Abstract

In this article, the defining properties of any valid measure of the dependence between two continuous random variables are revisited and complemented with two original ones, shown to imply other usual postulates. While other popular choices are proved to violate some of these requirements, a class of dependence measures satisfying all of them is identified. One particular measure, that we call the Hellinger correlation, appears as a natural choice within that class due to both its theoretical and intuitive appeal. A simple and efficient nonparametric estimator for that quantity is proposed, with its implementation publicly available in the R package HellCor. Synthetic and real-data examples illustrate the descriptive ability of the measure, which can also be used as test statistic for exact independence testing. Supplementary materials for this article are available online.

摘要

在本文中，对两个连续随机变量之间相关性的任何有效度量的定义属性进行了重新审视，并用两个原始属性进行了补充，以暗示其他通常的假设。虽然其他流行的选择被证明违反了其中一些要求，但确定了一类满足所有这些要求的依赖度量。由于其理论和直观的吸引力，我们称之为 Hellinger 相关性的一种特殊度量似乎是该类别中的自然选择。提出了一个简单而有效的非参数估计量，其实现可在 R 包 HellCor 中公开获得。合成和真实数据示例说明了度量的描述能力，也可以用作精确独立性测试的测试统计量。