Electronic Journal of Statistics

Honest adaptive confidence bands and self-similar functions

Adam D. Bull

Full-text: Open access

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.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 1490-1516.

Dates
First available in Project Euclid: 31 August 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1346421602

Digital Object Identifier
doi:10.1214/12-EJS720

Mathematical Reviews number (MathSciNet)
MR2988456

Zentralblatt MATH identifier
1295.62049

Subjects
Primary: 62G15: Tolerance and confidence regions
Secondary: 62G07: Density estimation 62G08: Nonparametric regression 62G20: Asymptotic properties

Keywords
Nonparametric statistics adaptation confidence sets supremum norm self-similar functions

Citation

Bull, Adam D. Honest adaptive confidence bands and self-similar functions. Electron. J. Statist. 6 (2012), 1490--1516. doi:10.1214/12-EJS720. https://projecteuclid.org/euclid.ejs/1346421602


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