Multivariate Phase-type Distributions in Stochastic Modelling

This thesis concerns univariate and multivariate phase-type distribution with a focus on their usage in statistics.
Chapters 1-3 serve as motivation and an introduction to the subject. We review phasetype distributions and their multivariate extension, MPH∗ distributions, as well as their discrete counterparts. Hereafter we consider the parameter estimation in both univariate and multivariate phase-type distributions, we direct attention towards an issue with the Expectation Maximization algorithm for such distributions.
Chapter 4 concerns a phase-type regression model and in particular parameter estimation in this model. Using the quasi-likelihood function and the Iterative Weighted Least Squares algorithm we suggest a faster approach for finding the maximum likelihood parameters. Chapter 5 and Chapter 6 focuses on univariate and multivariate exponential distributions. We find sufficient and necessary conditions for a phase-type representation to be of an exponential distribution, and use this to analyze previously published, bivariate exponential distributions with MPH∗ representations. For completion we also find the conditions under which a discrete phase-type representation becomes that of a geometric distribution.
Chapter 7 and Chapter 8 considers two different applications of multivariate phase-type distributions in a statistical analysis. The first application is a model for the conditional distribution for train delays at a station given already observed delays. The second application is a joint model for the number and size of claims to an insurance policy.