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

• Authors: Gábor Lugosi.
• Statistics
• Foundations of Computational Mathematics

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 both in 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 estimator are also reviewed. We give detailed proofs for the cornerstone results. We dedicate a section on statistical learning problems–in particular, regression function estimation–in the presence of possibly heavy-tailed data.