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August 2012 Adaptive covariance matrix estimation through block thresholding
T. Tony Cai, Ming Yuan
Ann. Statist. 40(4): 2014-2042 (August 2012). DOI: 10.1214/12-AOS999

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.

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T. Tony Cai. Ming Yuan. "Adaptive covariance matrix estimation through block thresholding." Ann. Statist. 40 (4) 2014 - 2042, August 2012. https://doi.org/10.1214/12-AOS999

Information

Published: August 2012
First available in Project Euclid: 30 October 2012

zbMATH: 1257.62060
MathSciNet: MR3059075
Digital Object Identifier: 10.1214/12-AOS999

Subjects:
Primary: 62H12
Secondary: 62F12 , 62G09

Keywords: adaptive estimation , block thresholding , Covariance matrix , Frobenius norm , minimax estimation , Optimal rate of convergence , spectral norm

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 4 • August 2012
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