Open Access
Translator Disclaimer
2012 Honest adaptive confidence bands and self-similar functions
Adam D. Bull
Electron. J. Statist. 6: 1490-1516 (2012). DOI: 10.1214/12-EJS720

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 Hölder 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 Hölder norm is similar at large and small scales. We show that such functions may be considered typical functions of a given Hölder class, and that the assumption of self-similarity is both necessary and sufficient for the construction of adaptive bands.

Citation

Download Citation

Adam D. Bull. "Honest adaptive confidence bands and self-similar functions." Electron. J. Statist. 6 1490 - 1516, 2012. https://doi.org/10.1214/12-EJS720

Information

Published: 2012
First available in Project Euclid: 31 August 2012

zbMATH: 1295.62049
MathSciNet: MR2988456
Digital Object Identifier: 10.1214/12-EJS720

Subjects:
Primary: 62G15
Secondary: 62G07 , 62G08 , 62G20

Keywords: Adaptation , Confidence sets , nonparametric statistics , self-similar functions , supremum norm

Rights: Copyright © 2012 The Institute of Mathematical Statistics and the Bernoulli Society

JOURNAL ARTICLE
27 PAGES


SHARE
Back to Top