Generalized hypergeometric, digamma and trigamma distributions
Generalized hypergeometric, digamma and trigamma distributions
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广义超几何、二伽玛和三伽玛分布
DOI：
10.1007/bf02480295
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发表时间：
1979
影响因子：
1
通讯作者：
M. Sibuya
中科院分区：
文献类型：
--
作者：
M. Sibuya
本文介绍并研究了定义在所有正整数集合上的新的概率分布--“digamma”和“trigamma”。它们作为截零的B3型广义超几何分布（逆Pólya-Eggenberger分布或负二项beta分布）的极限，也可以通过复合对数级数分布得到。双伽玛分布族以对数级数作为极限，以三伽玛作为另一极限。三伽马分布非常接近zeta（Zipf）分布。因此，当观测到的频率数据有一个很长的尾巴，不能用对数级数分布拟合时，我们的新分布可以作为对数级数的替代品。在开始部分，我们总结了B3型广义超几何分布的性质。它强调的是，分布是通过复合泊松分布的“伽马乘积比”分布。
SummaryIn this paper we introduce and study new probability distributions named “digamma” and “trigamma” defined on the set of all positive integers. They are obtained as limits of the zero-truncated Type B3 generalized hypergeometric distributions (inverse Pólya-Eggenberger or negative binomial beta distributions), and also by compounding the logarithmic series distributions.The family of digamma distributions has the logarithmic series as a limit and the trigamma as another limit. The trigamma distributions are very close to the zeta (Zipf) distributions. Thus, our new distributions are useful as substitutes of the logarithmic series when the observed frequency data have such a long tail that cannot be fitted by the latter distributions.In the beginning sections we summarize properties of the Type B3 generalized hypergeometric distributions. It is emphasized that the distributions are obtained by compounding a Poisson distribution by “gamma product-ratio” distributions.