Pointwise Adaptive M-estimation in Nonparametric Regression

TL;DR

This paper introduces an adaptive, robust nonparametric regression estimator that works under heteroscedastic noise with unknown error densities, using local polynomial M-estimation and Lepski's bandwidth selection.

Contribution

It develops a new adaptive estimator for heteroscedastic regression that does not require knowledge of error densities and is robust to extreme noise values.

Findings

Establishes new exponential inequalities for local M-estimators.

Constructs a minimax estimator that adapts over Hölder classes.

Demonstrates robustness to extreme noise values.

Abstract

This paper deals with the nonparametric estimation in heteroscedastic regression $ Y_i=f(X_i)+\xi_i, \: i=1,...,n $, with incomplete information, i.e. each real random variable $ \xi_i $ has a density $ g_{i} $ which is unknown to the statistician. The aim is to estimate the regression function $ f $ at a given point. Using a local polynomial fitting from M-estimator denoted $ \hat f^h $ and applying Lepski's procedure for the bandwidth selection, we construct an estimator $ \hat f^{\hat h} $ which is adaptive over the collection of isotropic H\"{o}lder classes. In particular, we establish new exponential inequalities to control deviations of local M-estimators allowing to construct the minimax estimator. The advantage of this estimator is that it does not depend on densities of random errors and we only assume that the probability density functions are symmetric and monotonically on $ \bR_+ $. It is important to mention that our estimator is robust compared to extreme values of the noise.