In this paper, a new three-parameter discrete family of distributions, the $r{\mathcal B}ell$ family, is introduced. The family is based on series expansion of the r-Bell polynomials. The proposed model generalises the classical Poisson and the recently proposed Bell and Bell–Touchard distributions. It exhibits interesting stochastic properties. Its probabilities can be computed by a recursive formula that allows us to calculate the probability function of the amount of aggregate claims in the collective risk model in terms of an integral equation. Univariate and bivariate regression models are presented. The former regression model is used to explain the number of out-of-use claims in an automobile insurance portfolio, by showing a good out-of-sample performance. The latter is used to describe the number of out-of-use and parking claims jointly. This family provides an alternative to other traditionally used distributions to describe count data such as the negative binomial and Poisson-inverse Gaussian models.

中文翻译：

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关于具有精算应用程序的分配的 ELL 系列

在本文中，一个新的三参数离散分布族，$r{\mathcal B}ell$家庭，介绍。该系列是基于系列扩展的r-贝尔多项式。所提出的模型概括了经典 Poisson 和最近提出的 Bell 和 Bell-Touchard 分布。它表现出有趣的随机特性。其概率可以通过递归公式计算，该公式允许我们根据积分方程计算集体风险模型中总索赔金额的概率函数。提出了单变量和双变量回归模型。前一个回归模型通过显示良好的样本外表现来解释汽车保险组合中的废弃索赔数量。后者用于联合描述停用和停车索赔的数量。