Integro-differential inequalities and the Poisson approximation
type: Item
instance of: scholarly article
title: Integro-differential inequalities and the Poisson approximation
zbMATH Open document ID: 0854.60022
author: V. Papathanasiou
author: S. A. Utev
published in: Siberian Advances in Mathematics
publication date: 1996-08-20T00:00:00Z
review text: \textit{T. Cacoullos} and the authors [Ann. Probab. 22, No. 3, 1607-1618 (1994; Zbl 0835.60023)] used Stein's method to establish bounds on the difference in total variation between the (absolutely continuous) distribution of a random variable \(X\) and the normal distribution with the same mean and variance, expressed in terms of the quantities \[ U_X = \sup_{g \in L_2} {\text{Var} g (X) \over \text{Var} XE \{[g'(X)]^2\}} \quad \text{and} \quad J_X = \inf_{g \in L_2} {\text{Var} g (X) \over \text{Var} XE^2[g'(X)]}. \] It is the case that \(U_X \geq 1\), and \(J_X \leq 1\), equality in each case characterizing the normal distribution. Analogous results are obtained for Poisson approximation in total variation of nonnegative integer valued random variables, with \(g'\) now being replaced by \(\Delta g\) in the definitions of \(U_X\) and \(J_X\). Applications to sums of independent Bernoulli and negative binomial random variables are given.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 918042
zbMATH Keywords: Stein's method
zbMATH Keywords: difference in total variation
zbMATH Keywords: Poisson approximation in total variation
zbMATH Keywords: negative binomial random variables
reviewed by: A. D. Barbour
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 918042