Automatic Optimal Batch Size Selection for Recursive Estimators of Time-Average Covariance Matrix. Issue 519 (3rd July 2017)

• Record Type: Journal Article
• Authors: Chan, Kin Wai; Yau, Chun Yip
• Is Part Of: Journal of the American Statistical Association. Volume 112:Issue 519(2017)
• Page Start: 1076
• Page End: 1089
• Publication Date: 2017-07-03
• Subjects: Batch means -- Dependence measures -- Long run variance -- Nonlinear time series -- Smoothness parameter; Statistics -- Periodicals; Statistics -- Periodicals; Statistiques -- Périodiques; États-Unis -- Statistiques -- Périodiques; 519.5
• DOI: 10.1080/01621459.2016.1189337
• Languages: English
• ISSNs: 0162-1459

Abstract

ABSTRACT: The time-average covariance matrix (TACM) Σ : = ∑ k ∈ Z Γ k, where Γ k is the auto-covariance function, is an important quantity for the inference of the mean of anR 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 ϒ β : = ∑ k ∈ Z | k | β Γ 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.