  This paper discusses tractable development and statistical estimation of a
continuous time stochastic process with a finite state space having non-Markov
property. The process is formed by a finite mixture of right-continuous Markov
jump processes moving at different speeds on the same finite state space,
whereas the speed regimes are assumed to be unobservable. The mixture was first
proposed by Frydman (J. Am. Stat. Assoc., 100, 1046-1053, 2005) in 2005 and
recently generalized in Surya (Stoch. Syst. 8, 29-44, 2018), in which
distributional properties and explicit identities of the process are given in
its full generality. The contribution of this paper is two fold. First, we
present Monte Carlo method for constructing the process and show distributional
equivalence between the simulated process and the actual process. Secondly, we
perform statistical inference on the distribution parameters of the process.
Under complete observation of the sample paths, maximum likelihood estimates
are given in explicit form in terms of sufficient statistics of the process.
Estimation under incomplete observation is performed using the EM algorithm.
The estimation results completely characterize the process in terms of the
initial probability of starting the process in any phase of the state space,
intensity matrices of the underlying Markov jump processes, and the switching
probability matrix of the process. Some numerical examples are given to test
the performance of the developed method. The proposed estimation generalizes
the existing statistical inferences for the Markov model by Albert (Ann. Math.
Statist., 38, p.727-753., 1961), the mover-stayer model by Frydman (J. Am.
Stat. Assoc.79, 632-638., 1984) and the Markov mixture model by Frydman (J. Am.
Stat. Assoc., 100, 1046-1053, 2005).
