The Annals of Statistics

Adaptive covariance matrix estimation through block thresholding

T. Tony Cai and Ming Yuan

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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.

Article information

Ann. Statist., Volume 40, Number 4 (2012), 2014-2042.

First available in Project Euclid: 30 October 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 62F12: Asymptotic properties of estimators 62G09: Resampling methods

Adaptive estimation block thresholding covariance matrix Frobenius norm minimax estimation optimal rate of convergence spectral norm


Cai, T. Tony; Yuan, Ming. Adaptive covariance matrix estimation through block thresholding. Ann. Statist. 40 (2012), no. 4, 2014--2042. doi:10.1214/12-AOS999.

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