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.

翻译：负多项分布在许多应用领域中出现，例如极化图像处理和纵向计数数据分析。在以往研究中，Mosimann（1963）推导了负多项分布下降阶乘矩的通用公式，而Withers与Nadarajah（2014）则获得了其累积量的表达式。尽管矩生成函数已知，但至今尚未计算出全面的矩公式。本文通过给出负多项分布中心矩和非中心矩的通用公式来填补这一空白。这些公式以二项式系数和第二类斯特林数表示。利用这些公式，我们提供了所有四阶及以下中心矩和所有八阶及以下非中心矩的显式表达式。