Sums, Trimmed Sums and Extremes

ISBN-13: 9781468467956 / Angielski / Miękka / 2012 / 418 str.

Hahn

Opis

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.

Spis treści

I Approaches to Trimming and Self-normalization Based on Analytic Methods.- Asymptotic Behavior of Partial Sums: A More Robust Approach Via Trimming and Self-Normalization.- Weak Convergence of Trimmed Sums.- Invariance Principles and Self-Normalizations for Sums Trimmed According to Choice of Influence Function.- On Joint Estimation of an Exponent of Regular Variation and an Asymmetry Parameter for Tail Distributions.- Center, Scale and Asymptotic Normality for Censored Sums of Independent, Nonidentically Distributed Random Variables.- A Review of Some Asymptotic Properties of Trimmed Sums of Multivariate Data.- II The Quantile-Transform-Empirical-Process Approach to Trimming.- The Quantile-Transform-Empirical-Process Approach to Limit Theorems for Sums of Order Statistics.- A Note on Weighted Approximations to the Uniform Empirical and Quantile Processes.- Limit Theorems for the Petersburg Game.- A Probabilistic Approach to the Tails of Infinitely Divisible Laws.- The Quantile-Transform Approach to the Asymptotic Distribution of Modulus Trimmed Sums.- On the Asymptotic Behavior of Sums of Order Statistics from a Distribution with a Slowly Varying Upper Tail.- Limit Results for Linear Combinations.- Non-Normality of a Class of Random Variables.