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.

我们考虑测试属于希尔伯特空间的函数变量和标量变量之间的关联的问题。特别是，我们提出了一种基于Kendall的Tau的无分布检验统计量，这是一种确定两个随机变量之间关联的流行方法。独立性的零假设下的检验统计量的分布是使用在希尔伯特空间中取值的U统计量的理论建立的。我们还考虑了稀疏地观察功能数据的情况，这种情况在许多应用中都会出现。仿真表明，该方法在多种设置下均优于其他方法，证明了我们方法的有效性和鲁棒性。我们提供的数据应用程序进一步展示了我们方法的实用性。