This paper focuses on investigating Stein's invariant shrinkage estimators for large sample covariance matrices and precision matrices in high-dimensional settings. We consider models that have nearly arbitrary population covariance matrices, including those with potential spikes. By imposing mild technical assumptions, we establish the asymptotic limits of the shrinkers for a wide range of loss functions. A key contribution of this work, enabling the derivation of the limits of the shrinkers, is a novel result concerning the asymptotic distributions of the non-spiked eigenvectors of the sample covariance matrices, which can be of independent interest.

翻译：本文聚焦于高维背景下大样本协方差矩阵与精度矩阵的Stein不变收缩估计量研究。我们考虑具有近乎任意总体协方差矩阵的模型，包括存在潜在谱峰的情况。通过施加温和的技术假设，我们建立了广泛损失函数下收缩估计量的渐近极限。本工作的关键贡献在于推导出收缩估计量极限的新颖结论——样本协方差矩阵非谱峰特征向量的渐近分布定理，该结果本身即具有独立研究价值。