Continuous scaled phase-type distributions

TL;DR

This paper investigates continuous scaled phase-type distributions, deriving their properties, developing an EM algorithm for inference, and demonstrating their heavy-tailed nature and analytical tractability with real data.

Contribution

It introduces continuous scaling of phase-type distributions, providing closed-form formulas and an EM algorithm for statistical inference, expanding the applicability of phase-type models.

Findings

Distributions are often heavy-tailed.

Closed-form formulas facilitate analysis and implementation.

Mixture distributions retain phase-type properties.

Abstract

Products between phase-type distributed random variables and any independent, positive and continuous random variable are studied. Their asymptotic properties are established, and an expectation-maximization algorithm for their effective statistical inference is derived and implemented using real-world datasets. In contrast to discrete scaling studied in earlier literature, in the present continuous case closed-form formulas for various functionals of the resulting distributions are obtained, which facilitates both their analysis and implementation. The resulting mixture distributions are very often heavy-tailed and yet retain various properties of phase-type distributions, such as being dense (in weak convergence) on the set of distributions with positive support.