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

The time-average covariance matrix (TACM) Σ:=∑k∈ZΓk, where <b><i>Γ</i></b>_<i>k</i> is the auto-covariance function, is an important quantity for the inference of the mean of a Rd-valued stationary process (<i>d</i> ≥ 1). This paper proposes two recursive estimators for <b><i>Σ</i></b> 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 ϒβ:=∑k∈Z|k|βΓk, for some β. This paper also develops recursive estimators for <b><i>ϒ</i></b>_β. Combining these two estimators, we obtain a fully automatic procedure for optimal on-line estimation for <b><i>Σ</i></b>. Consistency and convergence rates of the proposed estimators are derived. Applications to confidence region construction and Markov Chain Monte Carlo convergence diagnosis are discussed.