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October 2011 On adaptive inference and confidence bands
Marc Hoffmann, Richard Nickl
Ann. Statist. 39(5): 2383-2409 (October 2011). DOI: 10.1214/11-AOS903

Abstract

The problem of existence of adaptive confidence bands for an unknown density f that belongs to a nested scale of Hölder classes over ℝ or [0, 1] is considered. Whereas honest adaptive inference in this problem is impossible already for a pair of Hölder balls Σ(r), Σ(s), rs, of fixed radius, a nonparametric distinguishability condition is introduced under which adaptive confidence bands can be shown to exist. It is further shown that this condition is necessary and sufficient for the existence of honest asymptotic confidence bands, and that it is strictly weaker than similar analytic conditions recently employed in Giné and Nickl [Ann. Statist. 38 (2010) 1122–1170]. The exceptional sets for which honest inference is not possible have vanishingly small probability under natural priors on Hölder balls Σ(s). If no upper bound for the radius of the Hölder balls is known, a price for adaptation has to be paid, and near-optimal adaptation is possible for standard procedures. The implications of these findings for a general theory of adaptive inference are discussed.

Citation

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Marc Hoffmann. Richard Nickl. "On adaptive inference and confidence bands." Ann. Statist. 39 (5) 2383 - 2409, October 2011. https://doi.org/10.1214/11-AOS903

Information

Published: October 2011
First available in Project Euclid: 30 November 2011

zbMATH: 1232.62072
MathSciNet: MR2906872
Digital Object Identifier: 10.1214/11-AOS903

Subjects:
Primary: 62G15
Secondary: 62G05, 62G10

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 5 • October 2011
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