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.
Highlights
Throughout this paper, Gamma(α, β) denotes the gamma distribution with density f (x) =xα−1 e−x/β βαΓ(α) for x > 0
To avoid analytical or computational difficulties, it is useful in certain applications to approximate the exact convolution by a single gamma distribution
Gamma approximation to the generalized Dickman distribution is considered in detail in Section 4.3, where three gamma approximations are proposed as alternatives to the one with the same mean and variance. (A brief account of this distribution is included as well.) Appendix A is devoted to proofs
Summary
Throughout this paper, Gamma(α, β) denotes the gamma distribution with density f (x). When all the βi are equal, S is gamma distributed, and no approximation is required. This case is important from a mathematical/theoretical point of view, and is not excluded. Convolutions of gamma distributions (or sums of independent gamma variables) occur often, in particular in practical applications. To avoid analytical or computational difficulties, it is useful in certain applications to approximate the exact convolution by a single gamma distribution. Gamma approximation to the generalized Dickman distribution is considered in detail, where three gamma approximations are proposed as alternatives to the one with the same mean and variance. Gamma approximation to the generalized Dickman distribution is considered in detail in Section 4.3, where three gamma approximations are proposed as alternatives to the one with the same mean and variance. (A brief account of this distribution is included as well.) Appendix A is devoted to proofs
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