Asymptotic confidence bands for the histogram regression estimator

TL;DR

This paper develops asymptotic uniform confidence bands for multivariate nonparametric regression using histogram estimators, suitable for irregular functions and avoiding reliance on extreme value distributions.

Contribution

It introduces a method to construct explicit asymptotic confidence bands for histogram regression estimators under flexible partition conditions, applicable to unsmooth functions.

Findings

Confidence bands are explicitly calculable without extreme value distribution dependence.

Method applies to heteroscedastic noise and functions with low H"older regularity.

Construction is valid under flexible partition conditions.

Abstract

Asymptotic uniform confidence bands are constructed for a multivariate nonparametric regression model with heteroscedastic noise, employing histogram estimators under flexible partition conditions. The construction is especially applicable to unsmooth regression functions of H\"older regularity less than one. While the radius of the confidence bands could be approximated via the Gumbel distribution, our construction does not depend on an extreme value distribution, but instead can be explicitly calculated for the chosen partition.