Phase-Type Distributions

Specialeforsvar ved Niels Moctezuma Krarup

Titel: Phase-Type Distributions

Abstract: In this thesis, we give an introduction to phase-type distributions which are constructed as the time until absorption of an underlying Markov chain or Markov jump process. In the discrete version we extend the basic theory to cover first occurrence of a general sub-sequence of iid discrete random variables and a basic urn experiment. In the continuous case, the well known result of denseness of the class of phase-type distributions on the non-negative real line, in the sense of weak convergence, is proved. Closure properties, with emphasis on order statistics is investigated. We give a characteri-zation of the probabilities of the n! different orderings of n independent but non-identically distributed phase-type distributions, relying on matrix operations rather than integration. Transformation via linear rewards of the underlying process is discussed, being funda-mental in the construction of the class of multivariate phase-type distributions denoted MPH*, which is briefly discussed. Finally, we investigate two methods of estimating func-tionals of the underlying process conditioned on the time of absorption, an importance sampler and a Markov Chain Monte Carlo method. Quantities of interest are implemented in R.

Vejleder: Mogens Bladt
Censor:   Bo Friis Nielsen, DTU