Moments of the negative multinomial distribution

TL;DR

This paper derives comprehensive formulas for the central and non-central moments of the negative multinomial distribution, filling a gap in the literature and enabling detailed statistical analysis.

Contribution

It provides the first complete formulas for moments of the negative multinomial distribution expressed with binomial coefficients and Stirling numbers.

Findings

Explicit formulas for moments up to 8th order

Expressions for central moments up to 4th order

Facilitates advanced statistical analysis of negative multinomial data

Abstract

The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, Mosimann (1963) derived general formulas for the falling factorial moments of the negative multinomial distribution, while Withers & Nadarajah (2014) obtained expressions for the cumulants. Despite the availability of the moment generating function, no comprehensive formulas for the moments have been calculated thus far. This paper addresses this gap by presenting general formulas for both central and non-central moments of the negative multinomial distribution. These formulas are expressed in terms of binomial coefficients and Stirling numbers of the second kind. Utilizing these formulas, we provide explicit expressions for all central moments up to the 4th order and all non-central moments up to the 8th order.