Title: A goodness-of-fit test for the functional linear model with functional response
Authors: Javier Alvarez-Liebana - University of Oviedo (Spain) [presenting]
Wenceslao Gonzalez-Manteiga - University of Santiago de Compostela (Spain)
Eduardo Garcia-Portugues - Carlos III University of Madrid (Spain)
Gonzalo Alvarez-Perez - University of Oviedo (Spain)
Abstract: Functional data analysis enables to exploit the complexity and richness of data measured over continuous domains. When two functional random variables are available, it may be useful to determine their relation using a regression model. If the regression function is a linear Hilbert-Schmidt operator between two $L_2$ spaces, we are under the functional linear model with a functional response. We propose a novel goodness-of-fit test for the null (composite) hypothesis of this model, against a general, unspecified alternative. The test statistic is formulated in terms of the quadratic norm over a doubly-projected empirical process. It is easy to compute, interpret and calibrate on its distribution via a wild bootstrap on the residuals. A flexible hybrid approach, involving LASSO regularization and linearly-constrained least-squares, is used to perform the selection of functional predictors when estimating the residuals. The finite sample behavior of the test, regarding power and size, is illustrated via a complete simulation study under varying scenarios. The test is applied to some real datasets to check the validity of the model.