Mean estimation and regression under heavy-tailed distributions--a survey
It addresses the problem of robust statistical estimation for researchers and practitioners dealing with heavy-tailed data, but it is incremental as it surveys existing methods rather than introducing new ones.
This survey reviews recent advances in mean and regression function estimation for heavy-tailed data, focusing on sub-Gaussian estimators like median-of-means and other methods such as trimmed mean and Catoni's estimator, with detailed proofs provided.
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.