A highly popular regularized (shrinkage) covariance matrix estimator is the shrinkage sample covariance matrix (SCM) which shares the same set of eigenvectors as the SCM but shrinks its eigenvalues toward the grand mean of the eigenvalues of the SCM. In this paper, a more general approach is considered in which the SCM is replaced by an M-estimator of scatter matrix and a fully automatic data adaptive method to compute the optimal shrinkage parameter with minimum mean squared error is proposed. Our approach permits the use of any weight function such as Gaussian, Huber's, Tyler's, or $t$ weight functions, all of which are commonly used in M-estimation framework. Our simulation examples illustrate that shrinkage M-estimators based on the proposed optimal tuning combined with robust weight function do not loose in performance to shrinkage SCM estimator when the data is Gaussian, but provide significantly improved performance when the data is sampled from an unspecified heavy-tailed elliptically symmetric distribution. Also, real-world and synthetic stock market data validate the performance of the proposed method in practical applications.

中文翻译：

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缩小协方差矩阵的 M 估计量的特征值

一个非常流行的正则化（收缩）协方差矩阵估计器是收缩样本协方差矩阵 (SCM)，它与 ​​SCM 共享相同的一组特征向量，但将其特征值向 SCM 的特征值的总平均值收缩。在本文中，考虑了一种更通用的方法，其中 SCM 被散射矩阵的 M 估计量代替，并提出了一种全自动数据自适应方法来计算具有最小均方误差的最佳收缩参数。我们的方法允许使用任何权重函数，例如 Gaussian、Huber's、Tyler's 或 $t$ 权重函数，所有这些都在 M 估计框架中常用。我们的模拟示例表明，当数据是高斯时，基于所提出的优化调整结合鲁棒权重函数的收缩 M 估计器在性能上不会比收缩 SCM 估计器松散，但是当数据是从未指定的重采样时提供显着提高的性能 -尾椭圆对称分布。此外，真实世界和合成股票市场数据验证了所提出方法在实际应用中的性能。