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 desiderata 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 $1$ 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.
翻译:对两个随机变量(例如Pearson's和Spearman's的关联、Kendall's $tau$和Gini's $\gamma$)的流行依赖性措施大多在美元和美元独立时消失。然而,消失依赖性措施并不一定意味着独立,而一个相当于1美元的措施也不意味着一个变量是另一个变量的可衡量功能。然而,两种属性都是自然的,可以令人信服的依赖性衡量尺度。在本文中,我们提出了一个将特定依赖性措施转化为一个新的新措施的一般方法,该新措施的特征是独立和功能依赖性。我们的方法使用了由Hardy和Littlewood提出的单声调重新排列概念,适用于一系列广泛的措施。特别是,我们能够确定一个重新组合的Spearman's $rho$和一个重新组合的Kendall's $, $tlotau $,但有一个变量是另一个变量的可测量的功能。我们还用一个简单的估量性功能来说明其后视度性依赖性的方法。我们还提出一个简单的测度的模型,用来说明其精确的模型的特性。