Nonparametric estimators for Markov step processes

The distribution of a homogeneous, continuous-time Markov step process with values in an arbitrary state space is determined by the transition distribution and the mean holding time, which may depend on the state. We suppose that both are unknown, introduce a class of functionals which determines the transition distribution and the mean holding time up to equivalence, and construct estimators for the functionals. Assuming that the embedded Markov chain is Harris recurrent and uniformly ergodic, and that the mean holding time is bounded and bounded away from 0, we show that the estimators are asymptotically efficient, as the observation time increases. Then we consider the two submodels in which the mean holding time is assumed constant, and constant and known, respectively. We describe efficient estimators for the submodels. For finite state space, our results give efficiency of an estimator for the generator which was studied by Lange (1955) and Albert (1962).