稳健收缩协方差阵估计的高维分析和Optimization
Large Dimensional Analysis and Optimization of Robust Shrinkage Covariance Matrix Estimators
摘要：
本文研究了两个分散矩阵的正则化鲁棒估计，这两个估计是在（Chen等人，2011）和（Pascal等人，2013），基于泰勒的稳健M估计（泰勒，1987）和Ledoit和Wolf的收缩协方差阵估计（Ledoit和Wolf，2004）。这些混合估计具有以下优点：（i）对离群值或脉冲样本具有鲁棒性;（ii）对经典样本协方差阵估计具有小样本量充分性。我们在这里考虑i.i.d.的情况。在样本和总体规模都很大的情况下，椭圆零均值样本。我们证明，在这种情况下，研究中的估计渐近行为类似于众所周知的随机矩阵模型。这种表征允许我们导出最佳收缩策略来估计群体散布矩阵，从而显著改进在（Chen等人，2011年）。
英文摘要：
This article studies two regularized robust estimators of scatter matrices proposed (and proved to be well defined) in parallel in (Chen et al., 2011) and (Pascal et al., 2013), based on Tyler's robust M-estimator (Tyler, 1987) and on Ledoit and Wolf's shrinkage covariance matrix estimator (Ledoit and Wolf, 2004). These hybrid estimators have the advantage of conveying (i) robustness to outliers or impulsive samples and (ii) small sample size adequacy to the classical sample covariance matrix estimator. We consider here the case of i.i.d. elliptical zero mean samples in the regime where both sample and population sizes are large. We demonstrate that, under this setting, the estimators under study asymptotically behave similar to well-understood random matrix models. This characterization allows us to derive optimal shrinkage strategies to estimate the population scatter matrix, improving significantly upon the empirical shrinkage method proposed in (Chen et al., 2011).
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