Adaptive covariance matrix estimation through block thresholding

Abstract

Estimation of large covariance matrices has drawn considerable recent attention, and the theoretical focus so far has mainly been on developing a minimax theory over a fixed parameter space. In this paper, we consider adaptive covariance matrix estimation where the goal is to construct a single procedure which is minimax rate optimal simultaneously over each parameter space in a large collection. A fully data-driven block thresholding estimator is proposed. The estimator is constructed by carefully dividing the sample covariance matrix into blocks and then simultaneously estimating the entries in a block by thresholding. The estimator is shown to be optimally rate adaptive over a wide range of bandable covariance matrices. A simulation study is carried out and shows that the block thresholding estimator performs well numerically. Some of the technical tools developed in this paper can also be of independent interest.

Published in at http://dx.doi.org/10.1214/12-AOS999 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Regularized estimation of large covariance matrices
• Covariance regularization by thresholding
• Sparse permutation invariant covariance estimation
• Sparsistency and rates of convergence in large covariance matrix estimation
• Optimal rates of convergence for covariance matrix estimation
• Operator norm consistent estimation of large-dimensional sparse covariance matrices

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