The Annals of Statistics

Confidence bands in density estimation

Evarist Giné and Richard Nickl

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Abstract

Given a sample from some unknown continuous density f : ℝ→ℝ, we construct adaptive confidence bands that are honest for all densities in a “generic” subset of the union of t-Hölder balls, 0<tr, where r is a fixed but arbitrary integer. The exceptional (“nongeneric”) set of densities for which our results do not hold is shown to be nowhere dense in the relevant Hölder-norm topologies. In the course of the proofs we also obtain limit theorems for maxima of linear wavelet and kernel density estimators, which are of independent interest.

Article information

Source
Ann. Statist., Volume 38, Number 2 (2010), 1122-1170.

Dates
First available in Project Euclid: 19 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1266586625

Digital Object Identifier
doi:10.1214/09-AOS738

Mathematical Reviews number (MathSciNet)
MR2604707

Zentralblatt MATH identifier
1183.62062

Subjects
Primary: 62G07: Density estimation
Secondary: 60F05: Central limit and other weak theorems

Keywords
Adaptive estimation limit theorem density estimation extremes Gaussian processes wavelet estimators kernel estimators

Citation

Giné, Evarist; Nickl, Richard. Confidence bands in density estimation. Ann. Statist. 38 (2010), no. 2, 1122--1170. doi:10.1214/09-AOS738. https://projecteuclid.org/euclid.aos/1266586625


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