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

Shai Covo, Amir Elalouf

Research output: Contribution to journalArticlepeer-review

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 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.

Original languageEnglish
Pages (from-to)894-926
Number of pages33
JournalElectronic Journal of Statistics
Volume8
Issue number1
DOIs
StatePublished - 2014

Keywords

  • Approximation
  • Gamma process
  • Generalized Dickman distribution
  • Infinitely divisible distributions
  • Lévy measure
  • Moment-matching method
  • Negative binomial distribution
  • Sum of independent gamma variables

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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