This chapter reviews methods for linear shrinkage of the sample covariance matrix (SCM) and matrices (SCM-s) under elliptical distributions in single and multiple populations settings, respectively. In the single sample setting a popular linear shrinkage estimator is defined as a linear combination of the sample covariance matrix (SCM) with a scaled identity matrix. The optimal shrinkage coefficients minimizing the mean squared error (MSE) under elliptical sampling are shown to be functions of few key parameters only, such as elliptical kurtosis and sphericity parameter. Similar results and estimators are derived for multiple population setting and applications of the studied shrinkage estimators are illustrated in portfolio optimization.

翻译：本章综述了在单总体和多总体设定下，椭圆分布中样本协方差矩阵（SCM）及样本协方差矩阵组（SCM-s）的线性收缩方法。在单样本设定中，一种常用的线性收缩估计量被定义为样本协方差矩阵与缩放单位矩阵的线性组合。研究表明，在椭圆抽样下使均方误差（MSE）最小化的最优收缩系数仅为少数关键参数的函数，例如椭圆峰度和球形参数。针对多总体设定，本文推导了类似的结论与估计量，并通过投资组合优化实例展示了所研究收缩估计量的应用。