# Inhomogeneous phase--type distributions and heavy tails

## TL;DR

This paper introduces inhomogeneous phase-type distributions with heavy tails, providing a flexible framework for modeling heavy-tail phenomena and demonstrating its effectiveness on real-world insurance data.

## Contribution

It extends phase-type distributions to inhomogeneous cases, enabling better modeling of heavy tails with a unified probabilistic approach.

## Key findings

- The new distributions can model heavy tails effectively.
- Application to fire insurance data shows practical utility.
- The approach retains the fitting properties of traditional PH distributions.

## Abstract

We extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable to carry over fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach for the modelling of a real-world heavy-tailed fire insurance dataset.