  In this paper we consider estimation of sparse covariance matrices and
propose a thresholding procedure which is adaptive to the variability of
individual entries. The estimators are fully data driven and enjoy excellent
performance both theoretically and numerically. It is shown that the estimators
adaptively achieve the optimal rate of convergence over a large class of sparse
covariance matrices under the spectral norm. In contrast, the commonly used
universal thresholding estimators are shown to be sub-optimal over the same
parameter spaces. Support recovery is also discussed. The adaptive thresholding
estimators are easy to implement. Numerical performance of the estimators is
studied using both simulated and real data. Simulation results show that the
adaptive thresholding estimators uniformly outperform the universal
thresholding estimators. The method is also illustrated in an analysis on a
dataset from a small round blue-cell tumors microarray experiment. A supplement
to this paper which contains additional technical proofs is available online.
