On least squares estimation under heteroscedastic and heavy-tailed errors

Abstract

We consider least squares estimation in a general nonparametric regression model where the error is allowed to depend on the covariates. 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 error has a uniformly bounded conditional variance and has only finitely many moments. Our upper bound on the rate of convergence of the LSE depends on 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. We find sufficient conditions on the error distribution 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.