Abstract

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.

中文翻译：

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高斯均值的稳健估计中的组硬阈值估计器的过剩风险一致性

摘要

在这项工作中，当观测值被异常值破坏时，我们在估计高斯均值的设置中引入了超额风险的概念。已知样本均值在存在异常值时会失去其良好的性能[5，6]。另外，在多变量设置中，即使样本中位数也不是最小最大速率最优的。在这种情况下，最小最大风险的最佳比率由[1]确定。但是，即使这些最小最大速率优化结果也无法量化当污染率变为零时，受污染模型中的风险接近未受污染模型中的风险的速度。本文通过显示组硬阈值估计器的过度风险在腐败率接近零时变为零，从而迈出了填补这一空白的第一步。