This paper considers regularizing a covariance matrix of p variables
estimated from n observations, by hard thresholding. We show that the
thresholded estimate is consistent in the operator norm as long as the true
covariance matrix is sparse in a suitable sense, the variables are Gaussian or
sub-Gaussian, and (logp)/nβ0, and obtain explicit rates. The results are
uniform over families of covariance matrices which satisfy a fairly natural
notion of sparsity. We discuss an intuitive resampling scheme for threshold
selection and prove a general cross-validation result that justifies this
approach. We also compare thresholding to other covariance estimators in
simulations and on an example from climate data.Comment: Published in at http://dx.doi.org/10.1214/08-AOS600 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org