## The Annals of Statistics

#### 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.

#### Article information

Source
Ann. Statist., Volume 47, Number 1 (2019), 349-381.

Dates
Revised: January 2018
First available in Project Euclid: 30 November 2018

https://projecteuclid.org/euclid.aos/1543568591

Digital Object Identifier
doi:10.1214/18-AOS1690

Mathematical Reviews number (MathSciNet)
MR3909936

Zentralblatt MATH identifier
07036204

Subjects
Primary: 62G07: Density estimation 62G15: Tolerance and confidence regions

#### Citation

Patschkowski, Tim; Rohde, Angelika. Locally adaptive confidence bands. Ann. Statist. 47 (2019), no. 1, 349--381. doi:10.1214/18-AOS1690. https://projecteuclid.org/euclid.aos/1543568591

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#### Supplemental materials

• Supplement to “Locally adaptive confidence bands”. Supplement A is organized as follows. Section A.1 develops connections between the Weierstraß function and the Admissibility Condition 3.5. Further notation and auxiliary results from empirical process theory are provided in Section A.2, whereas Section A.3 provides a simulation study together with an algorithm for the calculation of the locally adaptive confidence band. Section A.4 presents the remaining proofs of the results of Section 3. We proceed with the proofs of the results of Section 4 in Section A.5. Auxiliary results are stated and proved in Section A.6.