Order statistics and multivariate discrete phase-type distributions

In this thesis, we present three different topics of research which are related to the theory of phase-type distributions. Those topics are explained next. The first research work is on order statistics from matrix-geometric distributions in the case of a sample of independent and non-identically distributed random variables.

We prove that order statistics from matrix-geometric distributions are matrix-geometric distributed and we provide representations for their distributions. The second research work is a study of the discrete version of multivariate phasetype distributions introduced by V. G. Kulkarni. We give an expression for the joint probability-generating function in the similar way than in the continuous time case and under this base we make an analysis of this class of distributions and present examples that are commonly found in the literature.

The third research work presented came out with the aim of relating the last two topics. That is, we found a problem which relates the concept of order statistics and multivariate phase-type distributions introduced by V. G. Kulkarni, the last in the case of continuous time. Thus, we present a research on concomitants of phase-type distributions.

We provide a procedure to calculate the density function of concomitants of phase-type distributions and we prove that concomitants of phase-type distributions are phase-type distributed.