmas on Expansions.- Appendix 3. Further Results on Distances of Measures.- Author Index.