We extend the scope of a recently introduced dependence coefficient between a scalar response $Y$ and a multivariate covariate $X$ to the case where $X$ takes values in a general metric space. Particular attention is paid to the case where $X$ is a curve. While on the population level, this extension is straight forward, the asymptotic behavior of the estimator we consider is delicate. It crucially depends on the nearest neighbor structure of the infinite-dimensional covariate sample, where deterministic bounds on the degrees of the nearest neighbor graphs available in multivariate settings do no longer exist. The main contribution of this paper is to give some insight into this matter and to advise a way how to overcome the problem for our purposes. As an important application of our results, we consider an independence test.

翻译：本文将最近提出的一种用于标量响应$Y$与多元协变量$X$之间的依赖性系数，推广至$X$取值于一般度量空间的情形。我们特别关注$X$为曲线的情况。虽然在总体层面上的推广是直接的，但我们考虑的估计量的渐近行为却十分微妙。这关键依赖于无穷维协变量样本的最近邻结构，而在多元情形下存在的最近邻图度数的确定界在此已不复存在。本文的主要贡献在于深入探讨这一问题，并针对我们的研究目的提出克服该困难的解决方案。作为我们结果的重要应用，我们考虑了一种独立性检验。