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Régression sur variable fonctionnelle: Estimation, tests de structure et Applications.

Abstract : Functional data analysis is a typical issue in modern statistics. During the last years, many papers have been devoted to theoretical results or applied studies on models involving functional data. In this manuscript, we focus on regression models where a real response variable depends on a functional random variable taking its values in an infinite dimensional space. We state various kinds of results linked with the asymptotic properties of the nonparametric kernel estimator generalized to the case of a functional explanatory variable. We firstly assume that the dataset under study is composed of strongmixing variables and focus on a nonparametric regression model. A first result gives asymptotic normality of the kernel estimator with explicit expressions of the asymptotic dominant bias and variance terms. On one hand, we propose from this result a way to construct asymptotic pointwise confidence bands and study their properties on simulation studies and apply our method on a dataset dealing with El Niño phenomenon. On the other hand, we decline from both asymptotical normality and uniform integrability results the explicit expressions of the asymptotic dominant terms of centered moments and Lp errors of the kernel estimator. We now assume that the dataset is composed of independent random variables and focus on structural testing procedures in regression on functional data. We construct a test statistic based on the comparison between the nonparametric kernel estimator and a particular one that must be chosen in accordance to the null hypothesis we want to test. We then state both asymptotic normality of our test statistic under the null hypothesis and its divergence under the alternative. Moreover, we prove our result under general conditions that enable to use our approach to construct innovative structural tests allowing to check if the regression operator is linear, single index, . . . Different bootstrap procedures are proposed and compared through various simulation studies. Finally, we focus on no effect tests and apply our testing procedures on two spectrometric datasets.
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https://tel.archives-ouvertes.fr/tel-00449806
Contributor : Laurent Delsol <>
Submitted on : Friday, January 22, 2010 - 4:34:20 PM
Last modification on : Thursday, March 5, 2020 - 6:49:20 PM
Long-term archiving on: : Thursday, June 17, 2010 - 11:13:31 PM

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  • HAL Id : tel-00449806, version 1

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Laurent Delsol. Régression sur variable fonctionnelle: Estimation, tests de structure et Applications.. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2008. Français. ⟨tel-00449806⟩

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