The minimum covariance determinant (MCD) estimator is ubiquitous in multivariate analysis, the critical step of which is to select a subset of a given size with the lowest sample covariance determinant. The concentration step (C-step) is a common tool for subset-seeking; however, it becomes computationally demanding for high-dimensional data. To alleviate the challenge, we propose a depth-based algorithm, termed as \texttt{FDB}, which replaces the optimal subset with the trimmed region induced by statistical depth. We show that the depth-based region is consistent with the MCD-based subset under a specific class of depth notions, for instance, the projection depth. With the two suggested depths, the \texttt{FDB} estimator is not only computationally more efficient but also reaches the same level of robustness as the MCD estimator. Extensive simulation studies are conducted to assess the empirical performance of our estimators. We also validate the computational efficiency and robustness of our estimators under several typical tasks such as principal component analysis, linear discriminant analysis, image denoise and outlier detection on real-life datasets. A R package \textit{FDB} and potential extensions are available in the Supplementary Materials.

翻译：最小协方差行列式（MCD）估计量在多元分析中应用广泛，其关键步骤是选取一个给定大小的子集，使其样本协方差行列式最小化。集中步骤（C-step）是子集搜索的常用工具，但在处理高维数据时计算量显著增加。为缓解这一挑战，我们提出一种基于深度的算法（记为\texttt{FDB}），该算法用统计深度诱导的截尾区域替代最优子集。我们证明，在特定深度概念（如投影深度）下，基于深度的区域与基于MCD的子集一致。采用这两种建议深度后，\texttt{FDB}估计量不仅计算效率更高，还达到了与MCD估计量同等的鲁棒性。通过大量仿真研究评估了所提出估计量的实证性能，并通过主成分分析、线性判别分析、图像去噪及真实数据集上的异常值检测等典型任务验证了其计算效率与鲁棒性。补充材料中提供了R包\textit{FDB}及可能的扩展应用。