In this paper, a new three-parameter discrete family of distributions, the $$r{\cal B}ell$$ 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.

在本文中，介绍了一个新的三参数离散分布族， $$ r {\ cal B} ell $$ 族。该家族基于R系列的扩展 -贝尔多项式。提出的模型概括了经典的Poisson和最近提出的Bell和Bell-Touchard分布。它表现出有趣的随机性质。它的概率可以通过递归公式来计算，该递归公式使我们能够根据积分方程来计算集体风险模型中总索赔额的概率函数。提出了单变量和双变量回归模型。前一种回归模型用于通过显示良好的样本外表现来解释汽车保险投资组合中未使用的索偿数量。后者用于共同描述停用和停车索赔的数量。