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 + ε moments for ε ∈ (0, 2). The associated convergence rates depend on ε.

中文翻译：

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高维协方差和精度矩阵的鲁棒估计


高维数据通常最有可能从具有复杂结构和某些或所有组件中尖峰态的分布中生成。协方差和精度矩阵提供了这种结构的有用总结，但流行的矩阵估计器的性能通常取决于亚高斯假设。本文提出了鲁棒的矩阵估计器，其性能可以保证更丰富的分布类别。所提出的估计器在有界四阶矩假设下实现了与亚高斯假设下的现有方法相同的极小极大收敛率。所提出的估计量的一致性也是在 ε ε (0, 2) 有界 2 + ε 矩的弱假设下建立的。相关的收敛速度取决于 ε。