Date of this Version
Probability Theory and Related Fields
Toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging, target detection, speech recognition, and communications systems. In this paper, we consider optimal estimation of large Toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spectral norm. The properties of the tapering and banding estimators are studied in detail and are used to obtain the minimax upper bound. The results also reveal a fundamental difference between the tapering and banding estimators over certain parameter spaces. The minimax lower bound is derived through a novel construction of a more informative experiment for which the minimax lower bound is obtained through an equivalent Gaussian scale model and through a careful selection of a finite collection of least favorable parameters. In addition, optimal rate of convergence for estimating the inverse of a Toeplitz covariance matrix is also established.
The final publication is available at Springer via http://dx.doi.org/10.1007/s00440-012-0422-7.
banding, covariance matrix, minimax lower bound, optimal rate of convergence, spectral norm, tapering, Toeplitz covariance matrix
Cai, T., Ren, Z., & Zhou, H. H. (2013). Optimal Rates of Convergence for Estimating Toeplitz Covariance Matrices. Probability Theory and Related Fields, 156 (1), 101-143. http://dx.doi.org/10.1007/s00440-012-0422-7
Date Posted: 27 November 2017
This document has been peer reviewed.