Estimating the order of continuous phase-type distributions and Markov-modulated Poisson processes

Phase-type distributions and Markov-modulated Poisson processes are two important models in applied probability. Estimation for such models has been considered in several papers in recent years; however, the order of the model has always been assumed to be fixed. In this paper we discuss model order selection via penalized likelihood methods and show that these methods asymptotically do not underestimate the order and also generate consistent estimates of the observed distribution.