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Comptes Rendus Mathématique
Volume 341, n° 6
pages 365-368 (septembre 2005)
Doi : 10.1016/j.crma.2005.06.027
Received : 7 October 2004 ;  accepted : 20 June 2005
Sélection automatique du paramètre de lissage pour lʼestimation non paramétrique de la régression pour des données fonctionnelles
Automatic smoothing parameter selection for the nonparametric regression estimation of functional data

Mustapha Rachdi a , Philippe Vieu b
a Université Pierre Mendès France, UFR SHS, BP. 47, 38040 Grenoble cedex 09, France 
b Université Paul Sabatier, LSP UMR CNRS 5583, 118, route de Narbonne, 31062 Toulouse cedex, France 


Dans cette Note, nous étudions lʼestimation de la régression quand le régresseur est de type fonctionnel. Lʼestimateur de la régression pour ce type de données a été récemment introduit. Il dépend dʼun paramètre de lissage qui contrôle sa vitesse de convergence, et le but de notre travail est de construire un critère de choix automatique de ce paramètre. Le critère est formulé sous la forme dʼune validation croisée fonctionnelle. Sous certaines hypothèses sur lʼopérateur de regression (inconnu), nous montrons que cette procédure est optimale. En plus, nous établissons lʼéquivalence asymptotique entre plusieurs mesures de risque pour lʼestimation non paramétrique de lʼopérateur de régression. Pour citer cet article : M. Rachdi, P. Vieu, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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We study regression estimation when the explanatory variable is functional. Nonparametric estimates of the regression operator have been recently introduced. They depend on a smoothing factor which controls its behaviour, and the aim of our Note is to construct some data-driven criterion for choosing this smoothing parameter. The criterion can be formulated in terms of a functional version of cross-validation ideas. Under mild assumptions on the unknown regression operator, it is seen that this rule is asymptotically optimal. As by-products of this result, we state asymptotic equivalences for several measures of accuracy for nonparametric estimate of the regression operator. To cite this article: M. Rachdi, P. Vieu, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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

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