The Annals of Statistics

Anti-concentration and honest, adaptive confidence bands

Victor Chernozhukov, Denis Chetverikov, and Kengo Kato

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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.

Article information

Ann. Statist., Volume 42, Number 5 (2014), 1787-1818.

First available in Project Euclid: 11 September 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G15: Tolerance and confidence regions

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


Chernozhukov, Victor; Chetverikov, Denis; Kato, Kengo. Anti-concentration and honest, adaptive confidence bands. Ann. Statist. 42 (2014), no. 5, 1787--1818. doi:10.1214/14-AOS1235.

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Supplemental materials

  • Supplementary material: Supplement to “Anti-concentration and honest, adaptive confidence bands”. This supplemental file contains additional proofs omitted in the main text, some results regarding nonwavelet projection kernel estimators, and a small-scale simulation study.