Testing independence and goodness-of-fit jointly for functional linear models

Abstract

A conventional regression model for functional data involves expressing a response variable in terms of the predictor function. Two assumptions, that (i) the predictor function and the error are independent and (ii) the relationship between the response variable and the predictor function takes functional linear model, are usually added to the model. Checking the validation of these two assumptions is fundamental to statistic inference and practical applications. We develop a test procedure to check these assumptions simultaneously based on generalized distance covariance. We establish the asymptotic theory for the proposed test under null and alternative hypotheses, and provide a bootstrap procedure to obtain the critical value of the test. The proposed test is consistent against all alternatives provided that the semimetrics related to the generalized distance are strong negative, and can be readily generalized to other functional regression models. We explore the finite sample performance of the proposed test by using both simulations and real data examples. The results illustrate that the proposed method has favorable performance compared with the competing method.