A Tractable Class of Multivariate Phase-Type Distributions for Loss Modeling

Abstract

Phase-type (PH) distributions are a popular tool for the analysis of univariate risks in numerous actuarial applications. Their multivariate counterparts (MPH∗), however, have not seen such a proliferation because of a 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 under the umbrella of MPH∗ class but enjoys explicit formulas that 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 subclass to have this property. In particular, the latter result provides a new short proof of the denseness of the MPH∗ class. In practice, this means that the mPH class allows for the modeling of bivariate risks with any given correlation or copula. We derive an expectation-maximization algorithm for its statistical estimation and illustrate it on bivariate insurance data. Extensions to more general settings are outlined.