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2014 A novel single-gamma approximation to the sum of independent gamma variables, and a generalization to infinitely divisible distributions
Shai Covo, Amir Elalouf
Electron. J. Statist. 8(1): 894-926 (2014). DOI: 10.1214/14-EJS914

Abstract

It is well known that the sum $S$ of $n$ independent gamma variables—which occurs often, in particular in practical applications—can typically be well approximated by a single gamma variable with the same mean and variance (the distribution of $S$ being quite complicated in general). In this paper, we propose an alternative (and apparently at least as good) single-gamma approximation to $S$. The methodology used to derive it is based on the observation that the jump density of $S$ bears an evident similarity to that of a generic gamma variable, $S$ being viewed as a sum of $n$ independent gamma processes evaluated at time $1$. This observation motivates the idea of a gamma approximation to $S$ in the first place, and, in principle, a variety of such approximations can be made based on it. The same methodology can be applied to obtain gamma approximations to a wide variety of important infinitely divisible distributions on $\mathbb{R}_{+}$ or at least predict/confirm the appropriateness of the moment-matching method (where the first two moments are matched); this is demonstrated neatly in the cases of negative binomial and generalized Dickman distributions, thus highlighting the paper’s contribution to the overall topic.

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Shai Covo. Amir Elalouf. "A novel single-gamma approximation to the sum of independent gamma variables, and a generalization to infinitely divisible distributions." Electron. J. Statist. 8 (1) 894 - 926, 2014. https://doi.org/10.1214/14-EJS914

Information

Published: 2014
First available in Project Euclid: 26 June 2014

zbMATH: 1348.62045
MathSciNet: MR3229102
Digital Object Identifier: 10.1214/14-EJS914

Subjects:
Primary: 62E17
Secondary: 60G50, 60G51

Rights: Copyright © 2014 The Institute of Mathematical Statistics and the Bernoulli Society

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Vol.8 • No. 1 • 2014
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