Missing data occur frequently in a wide range of applications. In this paper, we consider estimation of high-dimensional covariance matrices in the presence of missing observations under a general missing completely at random model in the sense that the missingness is not dependent on the values of the data. Based on incomplete data, estimators for bandable and sparse covariance matrices are proposed and their theoretical and numerical properties are investigated. Minimax rates of convergence are established under the spectral norm loss and the proposed estimators are shown to be rate-optimal under mild regularity conditions. Simulation studies demonstrate that the estimators perform well numerically. The methods are also illustrated through an application to data from four ovarian cancer studies. The key technical tools developed in this paper are of independent interest and potentially useful for a range of related problems in high-dimensional statistical inference with missing data.

中文翻译：

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具有不完整数据的高维协方差矩阵的极小极大速率最优估计

丢失数据在广泛的应用程序中经常发生。在本文中，我们考虑在一般完全随机缺失模型下存在缺失观测值的情况下估计高维协方差矩阵，因为缺失不依赖于数据的值。基于不完整数据，提出了带状和稀疏协方差矩阵的估计器，并研究了它们的理论和数值特性。在谱范数损失下建立了极大极小收敛速率，并且所提出的估计器在温和的规律性条件下被证明是速率最优的。模拟研究表明估计器在数值上表现良好。还通过对来自四项卵巢癌研究的数据的应用来说明这些方法。