Berry-Esseen Bounds and Moderate Deviations for Catoni-Type Robust Estimation

TL;DR

This work develops finite-sample Berry–Esseen bounds and Cramér-type moderate deviation principles for Catoni-type robust estimators under heavy-tailed data, without symmetry assumptions. It first treats univariate mean estimation with both known and unknown variance, including a self-normalized variant for the unknown-variance case, and then extends the framework to multivariate Catoni-type robust regression. The paper proves consistency and multivariate normal-approximation results for the regression estimator, and provides nonasymptotic and BE-type bounds for the regression setting. The results offer practical Gaussian-approximation guarantees for robust inference in heavy-tailed contexts, with implications for high-dimensional regression and stochastic algorithms.

Abstract

A powerful robust mean estimator introduced by Catoni (2012) allows for mean estimation of heavy-tailed data while achieving the performance characteristics of classical mean estimator for sub-Gaussian data. While Catoni's framework has been widely extended across statistics, stochastic algorithms, and machine learning, fundamental asymptotic questions regarding the Central Limit Theorem and rare event deviations remain largely unaddressed. In this paper, we investigate Catoni-type robust estimators in two contexts: (i) mean estimation for heavy-tailed data, and (ii) linear regression with heavy-tailed innovations. For the first model, we establish the Berry--Esseen bound and moderate deviation principles, addressing both known and unknown variance settings. For the second model, we demonstrate that the associated estimator is consistent and satisfies a multi-dimensional Berry-Esseen bound.

Theorems & Definitions (25)

• Theorem 2.1
• Remark 2.1
• Corollary 2.2
• Remark 2.2
• Corollary 2.3
• Theorem 2.4
• Corollary 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8