Abstract: Abstract: We propose a novel diagnostic test to check the goodness-of-fit for generalized functional regression models. The proposed test does not require a specification of the distribution, and can be applied to commonly employed functional regression models. Because it is based on independence in distribution, it includes mean-based and higher-order moment-based tests as special cases. In particular, we overcome the problem of the infinite dimensionality of the functional data by projecting functions along certain directions. Moreover, to avoid bias caused by the subjective selection of these directions, we integrate over the directions along which the functional variables project. As a result, the proposed test simultaneously enhances the local power and overcomes the infinite-dimensionality problem. A simple implementation procedure is developed. The performance of the proposed test is evaluated theoretically and using simulation studies. We apply the proposed procedure to analyze Canadian weather data and Chinese air pollution data, resulting in several interesting models that achieve higher interpretability and estimation accuracy than those of existing methods.

Key words and phrases: Distribution free, generalized functional regression, goodness-of-fit, local power, projection-based distribution test.