Approximate Distributions of Order Statistics

20.03.1989

Englisch

9780387968513

Springer US

Springer Series in Statistics

Nein

Rolf-Dieter Reiss

355

Gebundene Ausgabe

With Applications to Nonparametric Statistics

Mathematica, Random variable, statistics, statistical theory, Parametric statistics, probability

PBW - Angewandte Mathematik

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.

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