Adaptation Bounds for Confidence Bands under Self-Similarity

TL;DR

This paper establishes theoretical bounds on the adaptability of confidence bands for nonparametric functions under self-similarity, highlighting the influence of the self-similarity constant on the achievable adaptation.

Contribution

It provides the first explicit bounds on adaptation for confidence bands under self-similarity and constructs bands that nearly attain these bounds.

Findings

Adaptation is limited by the self-similarity constant.

Confidence bands can be constructed to nearly achieve the theoretical bounds.

The dependence on the self-similarity constant cannot be eliminated asymptotically.

Abstract

We derive bounds on the scope for a confidence band to adapt to the unknown regularity of a nonparametric function that is observed with noise, such as a regression function or density, under the self-similarity condition proposed by Gine and Nickl (2010). We find that adaptation can only be achieved up to a term that depends on the choice of the constant used to define self-similarity, and that this term becomes arbitrarily large for conservative choices of the self-similarity constant. We construct a confidence band that achieves this bound, up to a constant term that does not depend on the self-similarity constant. Our results suggest that care must be taken in choosing and interpreting the constant that defines self-similarity, since the dependence of adaptive confidence bands on this constant cannot be made to disappear asymptotically.