{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T19:58:46Z","timestamp":1791403126718,"version":"4.3.3"},"reference-count":35,"publisher":"Institute of Mathematical Statistics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2022,6,1]]},"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 \u03bb-level set L\u03bb 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\u03bb. Since these set-valued estimators approach L\u03bb, even the formulation of their weak convergence is nonstandard. We identify the joint limiting distribution of the symmetric difference of L\u03bb and each of the three estimators, at rate n\u22121\/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 \u201cricher\u201d limit behavior. Arguments rely on the boundary local empirical process, its cylinder representation, dimension-free concentration around the boundary of L\u03bb, and the set-valued argmax of a drifted Wiener process.<\/jats:p>","DOI":"10.1214\/21-aos2157","type":"journal-article","created":{"date-parts":[[2022,6,17]],"date-time":"2022-06-17T01:51:24Z","timestamp":1655430684000},"page":"1423-1446","source":"Crossref","is-referenced-by-count":1,"title":["Cube root weak convergence of empirical estimators of a density level set"],"prefix":"10.1214","volume":"50","author":[{"given":"Philippe","family":"Berthet","sequence":"first","affiliation":[{"name":"Institut de Math\u00e9matiques de Toulouse; UMR5219, Universit\u00e9 de Toulouse; CNRS"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"John H. J.","family":"Einmahl","sequence":"additional","affiliation":[{"name":"Department of Econometrics and OR and CentER, Tilburg University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"29","doi-asserted-by":"publisher","unstructured":"Schneider, R.<\/small> and Weil, W.<\/small> (2008). Stochastic and Integral Geometry<\/i>. Probability and Its Applications<\/i> (New York<\/i>). Springer, Berlin.","DOI":"10.1007\/978-3-540-78859-1"},{"key":"22","doi-asserted-by":"publisher","unstructured":"Polonik, W.<\/small> (1995). Measuring mass concentrations and estimating density contour clusters\u2014an excess mass approach. Ann. Statist.<\/i> 23<\/b> 855\u2013881.","DOI":"10.1214\/aos\/1176324626"},{"key":"23","doi-asserted-by":"publisher","unstructured":"Polonik, W.<\/small> (1997). Minimum volume sets and generalized quantile processes. Stochastic Process. 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