Estimation of Phase-Type Distributions

Abstract

This chapter deals with the estimation of phase-type distributions in a number of different circumstances. First, we consider their estimation when only absorption times are available, and provide both EM and MCMC approaches, which in turn complement each other, aiming at different purposes. Then we consider censored data of a different kind, which can be important in modeling data in survival analysis. When assuming phase-type distributions in a stochastic or statistical model, it is often due to their generality and tractability (if further stochastic analysis is to be performed posteriorly) rather than the underlying states having a physical interpretation. Actually, it would not be meaningful to assign a physical interpretation to the states (or phases) unless they can be observed, that is, at least partially. We will consider the latter situation by providing methods for estimating discretely observed phase-type distributions. As a byproduct, we also obtain a general method for estimating discretely observed Markov jump processes, which can be of interest on its own. Finally, the EM algorithm can be extended to approximate a given theoretical distribution by seeing it as empirical distribution functions based on an infinite sample.