论文标题
负多项式分布的矩
Moments of the negative multinomial distribution
论文作者
论文摘要
负多项式分布出现在许多应用领域，例如偏振图像处理和纵向计数数据的分析。在先前的研究中，Mosimann（1963）衍生出用于负多项式分布的下降阶乘时刻的一般公式，而Withers＆Nadarajah（2014）获得了累积物的表达式。尽管瞬间生成函数的可用性，但迄今为止尚未计算出矩时的全面公式。本文通过为负多项式分布的中央和非中心力矩提出一般公式来解决这一差距。这些公式以第二类的二项式系数和stirling数字表示。利用这些公式，我们为所有中心时刻提供明确的表达式，直到第四阶和所有非中心时刻，直到第8阶。
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.
