The Annals of Statistics

On adaptive inference and confidence bands

Marc Hoffmann and Richard Nickl

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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.

Article information

Ann. Statist., Volume 39, Number 5 (2011), 2383-2409.

First available in Project Euclid: 30 November 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G15: Tolerance and confidence regions
Secondary: 62G10: Hypothesis testing 62G05: Estimation

Adaptive confidence sets nonparametric hypothesis testing


Hoffmann, Marc; Nickl, Richard. On adaptive inference and confidence bands. Ann. Statist. 39 (2011), no. 5, 2383--2409. doi:10.1214/11-AOS903.

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