Confidence bands in density estimation

Abstract

Given a sample from some unknown continuous density $f:\mathbb{R}\to \mathbb{R}$, we construct adaptive confidence bands that are honest for all densities in a "generic" subset of the union of $t$-Hölder balls, $0<t\le r$, 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.

Published in at http://dx.doi.org/10.1214/09-AOS738 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• An adaptation theory for nonparametric confidence intervals
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• Adaptive confidence balls
• Uniform limit theorems for wavelet density estimators
• Weighted uniform consistency of kernel density estimators
• Confidence sets for nonparametric wavelet regression
• A simple adaptive estimator of the integrated square of a density
• Confidence balls in Gaussian regression

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