Zero biasing and a discrete central limit theorem
| dc.creator | Goldstein, Larry | |
| dc.creator | Xia, Aihua | |
| dc.date | 2005-09-20 | |
| dc.date | 2006-11-22 | |
| dc.date.accessioned | 2026-07-07T06:43:04Z | |
| dc.date.available | 2026-07-07T06:43:04Z | |
| dc.description | We introduce a new family of distributions to approximate $\mathbb {P}(W\in A)$ for $A\subset\{...,-2,-1,0,1,2,...\}$ and $W$ a sum of independent integer-valued random variables $ξ_1$, $ξ_2$, $...,$ $ξ_n$ 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 $\mathbb {E}(W)$ and variance $\operatorname {Var}(W)$, $\mathbb {P}(Z\in A)$ provides a good approximation to $\mathbb {P}(W\in A)$ for $A$ of the form $(-\infty,x]$. 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, nonnormal 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. | |
| dc.description | Published at http://dx.doi.org/10.1214/009117906000000250 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/math/0509444 | |
| dc.identifier | http://arxiv.org/abs/math/0509444 | |
| dc.identifier | Annals of Probability 2006, Vol. 34, No. 5, 1782-1806 | |
| dc.identifier | doi:10.1214/009117906000000250 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/102275 | |
| dc.subject | Probability | |
| dc.subject | 60F05 (Primary) 60G50 (Secondary) | |
| dc.title | Zero biasing and a discrete central limit theorem | |
| dc.type | text |