Efficient estimators for functionals of Markov chains with parametric marginals

Suppose we observe a geometrically ergodic Markov chain with a parametric model for the marginal, but no (further) information about the transition distribution. Then the empirical estimator for a linear functional of the joint law of two successive observations is no longer efficient. We construct an improved estimator and show that it is efficient. The construction is similar to a recent one for bivariate models with parametric marginals. The result applies to discretely observed parametric continuous-time processes. AMS 2000 subject classifications. Primary 62G20, 62G30, 62M05. Key words and Phrases. Least squares estimator, series estimator, orthonormal basis, effi-cient influence function, reversible Markov chain, discretely observed diffusion, constrained model. 1