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

ABSTRACT

The time-average covariance matrix (TACM) $\mathit{}\mathbf{\Sigma }:={\sum }_{k\in \mathbb{Z}}{\mathit{}\mathbf{\Gamma }}_k$, where Γ_k is the auto-covariance function, is an important quantity for the inference of the mean of an ${\mathbb{R}}^d$-valued stationary process (d ⩾ 1). This article proposes two recursive estimators for Σ 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 ${\mathit{}\mathbf{\Upsilon }}_{\beta }:={\sum }_{k\in \mathbb{Z}}{\left|k\right|}^{\beta }{\mathit{}\mathbf{\Gamma }}_k$, for some β. This article also develops recursive estimators for ϒ_β. Combining these two estimators, we obtain a fully automatic procedure for optimal online estimation for Σ. 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.

KEYWORDS:

• Batch means
• Dependence measures
• Long run variance
• Nonlinear time series
• Smoothness parameter

Funding

Supported in part by HKSAR-RGC-ECS 405012 and HKSAR-RGC-GRF 405113, 14601015.