Open Access
February 2008 Regularized estimation of large covariance matrices
Peter J. Bickel, Elizaveta Levina
Ann. Statist. 36(1): 199-227 (February 2008). DOI: 10.1214/009053607000000758


This paper considers estimating a covariance matrix of p variables from n observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse of the covariance. We show that these estimates are consistent in the operator norm as long as (log p)/n→0, and obtain explicit rates. The results are uniform over some fairly natural well-conditioned families of covariance matrices. We also introduce an analogue of the Gaussian white noise model and show that if the population covariance is embeddable in that model and well-conditioned, then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix. The results can be extended to smooth versions of banding and to non-Gaussian distributions with sufficiently short tails. A resampling approach is proposed for choosing the banding parameter in practice. This approach is illustrated numerically on both simulated and real data.


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Peter J. Bickel. Elizaveta Levina. "Regularized estimation of large covariance matrices." Ann. Statist. 36 (1) 199 - 227, February 2008.


Published: February 2008
First available in Project Euclid: 1 February 2008

zbMATH: 1132.62040
MathSciNet: MR2387969
Digital Object Identifier: 10.1214/009053607000000758

Primary: 62H12
Secondary: 62F12 , 62G09

Keywords: banding , Cholesky decomposition , Covariance matrix , regularization

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 1 • February 2008
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