A data-adaptive method for estimating density level sets under shape conditions
Alberto Rodríguez-Casal, Paula Saavedra-Nieves
The Annals of Statistics · 2022 · Volume 50 · Issue 3
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.
Keywords
60K35, 62G05, 62G07, Minimax level set estimation, r-convex hull, shape index
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