Efficient estimation of invariant distributions of some semiparametric Markov chain models

We characterize efficient estimators for the expectation of a function under the invariant distribution of a Markov chain and outline ways of constructing such estimators. We consider two models. The first is described by a parametric family of constraints on the transition distribution; the second is the heteroscedastic nonlinear autoregressive model. The efficient estimator for the first model adds a correction term to the empirical estimator. In the second model, the suggested efficient estimator is a one-step improvement of an initial estimator which might be obtained by a version of Markov chain Monte Carlo