On computing the constant in the Berry-Esseen inequality for a class of distributions
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 \(p\), \(0 \leq p \leq 1\).
scientific article; zbMATH DE number 578669
scientific article