TY - JOUR

T1 - A novel single-gamma approximation to the sum of independent gamma variables, and a generalization to infinitely divisible distributions

AU - Covo, Shai

AU - Elalouf, Amir

PY - 2014

Y1 - 2014

N2 - 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 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.

AB - 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 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.

KW - Approximation

KW - Gamma process

KW - Generalized Dickman distribution

KW - Infinitely divisible distributions

KW - Lévy measure

KW - Moment-matching method

KW - Negative binomial distribution

KW - Sum of independent gamma variables

UR - http://www.scopus.com/inward/record.url?scp=84904045013&partnerID=8YFLogxK

U2 - 10.1214/14-EJS914

DO - 10.1214/14-EJS914

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AN - SCOPUS:84904045013

SN - 1935-7524

VL - 8

SP - 894

EP - 926

JO - Electronic Journal of Statistics

JF - Electronic Journal of Statistics

IS - 1

ER -