A faster U-statistic for testing independence in the functional linear models

Testing the dependence between the response and the functional predictor in a functional linear model is of fundamental importance. In this paper, based on a U-statistic of order two, we develop a computationally more efficient test for lacking of dependence in functional linear regression model. By the martingale central limit theorem, we prove that the asymptotic normality of the proposed test statistic under some mild regularity conditions. Simulation results show that our proposed test can be tens or hundreds time faster than the FLUTE test by Hu et al. (2020) which uses a U-statistic of order four. We further demonstrate the superiority of our test by two real data applications.