User-Friendly Covariance Estimation for Heavy-Tailed Distributions
User-Friendly Covariance Estimation for Heavy-Tailed Distributions
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DOI：
10.1214/19-sts711
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发表时间：
2018-11
影响因子：
5.7
通讯作者：
Y. Ke;Stanislav Minsker;Zhao Ren;Qiang Sun;Wen-Xin Zhou
中科院分区：
文献类型：
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作者：
Y. Ke;Stanislav Minsker;Zhao Ren;Qiang Sun;Wen-Xin Zhou
我们对重尾分布的协方差估计的最新选定结果进行了调查。通过统一文献中出现的分散想法，我们提出了便于实际实施的用户友好方法。具体来说，我们引入了按元素和按谱截断运算符，以及它们的 $M$ 估计器对应项，以增强样本协方差矩阵。与通常以击穿特性为特征的经典鲁棒性概念不同，我们关注尾部鲁棒性，尾部鲁棒性由仅具有有限四阶矩的数据的估计器的非渐近偏差特性证明。关键的观察是，稳健性参数需要适应样本大小、维度和矩，以实现偏差和稳健性之间的最佳权衡。此外，为了促进其实际使用，我们提出了自动校准调整参数的免调整程序。重新审视了一系列高维结构化模型的应用，包括可带协方差估计、稀疏精度矩阵估计、低秩协方差估计以及因子模型下的协方差估计和多重检验。数值例子为我们提出的方法提供了强有力的支持。
We offer a survey of selected recent results on covariance estimation for heavy-tailed distributions. By unifying scattered ideas appeared in the literature, we propose user-friendly methods that facilitate practical implementation. Specifically, we introduce element-wise and spectrum-wise truncation operators, as well as their $M$-estimator counterparts, to robustify the sample covariance matrix. Different from the classical notion of robustness which is typically characterized by the breakdown property, we focus on the tail robustness that is evidenced by the nonasymptotic deviation property of the estimator for data with only finite fourth moments. The key observation is that the robustification parameter needs to adapt to the sample size, dimensionality and moment to achieve optimal tradeoff between bias and robustness. Furthermore, to facilitate their practical use, we propose tuning-free procedures that automatically calibrate the tuning parameters. Applications to a series of structured models in high dimensions, including the bandable covariance estimation, sparse precision matrix estimation, low-rank covariance estimation, and covariance estimation and multiple testing under factor models are revisited. Numerical examples lend strong support to our proposed methodology.