Statistical inference for a mixture of Markov jump processes

We estimate a general mixture of Markov jump processes. The key novel feature of the proposed mixture is that the transition intensity matrices of the Markov processes comprising the mixture are entirely unconstrained. The Markov processes are mixed with distributions that depend on the initial state of the mixture process. The new mixture is estimated from its continuously observed realizations using the EM algorithm, which provides the maximum likelihood (ML) estimates of the mixture's parameters. We derive the asymptotic properties of the ML estimators. To obtain estimated standard errors of the ML estimates of the mixture's parameters, an explicit form of the observed Fisher information matrix is derived. In its new form, the information matrix simplifies the conditional expectation of outer product of the complete-data score function in the Louis (1982) general matrix formula for the observed Fisher information matrix. Simulation study verifies the estimates' accuracy and confirms the consistency and asymptotic normality of the estimators. The developed methods are applied to a medical dataset, for which the likelihood ratio test rejects the constrained mixture in favor of the proposed unconstrained one. This application exemplifies the usefulness of a new unconstrained mixture for identification and characterization of homogeneous subpopulations in a heterogeneous population.