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Comptes Rendus Mathématique
Volume 340, n° 1
pages 59-62 (janvier 2005)
Doi : 10.1016/j.crma.2004.11.009
Received : 24 May 2004 ;  accepted : 3 November 2004
Inégalités dʼoracle pour lʼestimation dʼune densité de probabilité
Oracle inequalities for probability density estimations
 

Philippe Rigollet
Laboratoire de probabilités et modèles aléatoires, UMR CNRS 7599, université Paris 6, 4, place Jussieu, case 188, 75252 Paris cedex 05, France 

Résumé

Nous étudions le problème de lʼestimation dʼune densité de probabilité dans  . A partir dʼune formulation du risque quadratique intégré dans le domaine des fréquences de Fourier, nous montrons quʼil est proche du risque   dans le modèle de suite gaussienne. En appliquant alors une version modifiée de la méthode Stein par blocs, nous obtenons une inégalité dʼoracle sur les estimateurs linéaires monotones et une inégalité dʼoracle sur les estimateurs à noyau. Comme conséquence, lʼestimateur proposé est adaptatif au sens minimax exact (i.e. à la constante près) sur la famille de classes de Sobolev. Pour citer cet article : Ph. Rigollet, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

The full text of this article is available in PDF format.
Abstract

We study the problem of the nonparametric estimation of a probability density in  . Expressing the mean integrated squared error in the Fourier domain, we show that it is close to the mean squared error in the Gaussian sequence model. Then, applying a modified version of Steinʼs blockwise method, we obtain a linear monotone oracle inequality and a kernel oracle inequality. As a consequence, the proposed estimator is sharp minimax adaptive (i.e. up to a constant) on a scale of Sobolev classes of densities. To cite this article: Ph. Rigollet, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

The full text of this article is available in PDF format.


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