Excess-Risk Consistency of Group-hard Thresholding Estimator in Robust Estimation of Gaussian Mean

摘要

In this work we introduce the notion of the excess risk in the setup of estimation of the Gaussian mean when the observations are corrupted by outliers. It is known that the sample mean loses its good properties in the presence of outliers [5,6]. In addition, even the sample median is not minimax-rate-optimal in the multivariate setting. The optimal rate of the minimax risk in this setting was established by [1]. However, even these minimax-rate-optimality results do not quantify how fast the risk in the contaminated model approaches the risk in the uncontaminated model when the rate of contamination goes to zero. The present paper does a first step in filling this gap by showing that the group hard thresholding estimator has an excess risk that goes to zero when the corruption rate approaches zero.