Most of the popular dependence measures for two random variables $X$ and $Y$ (such as Pearson's and Spearman's correlation, Kendall's $\tau$ and Gini's $\gamma$) vanish whenever $X$ and $Y$ are independent. However, neither does a vanishing dependence measure necessarily imply independence, nor does a measure equal to 1 imply that one variable is a measurable function of the other. Yet, both properties are natural properties for a convincing dependence measure. In this paper, we present a general approach to transforming a given dependence measure into a new one which exactly characterizes independence as well as functional dependence. Our approach uses the concept of monotone rearrangements as introduced by Hardy and Littlewood and is applicable to a broad class of measures. In particular, we are able to define a rearranged Spearman's $\rho$ and a rearranged Kendall's $\tau$ which do attain the value $0$ if and only if both variables are independent, and the value $1$ if and only if one variable is a measurable function of the other. We also present simple estimators for the rearranged dependence measures, prove their consistency and illustrate their finite sample properties by means of a simulation study and a data example.

翻译：大多数用于两个随机变量$X$和$Y$的流行依赖度量（例如皮尔逊相关、斯皮尔曼相关、肯德尔$\tau$和基尼$\gamma$）在$X$与$Y$独立时均取值为零。然而，依赖度量取零并不必然意味着独立性，同样度量等于1也未必表示一个变量是另一个变量的可测函数。然而，这两个属性对于具有说服力的依赖度量而言是自然性质。本文提出了一种通用方法，可将给定的依赖度量转化为能够精确刻画独立性及函数依赖关系的新度量。该方法利用了哈代与利特尔伍德提出的单调重排概念，并适用于广泛的度量类别。特别地，我们定义了重排斯皮尔曼$\rho$和重排肯德尔$\tau$，它们满足：当且仅当两个变量独立时取值为0，当且仅当一个变量是另一个变量的可测函数时取值为1。我们还提出了重排依赖度量的简单估计量，证明了其一致性，并通过模拟研究和数据实例说明了其有限样本性质。