Locally Adaptive Confidence Bands

TL;DR

This paper introduces honest, locally adaptive confidence bands for probability densities that improve inference in cases of inhomogeneous smoothness, using novel concepts like local H"older regularity.

Contribution

It develops a new framework for local adaptivity in density estimation, including a statistical notion of local regularity and relaxed localization conditions.

Findings

Provides confidence bands that adapt to local smoothness.

Proves strong local adaptivity under relaxed conditions.

Identifies least favorable cases for honesty verification.

Abstract

We develop honest and locally adaptive confidence bands for probability densities. They provide substantially improved confidence statements in case of inhomogeneous smoothness, and are easily implemented and visualized. The article contributes conceptual work on locally adaptive inference as a straightforward modification of the global setting imposes severe obstacles for statistical purposes. Among others, we introduce a statistical notion of local H\"older regularity and prove a correspondingly strong version of local adaptivity. We substantially relax the straightforward localization of the self-similarity condition in order not to rule out prototypical densities. The set of densities permanently excluded from the consideration is shown to be pathological in a mathematically rigorous sense. On a technical level, the crucial component for the verification of honesty is the identification of an asymptotically least favorable stationary case by means of Slepian's comparison inequality.