We obtain the upper error bounds of robust estimators for mean vector, using the median-of-means (MOM) method. The method is designed to handle data with heavy tails and contamination, with only a finite second moment, which is weaker than many others, relying on the VC dimension rather than the Rademacher complexity to measure statistical complexity. This allows us to implement MOM in covariance estimation, without imposing conditions such as $L$-sub-Gaussian or $L_{4}-L_{2}$ norm equivalence. In particular, we derive a new robust estimator, the MOM version of the halfspace depth, along with error bounds for mean estimation in any norm.

翻译：我们利用均值中位数（MOM）方法，获得了均值向量鲁棒估计器的上误差界。该方法专为处理具有重尾和污染的数据而设计，仅需有限二阶矩条件，这一假设弱于许多其他方法。我们采用VC维度而非Rademacher复杂度来衡量统计复杂度，这使得我们能够在协方差估计中实现MOM方法，而无需施加$L$-次高斯或$L_{4}-L_{2}$范数等价等条件。特别地，我们推导出一种新的鲁棒估计器——半空间深度的MOM版本，并给出了任意范数下均值估计的误差界。