# Mean estimation and regression under heavy-tailed distributions--a survey

## TL;DR

This survey reviews recent advances in robust mean and regression estimation techniques suitable for heavy-tailed data, emphasizing median-of-means and related methods in various statistical settings.

## Contribution

It provides a comprehensive overview of recent methods for robust estimation under heavy tails, including detailed proofs and applications to regression problems.

## Key findings

- Median-of-means estimators achieve sub-Gaussian performance with heavy-tailed data
- Other robust estimators like trimmed mean and Catoni's estimator are effective
- The survey highlights applications in statistical learning and regression with heavy tails

## Abstract

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.