Mean Estimation and Regression Under Heavy-Tailed Distributions: A Survey

We survey some of the recent advances in mean estimation and regression function estimation. in particular, we describe sub-gaussian mean estimators for possibly heavy-tailed data in both the univariate and multivariate settings. we focus on estimators based on median-of-means techniques, but other methods such as the trimmed-mean and catoni’s estimators are also reviewed. we give detailed proofs for the cornerstone results. we dedicate a section to statistical learning problems—in particular, regression function estimation—in the presence of possibly heavy-tailed data.