Testing for lack of fit in functional regression models

Abstract:

We consider regression models with a response variable taking values in a Hilbert space, of finite or infinite dimension, and hybrid covariates. That means two sets of regressors are allowed, one of finite dimension and a second one functional with values in a Hilbert space. The problem we address is the test on the effect of the functional covariates. This problem could occur in many functional data models. We propose new tests based on univariate kernel smoothing. The test statistics are asymptotically standard normal under the null hypothesis provided the smoothing parameter tends to zero at a suitable rate. The one-sided tests are consistent against any fixed alternative and detect local alternatives a la Pitman approaching the null hypothesis at suitable rate. In particular we show that neither the dimension of the outcome nor the dimension of the functional covariates influences the theoretical power of the tests against such local alternatives.