Last decade witnesses significant methodological and theoretical advances in estimating large precision matrices. In particular, there are scientific applications such as longitudinal data, meteorology and spectroscopy in which the ordering of the variables can be interpreted through a bandable structure on the Cholesky factor of the precision matrix. However, the minimax theory has still been largely unknown, as opposed to the well established minimax results over the corresponding bandable covariance matrices. In this thesis, we focus on two commonly used types of parameter spaces, and 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 corresponding two types of bandable covariance matrices under both norms. This fundamental difference is established by carefully constructing the corresponding minimax lower bounds. 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. Lepski's method is considered to achieve optimal adaptation. We further establish rate optimality in the nonparanormal model, by applying our local cropping procedure to the rank-based estimators. Numerical studies are carried out to confirm our theoretical findings.

中文翻译：

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具有可带 Cholesky 因子的大精度矩阵的极大极小估计

过去十年见证了在估计大型精度矩阵方面的重大方法论和理论进步。特别是在诸如纵向数据、气象学和光谱学等科学应用中，可以通过精度矩阵的 Cholesky 因子上的带状结构来解释变量的排序。然而，与在相应的可带协方差矩阵上建立的极小极大结果相反，极大极小理论在很大程度上仍然是未知的。在本论文中，我们关注两种常用的参数空间类型，并在算子范数和 Frobenius 范数下开发最佳收敛速度。发现了一个惊人的现象：两类参数空间在算子范数下根本不同，但在 Frobenius 范数下具有相同的速率最优性，这与两种范数下对应的两类可带协方差矩阵的等价性形成鲜明对比。这个根本区别是通过仔细构建相应的极小极大下界来建立的。开发了两个新的估计程序：对于算子范数，我们的最佳程序基于一种新颖的局部裁剪估计器，该估计器针对精度矩阵的所有主子矩阵，而对于 Frobenius 范数，我们的最佳程序依赖于一个微妙的基于回归的块-阈值规则。Lepski 的方法被认为是实现最佳适应。我们进一步在非超自然模型中建立速率最优性，通过将我们的局部裁剪程序应用于基于等级的估计器。进行了数值研究以证实我们的理论发现。