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October 2014 Anti-concentration and honest, adaptive confidence bands
Victor Chernozhukov, Denis Chetverikov, Kengo Kato
Ann. Statist. 42(5): 1787-1818 (October 2014). DOI: 10.1214/14-AOS1235

Abstract

Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the classical Smirnov–Bickel–Rosenblatt (SBR) condition; see, for example, Giné and Nickl [Probab. Theory Related Fields 143 (2009) 569–596]. This condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process (equivalently, for the supremum of a Gaussian process with the same covariance function as that of the studentized empirical process). The principal contribution of this paper is to remove the need for this classical condition. We show that a considerably weaker sufficient condition is derived from an anti-concentration property of the supremum of the approximating Gaussian process, and we derive an inequality leading to such a property for separable Gaussian processes. We refer to the new condition as a generalized SBR condition. Our new result shows that the supremum does not concentrate too fast around any value.

We then apply this result to derive a Gaussian multiplier bootstrap procedure for constructing honest confidence bands for nonparametric density estimators (this result can be applied in other nonparametric problems as well). An essential advantage of our approach is that it applies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist (or is unknown). This is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data-driven fashion, which is needed for adaptive constructions of the confidence bands. Finally, of independent interest is our introduction of a new, practical version of Lepski’s method, which computes the optimal, nonconservative resolution levels via a Gaussian multiplier bootstrap method.

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Victor Chernozhukov. Denis Chetverikov. Kengo Kato. "Anti-concentration and honest, adaptive confidence bands." Ann. Statist. 42 (5) 1787 - 1818, October 2014. https://doi.org/10.1214/14-AOS1235

Information

Published: October 2014
First available in Project Euclid: 11 September 2014

zbMATH: 1305.62161
MathSciNet: MR3262468
Digital Object Identifier: 10.1214/14-AOS1235

Subjects:
Primary: 62G07 , 62G15

Keywords: Anti-concentration of separable Gaussian processes , honest confidence bands , Lepski’s method , multiplier method , non-Donsker empirical processes

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 5 • October 2014
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