CUBE ROOT WEAK CONVERGENCE OF EMPIRICAL ESTIMATORS OF A DENSITY LEVEL SET

• 成果类型: Article
• 署名作者: Berthet, Philippe; Einmahl, John H. J.
• 署名单位: Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite Federale Toulouse Midi-Pyrenees (ComUE); Institut National des Sciences Appliquees de Toulouse; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Centre National de la Recherche Scientifique (CNRS); Tilburg University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2157
• 发表日期: 2022
• 页码: 1423-1446
• 关键词: CONFIDENCE-REGIONS; brownian-motion; asymptotics; BOUNDARIES; contour

摘要

Given n independent random vectors with common density f on R-d, we study the weak convergence of three empirical-measure based estimators of the convex lambda-level set L-lambda 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 L-lambda. Since these set-valued estimators approach L-lambda, even the formulation of their weak convergence is nonstandard. We identify the joint limiting distribution of the symmetric difference of L-lambda 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 L-lambda, and the set-valued argmax of a drifted Wiener process.