Nonparametric testing of lack of dependence in functional linear models

An important inferential task in functional linear models is to test the dependence between the response and the functional predictor. The traditional testing theory was constructed based on the functional principle component analysis which requires estimating the covariance operator of the functional predictor. Due to the intrinsic high-dimensionality of functional data, the sample is often not large enough to allow accurate estimation of the covariance operator and hence causes the follow-up test underpowered. To avoid the expensive estimation of the covariance operator, we propose a nonparametric method called Functional Linear models with U-statistics TEsting (FLUTE) to test the dependence assumption. We show that the FLUTE test is more powerful than the current benchmark method (Kokoszka P,2008; Patilea V,2016) in the small or moderate sample case. We further prove the asymptotic normality of our test statistic under both the null hypothesis and a local alternative hypothesis. The merit of our method is demonstrated by both simulation studies and real examples.