This paper considers the estimation of large precision matrices. We focus on two commonly used types of parameter spaces with a bandable structure on the Cholesky factor of precision matrices. Last decade witnesses significant methodological advances and wide scientific applications in meteorology and spectroscopy. However, the minimax theory has still been largely unknown, as opposed to the well established minimax results over the corresponding two types of bandable covariance matrices. In this paper, we develop the optimal rates of convergence under both the operator norm and the Frobenius norm. A striking phenomenon is found: two types of parameter spaces are fundamentally different under the operator norm but enjoy the same rate optimality under the Frobenius norm, which is in sharp contrast to the equivalence of two types of bandable covariance matrices under both norms. This fundamental difference is largely established by carefully constructing the corresponding minimax lower bounds using the Le Cam and Assouad methods. Two new estimation procedures are developed: for the operator norm, our optimal procedure is based on a novel local cropping estimator targeting on all principle submatrices of the precision matrix while for the Frobenius norm, our optimal procedure relies on a delicate regression-based block-thresholding rule. We further establish rate optimality in the nonparanormal model by applying our local cropping procedure to the rank-based estimators. In the end, we carry out numerical studies to demonstrate the performance of our approach as well as illustrate the fundamental difference between two types of parameter spaces.
View on arXiv