Hidden Markov models with arbitrary state dwell-time distributions

Langrock and Zucchini have a new paper in Computational Statistics and Data Analysis. This develops models methodology for fitting discrete-time hidden semi-Markov models. They demonstrate that hidden semi-Markov models can be approximated through hidden Markov models with aggregate blocks of states. Dwell-time in a particular state is made up of a series of latent states. An individual enters a state in latent state 1. After k steps in a state where k < m , a subject either leaves the state with probability c(k) or progresses to the next latent state with probability 1-c(k). c(k) therefore represents the hazard rate of the dwell time distribution. If time m in the state is reached then the subject may either stay in latent state m with some probability c(m) or else leave the state. Hence the tail of the dwell-time distribution is constrained to be geometric. By choosing m sufficiently large a good approximation to the desired distribution can be found.