Blockwise Empirical Likelihood and Efficiency for Markov Chains

Suppose we observe an ergodic Markov chain on an arbitrary state space. The usual nonparametric estimator of a linear functional of the stationary distribution is the empirical estimator. If the stationary distribution obeys finitely many known linear constraints, we can improve the empirical estimator by empirical likelihood weights. Since the observations are dependent, an optimal choice of weights is determined by weighting averages over disjoint blocks of observations with slowly increasing length. We show that the improved empirical estimator is efficient. We also introduce two additively corrected empirical estimators that are asymptotically equivalent to the weighted empirical estimator, hence also efficient.