Approximate Distributions of Order Statistics

1. Front Matter

   Pages i-xii

   PDF

2. Introduction

   1. Introduction

      • R.-D. Reiss

      Pages 1-8

3. Exact Distributions and Basic Tools

   1. Front Matter

      Pages 9-9

      PDF

   2. Distribution Functions, Densities, and Representations

      • R.-D. Reiss

      Pages 11-63

   3. Multivariate Order Statistics

      • R.-D. Reiss

      Pages 64-82

   4. Inequalities and the Concept of Expansions

      • R.-D. Reiss

      Pages 83-104

4. Asymptotic Theory

   1. Front Matter

      Pages 105-105

      PDF

   2. Approximations to Distributions of Central Order Statistics

      • R.-D. Reiss

      Pages 107-150

   3. Approximations to Distributions of Extremes

      • R.-D. Reiss

      Pages 151-205

   4. Other Important Approximations

      • R.-D. Reiss

      Pages 206-228

   5. Approximations in the Multivariate Case

      • R.-D. Reiss

      Pages 229-239

5. Statistical Models and Procedures

   1. Front Matter

      Pages 241-241

      PDF

   2. Evaluating the Quantile and Density Quantile Function

      • R.-D. Reiss

      Pages 243-271

   3. Extreme Value Models

      • R.-D. Reiss

      Pages 272-291

   4. Approximate Sufficiency of Sparse Order Statistics

      • R.-D. Reiss

      Pages 292-317

6. Back Matter

   Pages 318-355

   PDF

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.

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

• Fachbereich 6, Mathematik, Universität Gesamthochschule Siegen, Siegen, Federal Republic of Germany

  R.-D. Reiss

Policies and ethics