A DATA-ADAPTIVE METHOD FOR ESTIMATING DENSITY LEVEL SETS UNDER SHAPE CONDITIONS

• 成果类型: Article
• 署名作者: Rodriguez-Casal, Alberto; Saavedra-Nieves, Paula
• 署名单位: Universidade de Santiago de Compostela; Universidade de Santiago de Compostela
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2168
• 发表日期: 2022
• 页码: 1653-1668
• 关键词: Bandwidth selection; rates; CONVERGENCE; asymptotics; regions; contour

摘要

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.