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A non-uniform bound for translated Poisson approximation


Barbour, A D; Choi, K (2004). A non-uniform bound for translated Poisson approximation. Electronic Journal of Probability, 9(2):18-36.

Abstract

Let $X_1, ldots , X_n$ be independent, integer valued random variables, with $p^{text{th}}$ moments, $p >2$, and let $W$ denote their sum. We prove bounds analogous to the classical non-uniform estimates of the error in the central limit theorem, but now, for approximation of $law(W)$ by a translated Poisson distribution. The advantage is that the error bounds, which are often of order no worse than in the classical case, measure the accuracy in terms of total variation distance. In order to have good approximation in this sense, it is necessary for $law(W)$ to be sufficiently smooth; this requirement is incorporated into the bounds by way of a parameter $al$, which measures the average overlap between $law(X_i)$ and $law(X_i+1), , 1 le i le n.$

Abstract

Let $X_1, ldots , X_n$ be independent, integer valued random variables, with $p^{text{th}}$ moments, $p >2$, and let $W$ denote their sum. We prove bounds analogous to the classical non-uniform estimates of the error in the central limit theorem, but now, for approximation of $law(W)$ by a translated Poisson distribution. The advantage is that the error bounds, which are often of order no worse than in the classical case, measure the accuracy in terms of total variation distance. In order to have good approximation in this sense, it is necessary for $law(W)$ to be sufficiently smooth; this requirement is incorporated into the bounds by way of a parameter $al$, which measures the average overlap between $law(X_i)$ and $law(X_i+1), , 1 le i le n.$

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:4 February 2004
Deposited On:16 Nov 2009 20:34
Last Modified:24 Sep 2019 16:16
Publisher:Institute of Mathematical Statistics
ISSN:1083-6489
OA Status:Green
Publisher DOI:https://doi.org/10.1214/EJP.v9-182
Official URL:http://www.emis.de/journals/EJP-ECP/_ejpecp/viewarticle671b.html?id=1425

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