Avella-Medina Marco, Battey Heather S, Fan Jianqing, Li Quefeng
Sloan School of Management, Massachusetts Institute of Technology, 30 Memorial Drive, Cambridge, Massachusetts 02142, U.S.A.
Department of Mathematics, Imperial College London, 545 Huxley Building, South Kensington Campus, London SW7 2AZ, U.K.
Biometrika. 2018 Jun 1;105(2):271-284. doi: 10.1093/biomet/asy011. Epub 2018 Mar 27.
High-dimensional data are often most plausibly generated from distributions with complex structure and leptokurtosis in some or all components. Covariance and precision matrices provide a useful summary of such structure, yet the performance of popular matrix estimators typically hinges upon a sub-Gaussianity assumption. This paper presents robust matrix estimators whose performance is guaranteed for a much richer class of distributions. The proposed estimators, under a bounded fourth moment assumption, achieve the same minimax convergence rates as do existing methods under a sub-Gaussianity assumption. Consistency of the proposed estimators is also established under the weak assumption of bounded 2 + moments for ∈ (0, 2). The associated convergence rates depend on .
高维数据通常最有可能由某些或所有分量具有复杂结构和尖峰厚尾性的分布生成。协方差矩阵和精度矩阵提供了这种结构的有用汇总，然而流行的矩阵估计器的性能通常取决于次高斯性假设。本文提出了鲁棒矩阵估计器，其性能对于更丰富的一类分布是有保证的。在有界四阶矩假设下，所提出的估计器实现了与次高斯性假设下现有方法相同的极小极大收敛速率。在所提出的估计器在 ∈ (0, 2) 的有界 2 + 矩的弱假设下也建立了一致性。相关的收敛速率取决于 。
