Phase-type (PH) distributions are a popular tool for the analysis of univariate risks in numerous actuarial applications. Their multivariate counterparts (MPH$^\ast$), however, have not seen such a proliferation, due to lack of explicit formulas and complicated estimation procedures. A simple construction of multivariate phase-type distributions -- mPH -- is proposed for the parametric description of multivariate risks, leading to models of considerable probabilistic flexibility and statistical tractability. The main idea is to start different Markov processes at the same state, and allow them to evolve independently thereafter, leading to dependent absorption times. By dimension augmentation arguments, this construction can be cast into the umbrella of MPH$^\ast$ class, but enjoys explicit formulas which the general specification lacks, including common measures of dependence. Moreover, it is shown that the class is still rich enough to be dense on the set of multivariate risks supported on the positive orthant, and it is the smallest known sub-class to have this property. In particular, the latter result provides a new short proof of the denseness of the MPH$^\ast$ class. In practice this means that the mPH class allows for modeling of bivariate risks with any given correlation or copula. We derive an EM algorithm for its statistical estimation, and illustrate it on bivariate insurance data. Extensions to more general settings are outlined.

翻译：阶段类型分布( PH) 是分析多种精算应用中的单向风险的流行工具。 但是,由于缺乏明确的公式和复杂的估算程序,它们的多变量对应方(MPH$ ⁇ ast$)没有看到这种扩散。 提议简单构建多变量类型分布( mPH),用于多变量风险的参数描述,从而产生相当的概率灵活性和统计可感性模型。 主要想法是在同一州启动不同的马可夫流程,并允许它们随后独立演变,导致依赖性吸收时间。 根据维度增强参数,这一构建可以投放到 MPH$ ⁇ ast 类的伞状中,但具有一般规格缺乏的明确公式,包括共同依赖度的衡量标准。 此外,还表明该类别仍然足够丰富,在支持正度或强度的多变量风险组合上仍然十分密集,而拥有这一属性的子类最小为已知的子类。 特别是,后一种结果为MPHH$ +Qast$ 类的密度模型提供了新的短期证据。 其分类和双级的统计模型的推导算。