Abstract
Acyclic phase-type distributions form a versatile model, serving as approximations to many probability distributions in various circumstances. They exhibit special properties and characteristics that usually make their applications attractive. Compared to acyclic continuous phase-type (ACPH) distributions, acyclic discrete phase-type (ADPH) distributions and their subclasses (ADPH family) have received less attention in the literature. In this paper, we present the definition, properties, characteristics and PH representations of ADPH distributions and their subclasses with finite state space. Based on the definitions of geometric and shifted geometric distributions, we propose a distinct classification for the ADPH subclasses analogous to ACPH family. We develop the PH representation for each ADPH subclass and prove them through their closure properties. The advantage of our proposed classifications is in applying precise representations of each subclass and preventing miscalculation of the probability mass function, by computing the ADPH family based on geometric and shifted geometric distributions.
Highlights
Phase-type (PH) distributions, introduced by Neuts (1975, 1981), form a very general class of distributions that have been successfully applied in a wide variety of stochastic disciplines for the last few decades
The complexity of overall system model can be controlled by Acyclic subsets of PH (APH) minimal representation (Cumani 1982)
The first exploration of acyclic discrete phase-type (ADPH) distributions is started by Bobbio et al (2003), and they show that similar to the continuous case (Cumani 1982), the ADPH class admits a unique minimal representation, called canonical form
Summary
Phase-type (PH) distributions, introduced by Neuts (1975, 1981), form a very general class of distributions that have been successfully applied in a wide variety of stochastic disciplines for the last few decades. The first exploration of acyclic discrete phase-type (ADPH) distributions is started by Bobbio et al (2003), and they show that similar to the continuous case (Cumani 1982), the ADPH class admits a unique minimal representation, called canonical form. The primary attempt to define the subclass of ADPH is given by Bobbio et al (2003) They propose three canonical forms to introduce the subclasses of ADPH and present the ML estimation algorithm for one of them.
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