  This paper considers the banding estimator proposed in Bickel and Levina
(2008) for estimation of large covariance matrices. We prove that the banding
estimator achieves rate-optimality under the operator norm, for a class of
approximately banded covariance matrices, improving the existing results in
Bickel and Levina (2008). In addition, we propose a Stein's unbiased risk
estimate (Sure)-type approach for selecting the bandwidth for the banding
estimator. Simulations indicate that the Sure-tuned banding estimator
outperforms competing estimators.
