# Matrix Mittag--Leffler distributions and modeling heavy-tailed risks

## TL;DR

This paper introduces matrix Mittag-Leffler distributions, a flexible family for modeling heavy-tailed risks, which can be interpreted through phase-type distributions or semi-Markov processes, and demonstrates their effectiveness on simulated and real insurance data.

## Contribution

It defines matrix Mittag-Leffler distributions and shows their advantages over existing methods in modeling heavy-tailed risks, addressing threshold selection issues.

## Key findings

- The class can be interpreted as inhomogeneous phase-type distributions with random scaling.
- It effectively models heavy-tailed risks in insurance data.
- Demonstrates advantages over traditional extreme value approaches.

## Abstract

In this paper we define the class of matrix Mittag-Leffler distributions and study some of its properties. We show that it can be interpreted as a particular case of an inhomogeneous phase-type distribution with random scaling factor, and alternatively also as the absorption time of a semi-Markov process with Mittag-Leffler distributed interarrival times. We then identify this class and its power transforms as a remarkably parsimonious and versatile family for the modelling of heavy-tailed risks, which overcomes some disadvantages of other approaches like the problem of threshold selection in extreme value theory. We illustrate this point both on simulated data as well as on a set of real-life MTPL insurance data that were modeled differently in the past.