Multivariate Phase-Type Distributions

Abstract

A (univariate) random variable is said to be of phase type if it can be represented as the time until absorption in a finite state absorbing Markov chain. Univariate phase type random variables are useful because they arise from processes that are often encountered in applications, they have densities that can be written in a closed form, they possess some useful closure properties, and they can approximate any nonnegative random variable. This paper introduces and discusses several extensions to the multivariate case. It shows that the multivariate random variables possess many of the properties of univariate phase type distributions and derives explicit formulas for various probabilistic quantities of interest. Some examples are included.

Markov multi-variate survival indicators for default simulation as a new characterization of the Marshall–Olkin law

“…The main practical implication of Theorem 2.2 is that the latter property is satisfied by MO, and moreover, MO is actually characterized by this property within the MCH family. Acknowledging previous results, it is known since [Assaf et al (1984)] that MO MPH, and it is also not difficult to observe that MO MCH. To the best of our knowledge, however, existing literature did neither realize that MO is essentially the only family of distributions satisfying the nested margining property, nor did it address the economic consequences thereof.…”

“…Neuts and co-workers, a survey can be found in [Bladt (2005)]. MPH are introduced in the context of default risk in [Assaf et al (1984)], see also [Cai, Li (2005)], defining a random vector explicitly via a Markov chain. Unfortunately, multi-variate phase type distributions, due to their generality, appear to be very difficult to work with in high-dimensional applications.…”

CMPH: a multivariate phase-type aggregate loss distribution

“…Certainly, the CMPH distribution bears resemblance with, but di ers from, the multivariate phase-type distribution studied by Assaf, Langberg, Savits and Shaked [1], and Kulkarni [20]. In particular, the latter paper introduces a continuous multivariate phase type distribution based on a continuous-time Markov chain (CTMC).…”

Multivariate Matrix-Exponential Distributions

“…We shall also need what has been termed reduced moments, Assaf et al (1984) introduced the class of multivariate phase-type distributions MPH by considering a special type of reward structure on a onedimensional phase-type representation. Consider a generator D of dimension m and a n-dimensional vector X of random variables.…”