{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T21:50:57Z","timestamp":1792792257464,"version":"4.5.0"},"reference-count":15,"publisher":"Institute of Mathematical Statistics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Inst. H. Poincar\u00e9 Probab. Statist."],"published-print":{"date-parts":[[2012,11,1]]},"abstract":"Nous pr\u00e9sentons de nouveaux M-estimateurs de la moyenne et de la variance d\u2019une variable al\u00e9atoire r\u00e9elle, fond\u00e9s sur des bornes PAC-Bay\u00e9siennes. Nous analysons les propri\u00e9t\u00e9s minimax non-asymptotiques des d\u00e9viations de ces estimateurs pour des distributions de l\u2019\u00e9chantillon soit de variance born\u00e9e, soit de variance et de kurtosis born\u00e9es. Sous ces hypoth\u00e8ses faibles, permettant des distributions \u00e0 queue lourde, nous montrons que les d\u00e9viations de la moyenne empirique sont dans le pire des cas sous-optimales. Nous prouvons en effet que pour tout niveau de confiance, il existe un M-estimateur dont les d\u00e9viations sont du m\u00eame ordre que les d\u00e9viations de la moyenne empirique d\u2019un \u00e9chantillon Gaussien, m\u00eame dans le cas o\u00f9 la v\u00e9ritable distribution de l\u2019\u00e9chantillon a une queue lourde. Le comportement exp\u00e9rimental de ces nouveaux estimateurs est du reste encore meilleur que ce que les bornes th\u00e9oriques laissent pr\u00e9voir, montrant que la fonction quantile des d\u00e9viations est constamment en dessous de celle de la moyenne empirique pour des \u00e9chantillons non Gaussiens aussi simples que des m\u00e9langes de deux distributions Gaussiennes.<\/jats:p>","DOI":"10.1214\/11-aihp454","type":"journal-article","created":{"date-parts":[[2012,11,16]],"date-time":"2012-11-16T15:40:55Z","timestamp":1353080455000},"page":"1148-1185","source":"Crossref","is-referenced-by-count":185,"title":["Challenging the empirical mean and empirical variance: A deviation study"],"prefix":"10.1214","volume":"48","author":[{"given":"Olivier","family":"Catoni","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"10","doi-asserted-by":"crossref","unstructured":"[10] P. J. Huber. Robust Statistics. Wiley Series in Probability and Mathematical Statistics<\/i>. Wiley-Interscience, New York, 1981.","DOI":"10.1002\/0471725250"},{"key":"12","doi-asserted-by":"crossref","unstructured":"[12] D. A. McAllester. PAC-Bayesian model averaging. In Proceedings of the 12th Annual Conference on Computational Learning Theory<\/i>. Morgan Kaufmann, New York, 1999.","DOI":"10.1145\/307400.307435"},{"key":"1","doi-asserted-by":"publisher","unstructured":"[1] P. Alquier. PAC-Bayesian bounds for randomized empirical risk minimizers. Math. Methods Statist.<\/i> 17<\/b> (2008) 279\u2013304.","DOI":"10.3103\/S1066530708040017"},{"key":"2","unstructured":"[2] J.-Y. Audibert. A better variance control for PAC-Bayesian classification. Preprint n.905bis, Laboratoire de Probabilit\u00e9s et Mod\u00e8les Al\u00e9atoires, Universit\u00e9s Paris 6 and Paris 7, 2004. Available at http:\/\/www.proba.jussieu.fr\/mathdoc\/textes\/PMA-905Bis.pdf<\/a>."},{"key":"3","doi-asserted-by":"publisher","unstructured":"[3] J.-Y. Audibert and O. Catoni. Robust linear least squares regression. Ann. Statist.<\/i> 39<\/b> (2011) 2766\u20132794.","DOI":"10.1214\/11-AOS918"},{"key":"4","unstructured":"[4] J.-Y. Audibert and O. Catoni. Robust linear regression through PAC-Bayesian truncation. Unpublished manuscript, 2010. Available at http:\/\/hal.inria.fr\/hal-00522536<\/a>."},{"key":"5","doi-asserted-by":"publisher","unstructured":"[5] R. Beran. An efficient and robust adaptive estimator of location. Ann. Statist.<\/i> 6<\/b> (1978) 292\u2013313.","DOI":"10.1214\/aos\/1176344125"},{"key":"6","doi-asserted-by":"publisher","unstructured":"[6] P. J. Bickel. On adaptive estimation. Ann. Statist.<\/i> 10<\/b> (1982) 647\u2013671.","DOI":"10.1214\/aos\/1176345863"},{"key":"7","unstructured":"[7] O. Catoni. Statistical Learning Theory and Stochastic Optimization: \u00c9cole d\u2019\u00c9t\u00e9 de Probabilit\u00e9s de Saint-Flour XXXI \u2013 2001. Lecture Notes in Math.<\/i> 1851<\/b>. Springer, Berlin, 2004."},{"key":"8","unstructured":"[8] O. Catoni. PAC-Bayesian Supervised Classification: The Thermodynamics of Statistical Learning. IMS Lecture Notes Monogr. Ser.<\/i> 56<\/b>. Institute of Mathematical Statistics, Beachwood, OH, 2007."},{"key":"9","doi-asserted-by":"publisher","unstructured":"[9] P. J. Huber. Robust estimation of a location parameter. Ann. Math. Statist.<\/i> 35<\/b> (1964) 73\u2013101.","DOI":"10.1214\/aoms\/1177703732"},{"key":"11","doi-asserted-by":"crossref","unstructured":"[11] O. Lepski. Asymptotically minimax adaptive estimation I: Upper bounds. Optimally adaptive estimates. Theory Probab. Appl.<\/i> 36<\/b> (1991) 682\u2013697.","DOI":"10.1137\/1136085"},{"key":"13","doi-asserted-by":"crossref","unstructured":"[13] D. A. McAllester. Some PAC-Bayesian theorems. Mach. Learn.<\/i> 37<\/b> (1999) 355\u2013363.","DOI":"10.1023\/A:1007618624809"},{"key":"14","doi-asserted-by":"crossref","unstructured":"[14] D. A. McAllester. PAC-Bayesian stochastic model selection. Mach. Learn.<\/i> 51<\/b> (2003) 5\u201321.","DOI":"10.1023\/A:1021840411064"},{"key":"15","doi-asserted-by":"publisher","unstructured":"[15] C. J. Stone. Adaptive maximum likelihood estimators of a location parameter. Ann. Statist.<\/i> 3<\/b> (1975) 267\u2013284.","DOI":"10.1214\/aos\/1176343056"}],"container-title":["Annales de l'Institut Henri Poincar\u00e9, Probabilit\u00e9s et Statistiques"],"original-title":[],"link":[{"URL":"http:\/\/projecteuclid.org\/download\/pdfview_1\/euclid.aihp\/1353098444","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T18:35:07Z","timestamp":1791570907000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/annales-de-linstitut-henri-poincare-probabilites-et-statistiques\/volume-48\/issue-4\/Challenging-the-empirical-mean-and-empirical-variance--A-deviation\/10.1214\/11-AIHP454.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11,1]]},"references-count":15,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2012,11,1]]}},"URL":"https:\/\/doi.org\/10.1214\/11-aihp454","relation":{},"ISSN":["0246-0203"],"issn-type":[{"value":"0246-0203","type":"print"}],"subject":[],"published":{"date-parts":[[2012,11,1]]}}}