F-type testing in functional linear models

In this paper, an F-type test is proposed for the functional association between a functional covariate and a scalar response in functional linear models (FLM). The proposed F test is based on the order statistics of functional principal components (fPCs), which is motivated by the phenomena that the power of the classical F test is sensitive to the choice of thresholds. Theoretical properties of the proposed test are established under both the null and alternative hypotheses, especially showing that it can detect the local alternative of order (Formula presented.), that is, the fastest rate in the high-dimensional linear model, where n and S_n are the sample size and the number of fPCs, respectively. Extensive simulation studies are conducted to demonstrate the finite sample performance, showing advantages to the existing tests. Moreover, our procedure is applied to a real data analysis related to the diffusion tensor imaging tractography study.