Cube root weak convergence of empirical estimators of a density level set
Berthet,Philippe Einmahl,John
Berthet,Philippe
Einmahl,John
Abstract
Given n independent random vectors with common density f on Rd, we study the weak convergence of three empirical-measure based estimators of the convex λ-level set Lλ of f, namely the excess mass set, the minimum volume set and the maximum probability set, all selected from a class of convex sets A that contains Lλ. Since these set-valued estimators approach Lλ, even the formulation of their weak convergence is non-standard. We identify the joint limiting distribution of the symmetric difference of Lλ and each of the three estimators, at rate n−1/3. It turns out that the minimum volume set and the maximum probability set estimators are asymptotically indistinguishable, whereas the excess mass set estimator exhibits "richer" limit behavior. Arguments rely on the boundary local empirical process, its cylinder representation, dimension-free concentration around the boundary of Lλ, and the set-valued argmax of a drifted Wiener process.
Description
Funding Information: We thank the Editor, the Associate Editor, and two referees for many useful comments that greatly improved the paper. John Einmahl holds the Arie Kapteyn Chair 2019–2022 and gratefully acknowledges the corresponding research support. Publisher Copyright: © Institute of Mathematical Statistics, 2022
Date
2022-06
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Research Projects
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Keywords
Argmax drifted Wiener process, cube root asymptotics, density level set, excess mass, local empirical process, minimum volume set, set-valued estimator
Citation
Berthet, P & Einmahl, J 2022, 'Cube root weak convergence of empirical estimators of a density level set', Annals of Statistics, vol. 50, no. 3, pp. 1423-1446. https://doi.org/10.1214/21-AOS2157
License
info:eu-repo/semantics/closedAccess