Adaptive confidence bands in the nonparametric fixed design regression model

TL;DR

This paper investigates the existence of adaptive confidence bands in the fixed design nonparametric regression model, extending previous ideas and demonstrating the existence of sup-norm adaptive estimators.

Contribution

It adapts and extends Hoffmann and Nickl's methods to the regression setting, establishing the existence of adaptive confidence bands and estimators.

Findings

Adaptive confidence bands exist in fixed design regression.

Sup-norm adaptive estimators are shown to exist.

The approach extends previous nonparametric confidence band results.

Abstract

In this note, we consider the problem of existence of adaptive confidence bands in the fixed design regression model, adapting ideas in Hoffmann and Nickl (2011) to the present case. In the course of the proof, we show that sup-norm adaptive estimators exist as well in regression.