A Goodness-of-Fit Test for the Functional Linear Model with Scalar Response

This article proposes a goodness-of-fit test for the null hypothesis of a functional linear model with scalar response. The test is based on a generalization to the functional framework of a previous one, designed for the goodness-of-fit of regression models with multivariate covariates using random projections. The test statistic is easy to compute using geometrical and matrix arguments, and simple to calibrate in its distribution by a wild bootstrap on the residuals. The finite sample properties of the test are illustrated by a simulation study for several types of basis and under different alternatives. Finally, the test is applied to two datasets for checking the assumption of the functional linear model and a graphical tool is introduced. Supplementary materials are available online.

[1]  Manuel Febrero-Bande,et al.  Statistical Computing in Functional Data Analysis: The R Package fda.usc , 2012 .

[2]  J. O. Ramsay,et al.  Functional Data Analysis (Springer Series in Statistics) , 1997 .

[3]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[4]  W. González-Manteiga,et al.  A General View of the Goodness-of-Fit Tests for Statistical Models , 2011 .

[5]  Frédéric Ferraty,et al.  Nonparametric Functional Data Analysis: Theory and Practice (Springer Series in Statistics) , 2006 .

[6]  Juan Antonio Cuesta-Albertos,et al.  The random projection method in goodness of fit for functional data , 2007, Comput. Stat. Data Anal..

[7]  Joel L. Horowitz,et al.  Methodology and convergence rates for functional linear regression , 2007, 0708.0466.

[8]  J. Zheng,et al.  A consistent test of functional form via nonparametric estimation techniques , 1996 .

[9]  E. Mammen,et al.  Comparing Nonparametric Versus Parametric Regression Fits , 1993 .

[10]  P. Sarda,et al.  Functional linear model , 1999 .

[11]  Philippe Vieu,et al.  Structural test in regression on functional variables , 2011, J. Multivar. Anal..

[12]  F. Ferraty,et al.  The Oxford Handbook of Functional Data Analysis , 2011, Oxford Handbooks Online.

[13]  Jeng-Min Chiou,et al.  Diagnostics for functional regression via residual processes , 2007, Comput. Stat. Data Anal..

[14]  Winfried Stute,et al.  Bootstrap Approximations in Model Checks for Regression , 1998 .

[15]  C. Sánchez-Sellero,et al.  Projection-based nonparametric goodness-of-fit testing with functional covariates , 2012, 1205.5578.

[16]  P. Hall,et al.  On properties of functional principal components analysis , 2006 .

[17]  J. Escanciano.  A CONSISTENT DIAGNOSTIC TEST FOR REGRESSION MODELS USING PROJECTIONS , 2006, Econometric Theory.

[18]  Gilbert Saporta,et al.  Régression PLS sur un processus stochastique , 2001 .

[19]  André Mas,et al.  Testing hypotheses in the functional linear model , 2003 .

[20]  Eduardo Garc'ia-Portugu'es,et al.  Bootstrap independence test for functional linear models , 2012, 1210.1072.

[21]  Wenceslao González-Manteiga,et al.  Measures of influence for the functional linear model with scalar response , 2010, J. Multivar. Anal..

[22]  Valentin Patilea,et al.  Breaking the curse of dimensionality in nonparametric testing , 2008 .

[23]  R. Fraiman,et al.  Trimmed means for functional data , 2001 .