The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. General formulas for the falling factorial moments of the negative multinomial distribution have been obtained in the past by Mosimann (1963), and similarly for cumulants by Withers & Nadarajah (2014). However, to the best of our knowledge, no one has ever calculated general formulas for the moments (although the moment generating function is known, see, e.g., Chapter~36 of Johnson et al. (1997), it is unpractical). In this paper, we fill this gap by providing general formulas for the central and non-central moments of the negative multinomial distribution in terms of binomial coefficients and Stirling numbers of the second kind. We use the formulas to give explicit expressions for all central moments up to the $4^{\text{th}}$ order and all non-central moments up to the $8^{\text{th}}$ order.

翻译：负多项分布在诸多应用领域出现，如偏振图像处理和纵向计数数据分析。既往Mosimann（1963）已推导出负多项分布下降阶乘矩的通用公式，Withers与Nadarajah（2014）亦给出了其累积量的类似公式。然而，据我们所知，尚无学者计算出该分布的矩通用公式（尽管已知其矩生成函数——参见Johnson等（1997）第36章——但该函数并不实用）。本文通过提供以二项式系数和第二类斯特林数表达的负多项分布中心矩与非中心矩通用公式填补了这一空白。我们利用这些公式给出了直至四阶的所有中心矩和直至八阶的所有非中心矩的显式表达式。