The Poisson, geometric and Bernoulli distributions are special cases of a flexible count distribution, namely the Conway-Maxwell-Poisson (CMP) distribution – a two-parameter generalization of the Poisson distribution that can accommodate data over- or under-dispersion. This work further generalizes the ideas of the CMP distribution by considering sums of CMP random variables to establish a flexible class of distributions that encompasses the Poisson, negative binomial, and binomial distributions as special cases. This sum-of-Conway-Maxwell-Poissons (sCMP) class captures the CMP and its special cases, as well as the classical negative binomial and binomial distributions. Through simulated and real data examples, we demonstrate this model’s flexibility, encompassing several classical distributions as well as other count data distributions containing significant data dispersion.

中文翻译：

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计数数据的灵活分配类

Poisson分布，几何分布和Bernoulli分布是灵活计数分布的特殊情况，即Conway-Maxwell-Poisson（CMP）分布– Poisson分布的两参数泛化，可以容纳过度分散或分散的数据。这项工作通过考虑CMP随机变量的总和来进一步建立CMP分布的思想，以建立包括Poisson分布，负二项式分布和二项式分布作为特殊情况的灵活的分布类。此Conway-Maxway-Maxwell-Poissons（sCMP）类捕获CMP及其特殊情况，以及经典的负二项式和二项式分布。通过模拟和真实的数据示例，我们证明了该模型的灵活性，