Robust nonparametric estimation via wavelet median regression

TL;DR

This paper introduces a robust nonparametric regression method that uses wavelet median regression to adaptively estimate functions across various classes while handling heavy-tailed and unknown error distributions.

Contribution

It develops a wavelet-based estimator that is both adaptive over diverse function classes and robust to a wide range of error distributions, including heavy-tailed ones.

Findings

Achieves optimal convergence rates over Besov classes.

Automatically adapts to local smoothness of functions.

Provides a quantile coupling theorem for medians.

Abstract

In this paper we develop a nonparametric regression method that is simultaneously adaptive over a wide range of function classes for the regression function and robust over a large collection of error distributions, including those that are heavy-tailed, and may not even possess variances or means. Our approach is to first use local medians to turn the problem of nonparametric regression with unknown noise distribution into a standard Gaussian regression problem and then apply a wavelet block thresholding procedure to construct an estimator of the regression function. It is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. A key technical result in our development is a quantile coupling theorem which gives a tight bound for the quantile coupling between the sample medians and a normal variable. This median coupling inequality may be of independent interest.