Phase Type Distributions with Finite Support

TLDR: In this paper, the authors consider the problem of finite support phase type distributions (FSPH) and derive the EM algorithms for two classes of FSPH, the first of which is the class of matrix exponential distributions dense in (a, b).

Abstract: This research is motivated by the fact that many random variables of practical interest have a finite support. For fixed a < b, we consider the distribution of a random variable X = (a + Ymod(b − a)), where Y is a phase type (PH) random variable. We demonstrate that as we traverse for Y the entire set of PH distributions (or even any subset thereof like Coxian that is dense in the class of distributions on [0, ∞)), we obtain a class of matrix exponential distributions dense in (a, b). We call these Finite Support Phase Type Distributions (FSPH) of the first kind. A simple example shows that though dense, this class by itself is not very efficient for modeling; therefore, we introduce (and derive the EM algorithms for) two other classes of finite support phase type distributions (FSPH). The properties of denseness, connection to Markov chains, the EM algorithm, and ability to exploit matrix-based computations should all make these classes of distributions attractive not only for applied probability but als...