Zero biasing and a discrete central limit theorem

• 成果类型: Article
• 署名作者: Goldstein, Larry; Xia, Aihua
• 署名单位: University of Southern California; University of Melbourne
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000250
• 发表日期: 2006
• 页码: 1782-1806
• 关键词: steins method; approximation

摘要

We introduce a new family of distributions to approximate P(W epsilon A) for A subset of {...-2, -1, 0,1, 2...} and W a sum of independent integer-valued random variables xi(1), xi(2), ..., xi(n) with finite second moments, where, with large probability, W is not concentrated on a lattice of span greater than 1. The well-known Berry-Esseen theorem states that, for Z a normal random variable with mean E(W) and variance Var(W), P(Z epsilon A) provides a good approximation to P(W epsilon A) for A of the form (-infinity, x]. However, for more general A, such as the set of all even numbers, the normal approximation becomes unsatisfactory and it is desirable to have an appropriate discrete, normal distribution which approximates W in total variation, and a discrete version of the Berry-Esseen theorem to bound the error. In this paper, using the concept of zero biasing for discrete random variables (cf. Goldstein and Reinert [J. Theoret. Probab. 18 (2005) 237-260]), we introduce a new family of discrete distributions and provide a discrete version of the Berry-Esseen theorem showing how members of the family approximate the distribution of a sum W of integer-valued variables in total variation.