In this paper we introduce a bivariate distribution on $\mathbb{R}_{+} \times \mathbb{N}$ arising from a single underlying Markov jump process. The marginal distributions are phase-type and discrete phase-type distributed, respectively, which allow for flexible behavior for modeling purposes. We show that the distribution is dense in the class of distributions on $\mathbb{R}_{+} \times \mathbb{N}$ and derive some of its main properties, all explicit in terms of matrix calculus. Furthermore, we develop an effective EM algorithm for the statistical estimation of the distribution parameters. In the last part of the paper, we apply our methodology to an insurance dataset, where we model the number of claims and the mean claim sizes of policyholders, which is seen to perform favorably. An additional consequence of the latter analysis is that the total loss size in the entire portfolio is captured substantially better than with independent phase-type models.
翻译:在本文中, 我们引入了一个双变量分布 $mathbb{R\\\ times\ times\ time\ mathb{N}$ 。 边际分布为阶段类型和离散阶段类型, 允许为建模目的采取灵活的行为。 我们显示, 分配在 $\ mathbb{R\\\ \ time\ \ mathb{N}$ 的分布类别中非常密集, 并得出其主要属性中的一些属性, 都以矩阵计算为明确值。 此外, 我们开发了一个有效的 EM 算法, 用于对分布参数进行统计估计。 在本文最后一部分, 我们将我们的方法应用到保险数据集中, 在那里, 我们用模型来模拟政策持有者的索赔数量和平均索赔大小, 并被认为效果优异。 后一项分析的另一个结果是, 整个投资组合的总损失规模比独立阶段类型模型要好得多 。