Testing for lack-of-fit in functional regression models against general alternatives

Abstract

A lack-of-fit test for functional regression models is proposed. The test is based on the fact that checking the no-effect of a functional covariate is equivalent to checking the nullity of the conditional expectation of the error term given a sufficiently rich set of projections of that covariate. The idea then is to search the projection that is, in some sense, the least favorable for the null hypothesis. Finally, it remains to perform a nonparametric check of the nullity of the conditional expectation of the residuals of the regression given the selected least favorable projection. For the search of a least favorable projection and the nonparametric check we use a kernel-based approach. As a result, the test statistic is a quadratic form based on univariate kernel smoothing and the asymptotic critical values are given by the standard normal law. The test is able to detect general departures from the model. The error term of the regression could present heteroscedasticity of unknown form. The law of the functional covariate need not be known. The test could be implemented quite easily and performs well in simulations and real data applications.