Regularized estimation of large covariance matrices

Bickel, Peter J. ; Levina, Elizaveta

Ann. Statist., Tome 36 (2008) no. 1, p. 199-227 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2008-02-15
Classification:  Covariance matrix,  regularization,  banding,  Cholesky decomposition,  62H12,  62F12,  62G09