Spatial covariance matrix estimation is of great significance in many applications in climatology, econometrics and many other fields with complex data structures involving spatial dependencies. High dimensionality brings new challenges to this problem, and no theoretical optimal estimator has been proved for the spatial high-dimensional covariance matrix. Over the past decade, the method of regularization has been introduced to high-dimensional covariance estimation for various structured matrices, to achieve rate optimal estimators. In this paper, we aim to bridge the gap in these two research areas. We use a structure of block bandable covariance matrices to incorporate spatial dependence information, and study rate optimal estimation of this type of structured high dimensional covariance matrices. A double tapering estimator is proposed, and is shown to achieve the asymptotic minimax error bound. Numerical studies on both synthetic and real data are conducted showing the improvement of the double tapering estimator over the sample covariance matrix estimator.
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