On Least Squares Estimation under Heteroscedastic and Heavy-Tailed Errors

TL;DR

This paper investigates the convergence rates of least squares estimators in nonparametric regression models with heteroscedastic and heavy-tailed errors, providing finite-sample bounds under minimal moment conditions.

Contribution

It establishes new upper bounds on the convergence rates of the LSE under weaker error assumptions, including only finite moments and heteroscedasticity, extending previous sub-Gaussian results.

Findings

Convergence rates depend on error moments, function class complexity, and local structure.

Sufficient conditions identified for LSE to achieve sub-Gaussian-like rates.

Results are finite-sample and applicable to heteroscedastic, heavy-tailed errors.

Abstract

We consider least squares estimation in a general nonparametric regression model. The rate of convergence of the least squares estimator (LSE) for the unknown regression function is well studied when the errors are sub-Gaussian. We find upper bounds on the rates of convergence of the LSE when the errors have uniformly bounded conditional variance and have only finitely many moments. We show that the interplay between the moment assumptions on the error, the metric entropy of the class of functions involved, and the "local" structure of the function class around the truth drives the rate of convergence of the LSE. We find sufficient conditions on the errors under which the rate of the LSE matches the rate of the LSE under sub-Gaussian error. Our results are finite sample and allow for heteroscedastic and heavy-tailed errors.