Robust M-Estimation of Scatter Matrices via Precision Structure Shrinkage

Abstract

Tyler’s M-Estimator (TME) is a well-known method for robust covariance structure estimation, based on the idea of iteratively reweighting data points according to their Mahalanobis distances. Due to its simplicity, robustness, and efficiency, TME is widely used in fields such as robust statistics, anomaly detection, and signal processing – particularly in situations where data potentially contain outliers.
While TME appears to offer high robustness, it is known that TME is vulnerable to a specific type of outliers known as clustered outliers, where outliers are concentrated within a narrow area.
In this talk, we first report numerical experiments that suggest the source of the vulnerability lies in the process of estimating the inverse of covariance structure, i.e., the precision structure, within the TME algorithm. To address this issue, we propose an algorithm that targets the estimation of the precision structure. The proposed method explicitly incorporates the Efron–Morris (1976) approach for precision structure estimation of the TME algorithm.
We demonstrate through numerical experiments that the proposed method exhibits high robustness not only against clustered outliers, but also against outliers that are due to heavy-tailed population distributions such as the t-distribution with small degrees of freedom. We evaluate the robustness of the proposed method in terms of the breakdown point and provide interpretations from various perspectives. 

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