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Comptes Rendus Mathématique
Volume 342, n° 11
pages 873-876 (juin 2006)
Doi : 10.1016/j.crma.2006.03.012
Received : 15 April 2005 ;  accepted : 6 Mars 2006
Comportement asymptotique dʼestimateurs de la densité par projection tronqués
Asymptotic behavior of truncated projection density estimators
 

Jean-Baptiste Aubin
Université Paris 6, LSTA, boîte 158, 175, rue du Chevaleret, 75013 Paris, France 

Résumé

Nous étudions deux versions tronquées de lʼestimateur de la densité (par rapport à une mesure -finie μ ) par projection. Ces versions se basent sur des indices de troncature dépendants des données et elles seront étudiées dans divers contextes. Nous décrivons dʼabord le comportement asymptotique des indices de troncature. Nous montrons alors que les estimateurs correspondants atteignent une vitesse suroptimale au sens de lʼerreur quadratique intégrée sur un sous-ensemble   dense de  . De plus, nous déterminons les cas dans lesquels les estimateurs atteignent une vitesse quasi-optimale pour cette même erreur sur le complémentaire de  . Pour citer cet article : J.-B. Aubin, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

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

We present different versions of the truncated projection estimator of a density with respect to a -finite measure μ , where the traditional truncation index   is replaced by   (or  ) under various conditions. First, we describe the asymptotic behaviour of   (or  ). Next, we show that these estimators reach a superoptimal rate for the mean square error on a dense subset   of  . We finally state under which conditions these estimators reach quasioptimal rate of convergence when the unknown density f belongs to  . To cite this article: J.-B. Aubin, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

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


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