Saddlepoint Approximations for Stopped-Sum Distributions

Abstract

One of the common used classes of distributions is the stopped-sum class. This class includes Hermite distribution, Polya–Aeppli distribution, Poisson-Gamma distribution, and Neyman type A. This article introduces the saddlepoint approximations to the stopped-sum class in continuous and discrete settings. We discuss approximations for mass/density and cumulative distribution functions of stopped-sum distributions. Examples of continuous and discrete distributions from the Poisson stopped-sum class are presented. Comparisons between saddlepoint approximations and the exact calculations show the great accuracy of the saddlepoint methods.

Keywords:

• Poisson-Gamma stopped-sum
• Polya–Aeppli stopped-sum distribution
• Saddlepoint approximations
• stopped-sum distribution

Mathematics Subject Classification:

• 62E17
• 62E15
• 62E20

Notes

*Requires a computation at the mean.