General formulas for the central and non-central moments of the multinomial distribution

TL;DR

This paper derives comprehensive formulas for the central and non-central moments of the multinomial distribution, enabling explicit calculations up to high orders and advancing previous limited results.

Contribution

It provides the first general formulas for moments of the multinomial distribution, extending calculations to higher orders than prior work.

Findings

Explicit formulas for non-central moments up to order 8.

Explicit formulas for central moments up to order 4.

Significant extension of previous moment calculations.

Abstract

We present the first general formulas for the central and non-central moments of the multinomial distribution, using a combinatorial argument and the factorial moments previously obtained in Mosimann (1962). We use the formulas to give explicit expressions for all the non-central moments up to order 8 and all the central moments up to order 4. These results expand significantly on those in Newcomer (2008) and Newcomer et al. (2008), where the non-central moments were calculated up to order 4.