Automatic Optimal Batch Size Selection for Recursive Estimators of Time-Average Covariance Matrix

The time-average covariance matrix (TACM) Sigma := Sigma(k is an element of Z) Gamma(k), where Gamma(k) is the auto-covariance function, is an important quantity for the inference of the mean of an R-d-valued stationary process (d >= 1). This article proposes two recursive estimators for Sigma with optimal asymptotic mean square error (AMSE) under different strengths of serial dependence. The optimal estimator involves a batch size selection, which requires knowledge of a smoothness parameter Y-beta:= Sigma(k is an element of Z) |k|(beta) Gamma(k), for some beta. This article also develops recursive estimators for Y-beta. Combining these two estimators, we obtain a fully automatic procedure for optimal online estimation for Sigma. Consistency and convergence rates of the proposed estimators are derived. Applications to confidence region construction and Markov chain Monte Carlo convergence diagnosis are discussed. Supplementary materials for this article are available online.