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Abstract:
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This paper considers the estimation of the high dimensional precision matrices. We focus on the two commonly used parameter spaces with the banding structure on the Cholesky factor of precision matrices. These parameter spaces are of great importance in practice, such as in meteorology and spectroscopy. However, the minimax theory has not been fully established. We develop the optimal rates of convergence under both the operator norm and Frobenius norm. The rates on these two parameter spaces reveal a fundamental distinction in estimating precision matrices with bandable Cholesky factor from bandable covariance matrices. The lower bounds are constructed via a careful selection in parameter spaces using the Le Cam and Assouad methods. The optimal procedure is based on a novel optimal estimator of the diagonal submatrix of precision matrix. In the end, we provide a numerical study to illustrate the performance of this approach.
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