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Sums, Trimmed Sums and Extremes

  • Marjorie G. Hahn
  • David M. Mason
  • Daniel C. Weiner

Part of the Progress in Probability book series (PRPR, volume 23)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Approaches to Trimming and Self-normalization Based on Analytic Methods

  3. The Quantile-Transform-Empirical-Process Approach to Trimming

    1. Front Matter
      Pages 215-215
    2. Sándor Csörgő, Erich Haeusler, David M. Mason
      Pages 215-267
    3. Sándor Csörgő, Rossitza Dodunekova
      Pages 285-315
    4. Sándor Csörgő, David M. Mason
      Pages 317-335
    5. Sándor Csörgő, Erich Haeusler, David M. Mason
      Pages 337-353
    6. Galen R. Shorack
      Pages 377-391
    7. David M. Mason, Galen R. Shorack
      Pages 393-416
  4. Back Matter
    Pages 417-417

About this book

Introduction

The past decade has seen a resurgence of interest in the study of the asymp­ totic behavior of sums formed from an independent sequence of random variables. In particular, recent attention has focused on the interaction of the extreme summands with, and their influence upon, the sum. As ob­ served by many authors, the limit theory for sums can be meaningfully expanded far beyond the scope of the classical theory if an "intermediate" portion (i. e. , an unbounded number but a vanishingly small proportion) of the extreme summands in the sum are deleted or otherwise modified (''trimmed',). The role of the normal law is magnified in these intermediate trimmed theories in that most or all of the resulting limit laws involve variance-mixtures of normals. The objective of this volume is to present the main approaches to this study of intermediate trimmed sums which have been developed so far, and to illustrate the methods with a variety of new results. The presentation has been divided into two parts. Part I explores the approaches which have evolved from classical analytical techniques (condi­ tionin~, Fourier methods, symmetrization, triangular array theory). Part II is Msed on the quantile transform technique and utilizes weak and strong approximations to uniform empirical process. The analytic approaches of Part I are represented by five articles involving two groups of authors.

Keywords

Finite Parameter Random variable Variance Variation approximation convergence distribution function statistics theorem variable

Editors and affiliations

  • Marjorie G. Hahn
    • 1
  • David M. Mason
    • 2
  • Daniel C. Weiner
    • 3
  1. 1.Department of MathematicsTufts UniversityMedfordUSA
  2. 2.Department of Mathematical SciencesUniversity of DelwareNewarkUSA
  3. 3.Department of MathematicsBoston UniversityBostonUSA

Bibliographic information