A tractable class of multivariate phase-type distributions for loss modeling

TL;DR

This paper introduces a new, explicit, and flexible class of multivariate phase-type distributions (mPH) for risk modeling, with proven density and dependence properties, and demonstrates its practical estimation and application to insurance data.

Contribution

The paper proposes a simple, explicit construction of multivariate phase-type distributions that are dense in the space of multivariate risks and provides an EM algorithm for their estimation.

Findings

The mPH class is dense in the set of multivariate risks on the positive orthant.

Explicit formulas for dependence measures are derived for mPH distributions.

The EM algorithm effectively estimates parameters from insurance data.

Abstract

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.