Pointwise adaptive estimation for robust and quantile regression

TL;DR

This paper introduces a data-driven, locally adaptive nonparametric regression method that combines M-estimators and sequential testing, achieving rate-optimality and demonstrating strong finite sample performance in simulations and applications like image denoising.

Contribution

It presents a novel adaptive estimation procedure for robust and quantile regression that adjusts to local regularity and extends to image denoising tasks.

Findings

Achieves rate-optimal risk bounds under weak conditions.

Demonstrates good finite sample properties in simulations.

Successfully applied to CT scan image denoising.

Abstract

A nonparametric procedure for robust regression estimation and for quantile regression is proposed which is completely data-driven and adapts locally to the regularity of the regression function. This is achieved by considering in each point M-estimators over different local neighbourhoods and by a local model selection procedure based on sequential testing. Non-asymptotic risk bounds are obtained, which yield rate-optimality for large sample asymptotics under weak conditions. Simulations for different univariate median regression models show good finite sample properties, also in comparison to traditional methods. The approach is extended to image denoising and applied to CT scans in cancer research.