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2016 Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation
T. Tony Cai, Zhao Ren, Harrison H. Zhou
Electron. J. Statist. 10(1): 1-59 (2016). DOI: 10.1214/15-EJS1081

Abstract

This is an expository paper that reviews recent developments on optimal estimation of structured high-dimensional covariance and precision matrices. Minimax rates of convergence for estimating several classes of structured covariance and precision matrices, including bandable, Toeplitz, sparse, and sparse spiked covariance matrices as well as sparse precision matrices, are given under the spectral norm loss. Data-driven adaptive procedures for estimating various classes of matrices are presented. Some key technical tools including large deviation results and minimax lower bound arguments that are used in the theoretical analyses are discussed. In addition, estimation under other losses and a few related problems such as Gaussian graphical models, sparse principal component analysis, factor models, and hypothesis testing on the covariance structure are considered. Some open problems on estimating high-dimensional covariance and precision matrices and their functionals are also discussed.

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T. Tony Cai. Zhao Ren. Harrison H. Zhou. "Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation." Electron. J. Statist. 10 (1) 1 - 59, 2016. https://doi.org/10.1214/15-EJS1081

Information

Received: 1 August 2014; Published: 2016
First available in Project Euclid: 17 February 2016

zbMATH: 1331.62272
MathSciNet: MR3466172
Digital Object Identifier: 10.1214/15-EJS1081

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

Rights: Copyright © 2016 The Institute of Mathematical Statistics and the Bernoulli Society

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Vol.10 • No. 1 • 2016
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