Fitting inhomogeneous phase-type distributions to data: the univariate and the multivariate case

TL;DR

This paper introduces a new fitting procedure for inhomogeneous phase-type distributions, extending existing models to better capture heavy tails in univariate and multivariate data, with applications demonstrated on insurance datasets.

Contribution

It develops a novel fitting method for IPH distributions and extends Kulkarni's multivariate PH class to the inhomogeneous case, including parameter estimation and handling censored data.

Findings

Effective fitting algorithms demonstrated on simulated data.

Application to real insurance data shows improved modeling of heavy tails.

Amended procedures for censored data enhance practical usability.

Abstract

The class of inhomogeneous phase-type distributions (IPH) was recently introduced in Albrecher and Bladt (2019) as an extension of the classical phase-type (PH) distributions. Like PH distributions, the class of IPH is dense in the class of distributions on the positive halfline, but leads to more parsimonious models in the presence of heavy tails. In this paper we propose a fitting procedure for this class to given data. We furthermore consider an analogous extension of Kulkarni's multivariate phase-type class (Kulkarni, 1989) to the inhomogeneous framework and study parameter estimation for the resulting new and flexible class of multivariate distributions. As a by-product, we amend a previously suggested fitting procedure for the homogeneous multivariate phase-type case and provide appropriate adaptations for censored data. The performance of the algorithms is illustrated in several numerical examples, both for simulated and real-life insurance data.