The class of inhomogeneous phase-type (IPH) distributions was recently introduced as a dense extension of classical phase-type (PH) distributions that leads to more parsimonious models in the presence of heavy tails.

Distributional properties of the IPH distributions may be expressed explicitly in terms of functions of matrices, which naturally extend the analytic form of known densities like e.g. Pareto, Weibull and Gompertz—Makeham. Thus we may fit an IPH distribution to data, where the body of the distribution can have any shape and with a tail assumed to be of certain kind, and express the resulting distribution in a compact explicit form.

In this talk we provide an overview of the basic construction, properties and estimation procedures of IPH distributions together with their multivariate extensions.