 # Compound poisson approximation via information functionals

Barbour, A D; Johnson, O; Kontoyiannis, I; Madiman, M (2010). Compound poisson approximation via information functionals. Electronic Journal of Probability, 15(42):1344-1369.

## Abstract

An information-theoretic development is given for the problem of compound Poisson approximation, which parallels earlier treatments for Gaussian and Poisson approximation. Nonasymptotic bounds are derived for the distance between the distribution of a sum of independent integer-valued random variables and an appropriately chosen compound Poisson law. In the case where all summands have the same conditional distribution given that they are non-zero, a bound on the relative entropy distance between their sum and the compound Poisson distribution is derived, based on the data-processing property of relative entropy and earlier Poisson approximation results. When the summands have arbitrary distributions, corresponding bounds are derived in terms of the total variation distance. The main technical ingredient is the introduction of two "information functionals,'' and the analysis of their properties. These information functionals play a role analogous to that of the classical Fisher information in normal approximation. Detailed comparisons are made between the resulting inequalities and related bounds.

## Abstract

An information-theoretic development is given for the problem of compound Poisson approximation, which parallels earlier treatments for Gaussian and Poisson approximation. Nonasymptotic bounds are derived for the distance between the distribution of a sum of independent integer-valued random variables and an appropriately chosen compound Poisson law. In the case where all summands have the same conditional distribution given that they are non-zero, a bound on the relative entropy distance between their sum and the compound Poisson distribution is derived, based on the data-processing property of relative entropy and earlier Poisson approximation results. When the summands have arbitrary distributions, corresponding bounds are derived in terms of the total variation distance. The main technical ingredient is the introduction of two "information functionals,'' and the analysis of their properties. These information functionals play a role analogous to that of the classical Fisher information in normal approximation. Detailed comparisons are made between the resulting inequalities and related bounds.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Statistics and Probability Social Sciences & Humanities > Statistics, Probability and Uncertainty English 31 August 2010 23 Dec 2010 13:53 29 Jul 2020 23:08 Institute of Mathematical Statistics 1083-6489 Hybrid https://doi.org/10.1214/EJP.v15-799 http://www.math.washington.edu/~ejpecp/EjpVol15/paper42.abs.html http://arxiv.org/abs/1004.3692

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