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Approximation of the difference of two Poisson-like counts by Skellam

Part of: Distribution theory Limit theorems Markov processes

Published online by Cambridge University Press:  26 July 2018

H. L. Gan  and
Eric D. Kolaczyk
H. L. Gan*
Affiliation:
Northwestern University
Eric D. Kolaczyk*
Affiliation:
Boston University
*
* Postal address: Mathematics Department, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA. Email address: [email protected]
** Postal address: Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA 02215, USA. Email address: [email protected]
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Abstract

Poisson-like behavior for event count data is ubiquitous in nature. At the same time, differencing of such counts arises in the course of data processing in a variety of areas of application. As a result, the Skellam distribution – defined as the distribution of the difference of two independent Poisson random variables – is a natural candidate for approximating the difference of Poisson-like event counts. However, in many contexts strict independence, whether between counts or among events within counts, is not a tenable assumption. Here we characterize the accuracy in approximating the difference of Poisson-like counts by a Skellam random variable. Our results fully generalize existing, more limited, results in this direction and, at the same time, our derivations are significantly more concise and elegant. We illustrate the potential impact of these results in the context of problems from network analysis and image processing, where various forms of weak dependence can be expected.

Keywords

Skellam approximationStein's methodPoisson approximation

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60F05: Central limit and other weak theorems 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 416 - 430
Copyright
Copyright © Applied Probability Trust 2018 

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