The continuous-time hidden Markov model based on discretization. Properties of estimators and applications

In this paper we consider continuous-time hidden Markov processes (CTHMM). The model considered is a two-dimensional stochastic process $$(X_t,Y_t)$$ ( X t , Y t ) , with $$X_t$$ X t an unobserved (hidden) Markov chain defined by its generating matrix and $$Y_t$$ Y t an observed process whose distribution law depends on $$X_t$$ X t and is called the emission function. In general, we allow the process $$Y_t$$ Y t to take values in a subset of the q-dimensional real space, for some q. The coupled process $$(X_t,Y_t)$$ ( X t , Y t ) is a continuous-time Markov chain whose generator is constructed from the generating matrix of X and the emission distribution. We study the theoretical properties of this two-dimensional process using a formulation based on semi-Markov processes. Observations of the CTHMM are obtained by discretization considering two different scenarii. In the first case we consider that observations of the process Y are registered regularly in time, while in the second one, observations arrive at random. Maximum-likelihood estimators of the characteristics of the coupled process are obtained in both scenarii and the asymptotic properties of these estimators are shown, such as consistency and normality. To illustrate the model a real-data example and a simulation study are considered.