Anisotropic adaptive kernel deconvolution

Comte, F.; Lacour, C.

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 569-609.

Dans ce travail, nous considérons un modèle de convolution multidimensionnel, pour lequel nous proposons des estimateurs à noyau anisotropes pour reconstruire la densité $f$ d’un signal mesuré avec un bruit additif. Pour ce faire, nous généralisons les estimateurs de Fan (Ann. Statist. 19(3) (1991) 1257-1272) à un contexte multidimensionnel et nous appliquons une méthode de sélection de fenêtre dans l'esprit des idées récentes développées par Goldenshluger et Lepski (Ann. Statist. 39(3) (2011) 1608-1632) pour l’estimation de densité en l’absence de bruit. Nous considérons tout d’abord le problème de l’estimation ponctuelle, et nous étudions ensuite le risque global intégré. Nos estimateurs dépendent d’une fenêtre aléatoire sélectionnée de façon automatique. Nous considérons les cas où les composantes du bruit, supposées connues, peuvent être ordinairement ou super régulières. De plus, nous étudions des classes de fonctions $f$ à estimer aussi bien dans des espaces de Hölder anisotropes que dans des espaces de Sobolev. Nous prouvons des bornes de risque non asymptotiques ainsi que des vitesses de convergence asymptotiques pour nos estimateurs adaptatifs, en même temps que des bornes inférieures dans un grand nombre de cas. Des simulations illustrent la méthode en s’appuyant sur des algorithmes de transformation de Fourier rapide. En conclusion, nous proposons une extension de la méthode lorsque la loi du bruit n’est plus connue, mais remplacée par un échantillon préliminaire où le bruit seul est observé.

In this paper, we consider a multidimensional convolution model for which we provide adaptive anisotropic kernel estimators of a signal density $f$ measured with additive error. For this, we generalize Fan’s (Ann. Statist. 19(3) (1991) 1257-1272) estimators to multidimensional setting and use a bandwidth selection device in the spirit of Goldenshluger and Lepski's (Ann. Statist. 39(3) (2011) 1608-1632) proposal for density estimation without noise. We consider first the pointwise setting and then, we study the integrated risk. Our estimators depend on an automatically selected random bandwidth. We assume both ordinary and super smooth components for measurement errors, which have known density. We also consider both anisotropic Hölder and Sobolev classes for $f$. We provide nonasymptotic risk bounds and asymptotic rates for the resulting data driven estimator, together with lower bounds in most cases. We provide an illustrative simulation study, involving the use of Fast Fourier Transform algorithms. We conclude by a proposal of extension of the method to the case of unknown noise density, when a preliminary pure noise sample is available.

Classification : 62G07
Mots clés : adaptive kernel estimator, anisotropic estimation, deconvolution, density estimation, measurement errors, multidimensional

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