Confidence bands in density estimation

TL;DR

This paper develops adaptive confidence bands for unknown densities that are honest for a broad class of functions, with theoretical guarantees and new limit theorems for estimators.

Contribution

It introduces honest adaptive confidence bands for densities in a generic subset of Hölder classes, with proven properties and new limit theorems for estimators.

Findings

Confidence bands are honest for a generic set of densities.

Limit theorems for maxima of wavelet and kernel estimators.

Exceptional densities form a nowhere dense set.

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 $t$-H\"older 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 H\"older-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.