Empirical estimators for semi-Markov processes

A semi-Markov process stays in state x for a time s and then jumps to state y according to a transition distribution Q(x; dy; ds). A statistical model is described by a family of such transition distributions. We give conditions for a nonparametric version of local asymptotic normality as the observation time tends to infinity. Then we introduce `empirical' estimators for linear functionals of the distribution ß(dx)Q(x; dy; ds), with ß denoting the invariant distribution of the embedded Markov chain, and characterize the empirical estimators which are efficient for a given model. We discuss efficiency of several classical estimators, in particular the jump frequency, the proportion of visits to a given set, the proportion of time spent in a set, and an estimator for Q (x; fyg \Theta [0; t]) suggested by Moore and Pyke (1968) for countable state space. 1 Introduction A semi-Markov process Y = (Y t ) t0 is a process on the time interval [0; 1), with values in some arbitrary state spac..