1490 articles – 5641 Notices  [english version]
 HAL : hal-00708329, version 1
 arXiv : 1205.5578
 Projection-based nonparametric goodness-of-fit testing with functional covariates
 (24/05/2012)
 This paper studies the problem of nonparametric testing for the effect of a random functional covariate on a real-valued error term. The covariate takes values in $L^2[0,1]$, the Hilbert space of the square-integrable real-valued functions on the unit interval. The error term could be directly observed as a response or \emph{estimated} from a functional parametric model, like for instance the functional linear regression. Our test is based on the remark that checking the no-effect of the functional covariate is equivalent to checking the nullity of the conditional expectation of the error term given a sufficiently rich set of projections of the covariate. Such projections could be on elements of norm 1 from finite-dimension subspaces of $L^2[0,1]$. Next, the idea is to search a finite-dimension element of norm 1 that is, in some sense, the least favorable for the null hypothesis. Finally, it remains to perform a nonparametric check of the nullity of the conditional expectation of the error term given the scalar product between the covariate and the selected least favorable direction. For such finite-dimension search and nonparametric check we use a kernel-based approach. As a result, our test statistic is a quadratic form based on univariate kernel smoothing and the asymptotic critical values are given by the standard normal law. The test is able to detect nonparametric alternatives, including the polynomial ones. The error term could present heteroscedasticity of unknown form. We do no require the law of the covariate $X$ to be known. The test could be implemented quite easily and performs well in simulations and real data applications. We illustrate the performance of our test for checking the functional linear regression model.
 1 : Ecole Nationale de la Statistique et de l'Analyse de l'Information (ENSAI) Ensai, Ecole Nationale de la Statistique et de l'Analyse de l'Information 2 : Institut de Recherche Mathématique de Rennes (IRMAR) CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne 3 : Institut National des Sciences Appliquées de Rennes (INSA Rennes) Institut National des Sciences Appliquées (INSA) - Rennes
 Équipe de recherche : Statistique
 Domaine : Statistiques/Théorie
 Mots Clés : functional data regression – kernel smoothing – nonparametric testing
 Lien vers le texte intégral : http://fr.arXiv.org/abs/1205.5578
 hal-00708329, version 1 http://hal.archives-ouvertes.fr/hal-00708329 oai:hal.archives-ouvertes.fr:hal-00708329 Contributeur : Marie-Annick Guillemer <> Soumis le : Jeudi 14 Juin 2012, 17:07:36 Dernière modification le : Jeudi 14 Juin 2012, 17:07:36