An association test for functional data based on Kendall’s Tau

Abstract

We consider the problem of testing for association between a functional variable belonging to a Hilbert space and a scalar variable. Particularly, we propose a distribution-free test statistic based on Kendall’s Tau, which is a popular method for determining the association between two random variables. The distribution of the test statistic under the null hypothesis of independence is established using the theory of U-statistics taking values in a Hilbert space. We also consider the case where the functional data is sparsely observed, a situation that arises in many applications. Simulations show that the proposed method outperforms the alternatives under multiple settings, demonstrating the effectiveness and robustness of our approach. We provide data applications that further showcase the utility of our method.