June 2022 A data-adaptive method for estimating density level sets under shape conditions
Alberto Rodríguez-Casal, Paula Saavedra-Nieves
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Ann. Statist. 50(3): 1653-1668 (June 2022). DOI: 10.1214/21-AOS2168


Given a random sample of points from some unknown density, we propose a method for estimating density level sets, for a positive threshold t, under the r-convexity assumption. This shape condition generalizes the convexity property and allows to consider level sets with more than one connected component. The main problem in practice is that r is an unknown geometric characteristic of the set related to its curvature, which may depend on t. A stochastic algorithm is proposed for selecting its value from data. The resulting reconstruction of the level set is able to achieve minimax rates for Hausdorff metric and distance in measure uniformly on the level t.

Funding Statement

This work was supported by the Government of Galicia through the ERDF (Grupos de Referencia Competitiva) ED431C 2021/24, by the Spanish Ministry of Science and Innovation through projects PID2020-118101GBI00 and PID2020-116587GBI00 and by the Spanish Ministry of Economy and Competitiveness through projects MTM2016-76969-P and MTM2017-089422-P.


The authors are grateful to the referees and the Associate Editor for their useful comments. The authors also thank Antonio Cuevas for his help.


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Alberto Rodríguez-Casal. Paula Saavedra-Nieves. "A data-adaptive method for estimating density level sets under shape conditions." Ann. Statist. 50 (3) 1653 - 1668, June 2022. https://doi.org/10.1214/21-AOS2168


Received: 1 May 2021; Revised: 1 December 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441135
zbMATH: 07547945
Digital Object Identifier: 10.1214/21-AOS2168

Primary: 60K35 , 62G05
Secondary: 62G07

Keywords: Minimax level set estimation , r-convex hull , shape index

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • June 2022
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