Association of multivariate phase-type distributions, with applications to shock models

A random vector is said to be of (multivariate) phase-type if it can be represented as the vector of random times until absorptions into various stochastically closed subsets of the finite state space in an absorbing Markov chain. The phase-type distributions are useful since Markovian methods may be applicable in the situations where one adopts a (univariate or multivariate) phase-type distribution for time intervals that are needed in setting up a stochastic model. This paper studies the dependence nature of multivariate phase-type distributions and some related shock models, and it shows that under some mild conditions, the multivariate phase-type distributions are positively associated. The association properties for the lifetimes of components operating in some common shock environments are also obtained.