Regularized estimation of large covariance matrices

Understand

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 $(\log p)/n\to0$, 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.