We introduce a novel class of bivariate common-shock discrete phase-type (CDPH) distributions to describe dependencies in loss modeling, with an emphasis on those induced by common shocks. By constructing two jointly evolving terminating Markov chains that share a common evolution up to a random time corresponding to the common shock component, and then proceed independently, we capture the essential features of risk events influenced by shared and individual-specific factors. We derive explicit expressions for the joint distribution of the termination times and prove various class and distributional properties, facilitating tractable analysis of the risks. Extending this framework, we model random sums where aggregate claims are sums of continuous phase-type random variables with counts determined by these termination times, and show that their joint distribution belongs to the multivariate phase-type or matrix-exponential class. We develop estimation procedures for the CDPH distributions using the expectation-maximization algorithm and demonstrate the applicability of our models through simulation studies and an application to bivariate insurance claim frequency data.

翻译：本文引入一类新颖的双变量共同冲击离散相位型（CDPH）分布，用于描述损失建模中的相依性，尤其关注由共同冲击引发的相依结构。通过构建两个联合演化的终止马尔可夫链，它们在对应于共同冲击分量的随机时间之前共享相同的演化路径，随后独立进行，从而捕捉了受共享因素和个体特定因素影响的风险事件本质特征。我们推导了终止时间联合分布的显式表达式，并证明了该分布类的多种性质，为风险的可处理分析提供了便利。通过扩展该框架，我们对随机和进行建模，其中聚合索赔是连续相位型随机变量的总和，其计数由这些终止时间决定，并证明其联合分布属于多元相位型或矩阵指数类。我们利用期望最大化算法为CDPH分布开发了估计程序，并通过模拟研究及对双变量保险索赔频率数据的应用，展示了所提模型的适用性。