On non-uniqueness of representations of phase-type distributions

Abstract

We introduce two properties which are useful in addressing the question of non-uniqueness of representations of phase-type distributions. One property, called phase-type simplicity, concerns the possibility that a given phase-type distribution may have two distinct representations in terms of the same Markov chain. The second, called phase-type majorization, concerns the possibility that one Markov chain may provide representations for all the distributions represented by another Markov chain. We prove some characterizations of these properties, discuss their interrelationship, and relate them to some known results on mixtures of convolutions of exponential distributions.