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 concernยญ 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 approxiยญ 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 estimaยญ 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
CONTENT
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 -- 2 Multivariate Order Statistics -- 3 Inequalities and the Concept of Expansions -- II Asymptotic Theory -- 4 Approximations to Distributions of Central Order Statistics -- 5 Approximations to Distributions of Extremes -- 6 Other Important Approximations -- 7 Approximations in the Multivariate Case -- III Statistical Models and Procedures -- 8 Evaluating the Quantile and Density Quantile Function -- 9 Extreme Value Models -- 10 Approximate Sufficiency of Sparse Order Statistics -- Appendix 1. The Generalized Inverse -- Appendix 2. Two Technical Lemmas on Expansions -- Appendix 3. Further Results on Distances of Measures -- Author Index
