Minimax and Adaptive Estimation of Covariance Operator for Random Variables Observed on a Lattice Graph
Tony Cai and Ming Yuan
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.
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