In this paper, we propose the Minimum Regularized Covariance Trace (MRCT) estimator, a novel method for robust covariance estimation and functional outlier detection. The MRCT estimator employs a subset-based approach that prioritizes subsets exhibiting greater centrality based on the generalization of the Mahalanobis distance, resulting in a fast-MCD type algorithm. Notably, the MRCT estimator handles high-dimensional data sets without the need for preprocessing or dimension reduction techniques, due to the internal smoothening whose amount is determined by the regularization parameter $\alpha > 0$. The selection of the regularization parameter $\alpha$ is automated. The proposed method adapts seamlessly to sparsely observed data by working directly with the finite matrix of basis coefficients. An extensive simulation study demonstrates the efficacy of the MRCT estimator in terms of robust covariance estimation and automated outlier detection, emphasizing the balance between noise exclusion and signal preservation achieved through appropriate selection of $\alpha$. The method converges fast in practice and performs favorably when compared to other functional outlier detection methods.

翻译：本文提出最小正则化协方差迹（MRCT）估计器，这是一种用于稳健协方差估计和函数型异常值检测的新方法。MRCT估计器采用基于子集的方法，根据马氏距离的推广优先选择具有更高中心性的子集，从而形成一种fast-MCD型算法。值得注意的是，由于内部平滑处理（其程度由正则化参数$\alpha > 0$确定），MRCT估计器无需预处理或降维技术即可处理高维数据集。正则化参数$\alpha$的选择是自动化的。所提方法通过直接处理基系数有限矩阵，无缝适应稀疏观测数据。广泛的仿真研究证明了MRCT估计器在稳健协方差估计和自动异常值检测方面的有效性，强调了通过适当选择$\alpha$实现的噪声排除与信号保留之间的平衡。该方法在实践中收敛速度快，并与其他函数型异常值检测方法相比性能更优。