In this paper we consider both minimax and adaptive estimation of the covariance operator over collections of polynomially decaying and exponentially decaying parameter spaces. We first establish the minimax rates of convergence for estimating the covariance operator under the operator norm. The results show that the dimension of the lattice graph significantly affects the optimal rates convergence, often much more so than the dimension of the random variables. We then consider adaptive estimation of the covariance operator. A fully data driven block thresholding procedure is proposed and is shown to be adaptively rate optimal simultaneously over a wide range of polynomially decaying and exponentially decaying parameter spaces. The adaptive block thresholding procedure is easy to implement and numerical experiments are carried out to illustrate the merit of the procedure.

Cai, T., Zhang, C.-H. & Zhou, H. (2010).
Optimal rates of convergence for covariance matrix estimation.
The Annals of Statistics 38, 2118-2144.

Cai, T., Liu, W. & Luo, X. (2011).
A constrained l₁ minimization approach to sparse precision matrix estimation.
J. American Statistical Association 494, 594-607.

Cai, T. & Liu, W. (2011).
Adaptive thresholding for sparse covariance matrix estimation.
J. American Statistical Association 106, 672-684.

Cai, T. & Zhou, H. (2012).
Minimax estimation of large covariance matrices under l₁ norm (with discussion).
Statistica Sinica 22, 1319-1378.

Cai, T. & Zhou, H. (2012).
Optimal rates of convergence for sparse covariance matrix estimation.
The Annals of Statistics 40, 2389-2420.

Cai, T. & Yuan, M. (2012).
Adaptive covariance matrix estimation through block thresholding.
The Annals of Statistics 40, 2014-2042.