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：我们针对低样本场景下的协方差估计问题，采用具有结构协方差矩阵的收缩方法。该方法将低偏差/高方差的经验估计与有偏的正则化估计进行凸组合，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版本之间的风险，提供了正则化量的最优设置。此类估计量针对零均值统计量推导，并采用独立同分布的对角正则化矩阵仅考虑平均样本方差。我们将这些结果扩展至适用于中心化与非中心化样本中样本方差的正则化矩阵。针对非中心化情形，我们将真实均值的经验估计纳入收缩估计量。引入统计量置信权重进一步增强了估计量对异常值的鲁棒性。我们通过数值模拟和实际天文探测问题的求解，将所提估计量与其他收缩方法进行了比较。