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Solving the Stein Equation in compound poisson approximation

Part of: Limit theorems Distribution theory

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour  and
Sergey Utev
A. D. Barbour*
Affiliation:
Universität Zürich
Sergey Utev*
Affiliation:
La Trobe University and Novosibirsk University
*
∗ Postal address: Abteilung für Angewandte Mathematik, Universität Zürich-Irchel, Winterthurerstrasse 190, CH 8057 Zürich, Switzerland. Email address: adb@amath.unizh.ch
∗∗ Postal address: (1) School of Statistical Science, La Trobe University, Bundoora, Melbourne, Vic. 3083, Australia, (2) Institute of Mathematics, Novosibirsk University.
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Abstract

The accuracy of compound Poisson approximation can be estimated using Stein's method in terms of quantities similar to those which must be calculated for Poisson approximation. However, the solutions of the relevant Stein equation may, in general, grow exponentially fast with the mean number of ‘clumps’, leading to many applications in which the bounds are of little use. In this paper, we introduce a method for circumventing this difficulty. We establish good bounds for those solutions of the Stein equation which are needed to measure the accuracy of approximation with respect to Kolmogorov distance, but only in a restricted range of the argument. The restriction on the range is then compensated by a truncation argument. Examples are given to show that the method clearly outperforms its competitors, as soon as the mean number of clumps is even moderately large.

Keywords

Stein's methodcompound Poissondistributional approximation

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998 , pp. 449 - 475
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Supported in part by Schweizerischer Nationalfonds Projekte Nr. 20-31262.91 and 21-37354.93.

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