On computing the constant in the Berry-Esseen inequality for a class of distributions
type: Item
instance of: scholarly article
title: On computing the constant in the Berry-Esseen inequality for a class of distributions
zbMATH Open document ID: 0795.60018
DOI: 10.1007/BF00970674
author: A. Mitalauskas
published in: Lithuanian Mathematical Journal
publication date: 1994-07-21T00:00:00Z
review text: Let us consider a sequence of i.i.d. random variables \(\{\xi_ j\}\), \(j=1,2,\dots\), with expectations \(E \xi_ j=0\), variance \(D \xi_ j = \sigma^ 2\), and finite absolute moments \(E | \xi_ j |^ 3=\beta_ 3\). Recently, there has been a growing interest in the well- known Berry-Esseen inequality: \[ \Delta_ n=\sup_ x \left | P \left\{ {1 \over \sigma \sqrt n} \sum^ n_{j=1} \xi_ j<x \right\} - \Phi (x) \right | \leq {A \beta_ 3 \over \sigma^ 3 \sqrt n}. \] Sufficiently accurate general estimates of the constant in the inequality have already been obtained (for \(n=1\) and for \(n \to \infty)\). However, these estimates are not always satisfactory. They include cases of extremal distributions as well; hence they frequently appear to be too rough for distributions which are common in practice. Therefore, it is sensible to consider specific cases in order to obtain better estimates for the constant mentioned. The present paper is devoted to a more thorough investigation of the problem for a class of distributions, namely for those with the density function \[ p(x)=\begin{cases} p, & x \in [- 1,0], \\ 1-p, & x \in (0,1], \\ 0, & x \notin [-1,1] \end{cases} \] for various values of \(p\), \(0 \leq p \leq 1\).
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 578669
zbMATH Keywords: Berry-Esseen inequality
MaRDI profile type: Publication
cites work: Estimates of the proximity of a distribution to the normal law
cites work: Q3036382
cites work: Estimation of the constant in the Berry-Esseen inequality
full work available at URL: https://doi.org/10.1007/bf00970674
OpenAlex ID: W1967233456
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description: scientific article; zbMATH DE number 578669