Within the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD’s) to allow both overdispersion and underdispersion. Similarly to Kemp’s generalized hypergeometric probability distribution, which is based on hypergeometric functions, we analyze a class of WPD’s related to a generalization of Mittag–Leffler functions. The proposed class of distributions includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature.

在超分散计数数据的概率模型框架内，我们提出了广义分数泊松分布（gfPd），这是分数泊松分布（fPd）和标准泊松分布的自然概括。我们得出gfPd的某些属性，更具体地说，我们研究fPd的力矩，限制行为和其他特征。偏度表明fPd可以是左偏，右偏或对称的。这使得该模型在实践中具有灵活性和吸引力。我们将模型应用于实际的大计数数据，并使用最大似然估计模型参数。然后，我们转向非常普通的加权泊松分布（WPD's）类，以允许过度分散和欠分散。与Kemp的广义超几何概率分布相似，基于超几何函数，我们分析了一类与Mittag-Leffler函数泛化有关的WPD。提议的分布类别包括著名的COM-泊松模型和超泊松模型。我们在允许过度分散和分散不足的参数上表征条件，并分析了两个尚未在文献中出现的特殊情况。