We tackle covariance estimation in low-sample scenarios, employing a structured covariance matrix with shrinkage methods. These involve convexly combining a low-bias/high-variance empirical estimate with a biased regularization estimator, striking a bias-variance trade-off. Literature provides optimal settings of the regularization amount through risk minimization between the true covariance and its shrunk counterpart. Such estimators were derived for zero-mean statistics with i.i.d. diagonal regularization matrices accounting for the average sample variance solely. We extend these results to regularization matrices accounting for the sample variances both for centered and non-centered samples. In the latter case, the empirical estimate of the true mean is incorporated into our shrinkage estimators. Introducing confidence weights into the statistics also enhance estimator robustness against outliers. We compare our estimators to other shrinkage methods both on numerical simulations and on real data to solve a detection problem in astronomy.

翻译：本文针对低样本量场景下的协方差估计问题，采用结构化协方差矩阵结合收缩方法进行研究。该方法通过将低偏差/高方差的经验估计量与带偏差的正则化估计量进行凸组合，实现偏差-方差权衡。现有文献通过最小化真实协方差与其收缩估计量之间的风险函数，给出了正则化参数的最优设置。此类估计器最初是针对零均值统计量推导的，其独立同分布对角正则化矩阵仅考虑平均样本方差。我们将这些结果推广到同时适用于中心化与非中心化样本的正则化矩阵，其中后者将真实均值的经验估计纳入收缩估计器设计。通过在统计量中引入置信权重，进一步增强了估计器对异常值的鲁棒性。我们通过数值模拟和天文检测实际数据，将所提估计器与其他收缩方法进行了对比分析。