Abstract
This work focuses on recent advances on the way general structural testing procedures can be constructed in regression on functional variable. Our test statistic is constructed from an estimator adapted to the specific model to be checked and uses recent advances concerning kernel smoothing methods for functional data. A general theoretical result states the asymptotic normality of our test statistic under the null hypothesis and its divergence under local alternatives. This result opens interesting prospects about tests for no-effect, for linearity, or for reduction dimension of the covariate. Bootstrap methods are then proposed to compute the threshold value of our test. Finally, we present some applications to spectrometric datasets and discuss interesting prospects for the future.
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References
Ait-Sa¨ıdi, A., Ferraty, F., Kassa, R., Vieu, P.: Cross-validated estimations in the single functional index model. Statistics 42, 475–494 (2008)
Aneiros-Perez, G., Vieu, P.: Time series prediction: a semi-functional partial linear model. J. Multivariate Anal. 99, 834–857 (2008)
Borggaard, C., Thodberg, H.H.: Optimal minimal neural interpretation of spectra. Anal. Chem., 64, (5), 545–551 (1992)
Bosq, D.: Linear Processes in Function Spaces : Theory and Applications. Lecture Notes in Statistics, 149, Springer Verlag, New York (2000)
Cardot, H., Ferraty, F.,Mas, A., Sarda, P.: Testing Hypotheses in the Functional LinearModel. Scand. J. Stat. 30, 241–255 (2003)
Cardot, H., Ferraty, F., Sarda, P.: Functional Linear Model. Statist. Prob. Lett. 45, 11–22 (1999)
Cardot, H., Ferraty, F., Sarda, P.: Etude asymptotique d’un estimateur spline hybride pour le mod`ele lin´eaire fonctionnel. (French) [Asymptotic study of a hybrid spline estimator for the functional linear model] C. R. Acad. Sci. Ser. I 330 (6), 501–504 (2000)
Cardot, H., Goia, A., Sarda, P.: Testing for no effect in functional linear regression models, some computational approaches. Commun. Stat. Simulat. C. 33 (1), 179–199 (2004)
Cardot, H., Crambes, C., Kneip, A., Sarda, P.: Smoothing splines estimators in functional linear regression with errors-in-variables. Comput. Stat. Data An. 51 (10), 4832–4848 (2007)
Chiou, J.M.,M¨uller H.-G.: Diagnostics for functional regression via residual processes. Comput. Stat. Data An. 51 (10), 4849–4863 (2007)
Crambes, C., Kneip, A., Sarda, P.: Smoothing splines estimators for functional linear regression. Ann. Stat. 37, 35–72 (2009)
Delsol, L. (2007) R´egression non-param´etrique fonctionnelle : Expressions asymptotiques des moments. Annales de l’I.S.U.P., LI, (3), 43–67.
Delsol, L. (2008) R´egression sur variable fonctionnelle: Estimation, Tests de structure et Applications. Th`ese de doctorat de l’Universit´e de Toulouse.
Delsol, L.: Advances on asymptotic normality in nonparametric functional Time Series Analysis. Statistics 43 (1), 13–33 (2009)
Delsol, L., Ferraty, F., Vieu, P.: Structural test in regression on functional variables. J. Multivariate Anal. 102 (3), 422–447 (2011)
Ferraty F., Goia A., Vieu P.: Functional nonparametric model for time series : a fractal approach for dimension reduction. Test 11 (2), 317–344 (2002b)
Ferraty, F., Romain, Y.: The Oxford Handbook on Functional Data Analysis. Oxford University Press (2011)
Ferraty, F., Vieu, P.: Dimension fractale et estimation de la r´egression dans des espaces vectoriels semi-norm´es. C. R. Acad. Sci. Ser. I 330, 403–406 (2000)
Ferraty, F., Vieu, P.: Nonparametric Functional Data Analysis: Theory and Practice. Springer, New York (2006)
Gadiaga, D., Ignaccolo, R.: Test of no-effect hypothesis by nonparametric regression. Afr. Stat. 1 (1), 67–76 (2005)
Hall, P., Cai, T.T.: Prediction in functional linear regression. Ann. Stat. 34 (5), 2159–2179 (2006)
H¨ardle, W., Mammen, E.: Comparing Nonparametric Versus Parametric Regression Fits. Ann. Stat. 21 (4), 1926–1947 (1993)
Masry, E.: Nonparametric regression estimation for dependent functional data : asymptotic normality. Stoch. Process. Appl. 115 (1), 155–177 (2005)
M¨uller, H.-G., Stadtm¨uller, U.: Generalized functional linear models. Ann. Stat. 33 (2), 774–805 (2005)
Mammen, E.: Bootstrap and wild bootstrap for high-dimensional linear models. Ann. Stat. 21 (1), 255–285 (1993)
Preda, C., Saporta, G.: PLS regression on a stochastic process. Comput. Stat. Data An. 48 (1), 149–158 (2005)
Ramsay, J., Dalzell, C.: Some tools for functional data analysis. J. Roy. Stat. Soc. B 53, 539– 572 (1991)
Ramsay, J., Silverman, B.: Functional Data Analysis. Springer-Verlag, New York (1997) 29. Ramsay, J., Silverman, B.: Applied functional data analysis : Methods and case studies. Spinger Verlag, New York (2002)
Ramsay, J., Silverman, B.: Functional Data Analysis (Second Edition). Springer Verlag, New
York (2005)
Sood, A., James, G., Tellis, G.: Functional Regression: A New Model for Predicting Market Penetration of New Products. Marketing Science 28, 36–51 (2009)
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Delsol, L., Ferraty, F., Vieu, P. (2011). Structural Tests in Regression on Functional Variable. In: Ferraty, F. (eds) Recent Advances in Functional Data Analysis and Related Topics. Contributions to Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2736-1_12
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DOI: https://doi.org/10.1007/978-3-7908-2736-1_12
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