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 order 4 and all non-central moments up to order 8.
翻译：Mosimann(1963年)过去曾获得过负多部分布负因子时的普通公式,而Westers & Nadarajah(2014年)也曾获得过类似积分。然而,据我们所知,没有人曾计算过这些时的通用公式(尽管人们知道产生时的功能,例如,见Johnson等人(1997年)第~36章,这是不实际的)。在本文中,我们通过以二进制系数和第二类型恒定数字提供负多部分布中央和非中央时的通用公式来填补这一空白。我们用这些公式来明确表达所有中心时段的表达方式,以顺序排列4,所有非中央时段到8。我们用这些公式来表示所有中央时段的清晰表达方式,以顺序顺序排列4,所有非中央时段到8。
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