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