The distribution of a homogeneous, continuous-time Markov step process
with values in an arbitrary state space is determined by the transiti
on distribution and the mean holding time, which may depend on the sta
te. We suppose that both are unknown, introduce a class of functionals
which determines the transition distribution and the mean holding tim
e up to equivalence, and construct estimators for the functionals. Ass
uming that the embedded Markov chain is Harris recurrent and uniformly
ergodic, and that the mean holding time is bounded and bounded away f
rom 0, we show that the estimators are asymptotically efficient, as th
e observation time increases. Then we consider the two submodels in wh
ich 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 t
he generator which was studied by Lange (1955) and Albert (1962).