Distributional analysis of a generalization of the Polya process

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• 作者：Gordon E. Willmot
• 关键词：Nonhomogeneous birth process ; Negative binomial distribution ; Compound Poisson distribution ; Geometric distribution ; Completely monotone ; Mixture of geometrics ; Logarithmic series distribution ; STER distribution ; Incomplete gamma function ; Incomplete beta
• 刊名：Insurance: Mathematics and Economics
• 出版年：2010
• 出版时间：December 2010
• 年：2010
• 卷：47
• 期：3
• 页码：423-427
• 全文大小：228 K

文摘

A nonhomogeneous birth process generalizing the Polya process is analyzed, and the distribution of the transition probabilities is shown to be the convolution of a negative binomial distribution and a compound Poisson distribution, whose secondary distribution is a mixture of zero-truncated geometric distributions. A simplified form of the secondary distribution is obtained when the transition intensities have a particular structure, and may sometimes be expressed in terms of Stirling numbers and special functions such as the incomplete gamma function, the incomplete beta function, and the exponential integral. Conditions under which the compound Poisson form of the marginal distributions may be improved to a geometric mixture are also given.