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A novel single-gamma approximation to the sum of independent gamma variables, and a generalization to infinitely divisible distributions. (English) Zbl 1348.62045

Summary: 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.

MSC:

62E17 Approximations to statistical distributions (nonasymptotic)
60G50 Sums of independent random variables; random walks
60G51 Processes with independent increments; Lévy processes
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References:

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