Honest adaptive confidence bands and self-similar functions

TL;DR

This paper introduces a method for constructing honest adaptive confidence bands for self-similar functions, enabling optimal width contraction across H"older classes and addressing the undersmoothing problem.

Contribution

It demonstrates that self-similarity is both necessary and sufficient for adaptive confidence bands, improving inference in nonparametric density and regression models.

Findings

Self-similarity ensures the existence of adaptive confidence bands.

The proposed bands are honest for functions of any H"older norm.

The method addresses the undersmoothing issue in confidence band construction.

Abstract

Confidence bands are confidence sets for an unknown function f, containing all functions within some sup-norm distance of an estimator. In the density estimation, regression, and white noise models, we consider the problem of constructing adaptive confidence bands, whose width contracts at an optimal rate over a range of H\"older classes. While adaptive estimators exist, in general adaptive confidence bands do not, and to proceed we must place further conditions on f. We discuss previous approaches to this issue, and show it is necessary to restrict f to fundamentally smaller classes of functions. We then consider the self-similar functions, whose H\"older norm is similar at large and small scales. We show that such functions may be considered typical functions of a given H\"older class, and that the assumption of self-similarity is both necessary and sufficient for the construction of adaptive bands. Finally, we show that this assumption allows us to resolve the problem of undersmoothing, creating bands which are honest simultaneously for functions of any H\"older norm.