Efficient estimation for semiparametric semi-Markov processes

We consider semiparametric models of semi-Markov processes with arbitrary state space. Assuming that the process is geometrically ergodic, we characterize efficient estimators, in the sense of Hájek and Le Cam, for arbitrary real-valued smooth functionals of the distribution of the embedded Markov renewal process. We construct efficient estimators of the parameter and of linear functionals of the distribution. In particular we treat the two cases in which we have a parametric model for the transition distribution of the embedded Markov chain and an arbitrary conditional distribution of the inter-jump times, and vice versa.