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February 2019 Locally adaptive confidence bands
Tim Patschkowski, Angelika Rohde
Ann. Statist. 47(1): 349-381 (February 2019). DOI: 10.1214/18-AOS1690

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 Hölder 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.

Citation

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Tim Patschkowski. Angelika Rohde. "Locally adaptive confidence bands." Ann. Statist. 47 (1) 349 - 381, February 2019. https://doi.org/10.1214/18-AOS1690

Information

Received: 1 August 2017; Revised: 1 January 2018; Published: February 2019
First available in Project Euclid: 30 November 2018

zbMATH: 07036204
MathSciNet: MR3909936
Digital Object Identifier: 10.1214/18-AOS1690

Subjects:
Primary: 62G07, 62G15

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 1 • February 2019
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