An exponential nonuniform Berry-Esseen bound for self-normalized sums
type: Item
instance of: scholarly article
title: An exponential nonuniform Berry-Esseen bound for self-normalized sums
zbMATH Open document ID: 0972.60011
author: Qiying Wang
author: Bing-Yi Jing
published in: The Annals of Probability
publication date: 2001-11-13T00:00:00Z
review text: Nonuniform Berry-Esseen bounds for self-normalized sums of independent random variables, with application to the Student \(t\)-statistic, are obtained. They are substantially better than, in fact in sharp contrast with, those available for simply standardized sums in that the dependence on the range point \(x\) is exponentially instead of polynomial, even under only finiteness of the third moments. These results apply in particular to moderate deviations. Here is one of the corollaries: If \(X_i\) are independent identically distributed symmetric random variables with \(E|X_i|< \infty\) for all \(i\), and if we let \(L_{3n}:= n^{-1/2}E|X_1|^3/(EX^2_1)^{3/2}\), then the absolute deviation of the distribution function \(F_n(x)\) of \(\sum^n_{i=1} X_i/(\sum^n_{i=1} X^2_i)^{1/2}\) from that of the normal, \(\Phi(x)\), satisfies \[ |F_n(x)- \Phi(x)|\leq A\min\{(1+|x|^3) L_{3n},1\} \exp(- x^2/2), \] where \(A\) is an absolute constant.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 62F03
zbMATH DE Number: 1496121
zbMATH Keywords: Berry-Esseen bound
zbMATH Keywords: nonuniform estimate
zbMATH Keywords: self-normalized sums
zbMATH Keywords: \(t\)-statistic
zbMATH Keywords: moderate deviations
reviewed by: Evarist Giné
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677562
OpenAlex ID: W4235425958
DOI: 10.1214/AOP/1022677562
description: scientific article; zbMATH DE number 1496121