Generalized Phase-Type Distribution and Competing Risks for Markov Mixtures Process

Abstract

Phase-type distribution has been an important probabilistic tool in the analysis of complex stochastic system evolution. It was introduced by Neuts \cite{Neuts1975} in 1975. The model describes the lifetime distribution of a finite-state absorbing Markov chains, and has found many applications in wide range of areas. It was brought to survival analysis by Aalen \cite{Aalen1995} in 1995. However, the model has lacks of ability in modeling heterogeneity and inclusion of past information which is due to the Markov property of the underlying process that forms the model. We attempt to generalize the distribution by replacing the underlying by Markov mixtures process. Markov mixtures process was used to model jobs mobility by Blumen \cite{Blumen} et al. in 1955. It was known as the mover-stayer model describing low-productivity workers tendency to move out of their jobs by a Markov chains, while those with high-productivity tend to stay in the job. Frydman \cite{Frydman2005} later extended the model to a mixtures of finite-state Markov chains moving at different speeds on the same state space. In general the mixtures process does not have Markov property. We revisit the mixtures model \cite{Frydman2005} for mixtures of multi absorbing states Markov chains, and propose generalization of the phase-type distribution under competing risks. The new distribution has two main appealing features: it has the ability to model heterogeneity and to include past information of the underlying process, and it comes in a closed form. Built upon the new distribution, we propose conditional forward intensity which can be used to determine rate of occurrence of future events (caused by certain type) based on available information. Numerical study suggests that the new distribution and its forward intensity offer significant improvements over the existing model.