June 2022 Cube root weak convergence of empirical estimators of a density level set
Philippe Berthet, John H. J. Einmahl
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Ann. Statist. 50(3): 1423-1446 (June 2022). DOI: 10.1214/21-AOS2157


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 nonstandard. We identify the joint limiting distribution of the symmetric difference of Lλ and each of the three estimators, at rate n1/3. 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.


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.


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Philippe Berthet. John H. J. Einmahl. "Cube root weak convergence of empirical estimators of a density level set." Ann. Statist. 50 (3) 1423 - 1446, June 2022. https://doi.org/10.1214/21-AOS2157


Received: 1 September 2021; Revised: 1 November 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441126
zbMATH: 07547936
Digital Object Identifier: 10.1214/21-AOS2157

Primary: 62G05 , 62G20
Secondary: 60F05 , 60F17

Keywords: Argmax drifted Wiener process , cube root asymptotics , Density level set , excess mass , local empirical process , minimum volume set , set-valued estimator

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • June 2022
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