Additivity test on the nonlinear part in partially linear models

Khalid Chokri; Djamal Louani

doi:10.1016/j.crma.2013.01.016

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Additivity test on the nonlinear part in partially linear models
[Test dʼadditivité de la partie non linéaire dans les modèles partiellement linéaires]

Présenté par : Paul Deheuvels

Khalid Chokri ¹ ; Djamal Louani ^1,²

¹ LSTA, université Paris-6, 175, rue du Chevaleret, 75013 Paris, France
² Université de Reims–Champagne–Ardennes, BP 1039, 51687 Reims cedex 2, France

Comptes Rendus. Mathématique, Volume 351 (2013) no. 3-4, pp. 143-148.

Résumé
Abstract

Cette Note a pour objet de construire un test dʼadditivité de la partie non linéaire du modèle partiellement linéaire. Plus précisément, en considérant un modèle de la forme $Y = Z^{⊤} β + m (X) + ε$ , où $m (X)$ correspond à la partie non linéaire et X est un vecteur aléatoire prenant ses valeurs dans $R^{d}$ , il sʼagit alors de construire une procédure de test permettant de vérifier si lʼhypothèse selon laquelle la fonction m peut sʼécrire sous la forme $m (x) = \sum_{l = 1}^{d} m_{l} (x_{l})$ , où les $m_{i}$ sont des fonctions définie sur $R$ , est vraisemblable.

Our aim in this Note is to build an additivity test relative to the nonlinear part of the partially linear model. More precisely, considering the model of the form $Y = Z^{⊤} β + m (X) + ε$ , where $m (X)$ stands as the nonlinear part and X is a random vector taking values in the space $R^{d}$ , our goal is to construct a testing procedure that allows us to test the validity of the hypothesis according to which the function m may be written with the shape $m (x) = \sum_{l = 1}^{d} m_{l} (x_{l})$ , where the $m_{l}$ ʼs are functions defined on $R$ .

Reçu le : 2011-12-20
Accepté le : 2013-01-31
Publié le : 2013-02-01

DOI : 10.1016/j.crma.2013.01.016

Affiliations des auteurs :

Khalid Chokri ¹ ; Djamal Louani ^{1, 2}

¹ LSTA, université Paris-6, 175, rue du Chevaleret, 75013 Paris, France
² Université de Reims–Champagne–Ardennes, BP 1039, 51687 Reims cedex 2, France

@article{CRMATH_2013__351_3-4_143_0,
     author = {Khalid Chokri and Djamal Louani},
     title = {Additivity test on the nonlinear part in partially linear models},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {143--148},
     publisher = {Elsevier},
     volume = {351},
     number = {3-4},
     year = {2013},
     doi = {10.1016/j.crma.2013.01.016},
     language = {en},
}

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%0 Journal Article
%A Khalid Chokri
%A Djamal Louani
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Khalid Chokri; Djamal Louani. Additivity test on the nonlinear part in partially linear models. Comptes Rendus. Mathématique, Volume 351 (2013) no. 3-4, pp. 143-148. doi : 10.1016/j.crma.2013.01.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.01.016/

Bibliographie
Cité par

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[2] C. Camlong-Viot Vers un test dʼadditivité en régression non paramétrique sous des conditions de mélange, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2001), pp. 1105-1109

[3] K. Chokri; D. Louani Asymptotic results for the linear parameter estimate in partially linear additive regression model, C. R. Acad. Sci. Paris, Ser. I, Volume 333 (2011), pp. 877-880

[4] L. Delsol; F. Ferraty; P. Vieu Structural tests in regression on functional variable, Recent Advances in Functional Data Analysis and Related Topics, Physica-Verlag/Springer, Heidelberg, Germany, 2011, pp. 77-83

[5] S. Derbort; H. Dette; A. Munk A test for additivity in nonparametric regression, Ann. Inst. Stat. Math., Volume 54 (2002), pp. 60-82

[6] H. Dette; C. von Lieres und Wilkau Testing additivity by kernel-based methods: what is a reasonable test?, Bernoulli, Volume 7 (2001), pp. 669-697

[7] F. Ferraty; P. Vieu Additive prediction and boosting for functional data, Comput. Stat. Data Anal., Volume 53 (2009), pp. 1400-1413

[8] W. González-Manteiga; G. Aneiros-Pérez Testing in partial linear regression models with dependent errors, J. Nonparametr. Stat., Volume 15 (2003), pp. 93-111

[9] P. Gozalo; O. Linton Testing additivity in generalized nonparametric regression models with estimated parameters, J. Econom., Volume 104 (2001), pp. 1-48

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