The Annals of Statistics

Regularized estimation of large covariance matrices

Peter J. Bickel and Elizaveta Levina

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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.

Article information

Ann. Statist. Volume 36, Number 1 (2008), 199-227.

First available in Project Euclid: 1 February 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Covariance matrix regularization banding Cholesky decomposition


Bickel, Peter J.; Levina, Elizaveta. Regularized estimation of large covariance matrices. Ann. Statist. 36 (2008), no. 1, 199--227. doi:10.1214/009053607000000758.

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