In this paper, we propose a shrinkage framework for jointly estimating multiple covariance matrices by shrinking the sample covariance matrices towards the pooled sample covariance matrix. This framework allows us to borrow information across different groups. We derive the optimal shrinkage parameters under the Stein and quadratic loss functions, and prove that our derived estimators are asymptotically optimal when the sample size or the number of groups tends to infinity. Simulation studies demonstrate that our proposed shrinkage method performs favorably compared to the existing methods.

中文翻译：

---

多协方差矩阵联合估计的收缩方法

在本文中，我们提出了一个收缩框架，用于通过将样本协方差矩阵向合并样本协方差矩阵收缩来联合估计多个协方差矩阵。这个框架允许我们跨不同的群体借用信息。我们在 Stein 和二次损失函数下推导出最优收缩参数，并证明当样本大小或组数趋于无穷大时，我们导出的估计量是渐近最优的。模拟研究表明，与现有方法相比，我们提出的收缩方法表现良好。