Inverse moments of univariate discrete distributions via the Poisson expansion

TL;DR

This paper introduces a series expansion method for calculating inverse moments of non-negative discrete distributions using factorial cumulants and Poisson-Charlier expansion, with applications to the positive binomial distribution.

Contribution

It provides a novel series expansion approach for inverse moments of discrete distributions, including convergence guarantees and error bounds.

Findings

Derived a convergent series for inverse moments of the positive binomial distribution.

Established uniform error bounds across the entire parameter interval.

Demonstrated applicability of the method to discrete distributions using factorial cumulants.

Abstract

In this note we present a series expansion of inverse moments of a non-negative discrete random variate in terms of its factorial cumulants, based on the Poisson-Charlier expansion of a discrete distribution. We apply the general method to the positive binomial distribution and obtain a convergent series for its inverse moments with an error residual that is uniformly bounded on the entire interval 0<=p<=1.