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:两种随机变量(例如Pearson's and Spearman's related, Kendall's $\tau$ and 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性,而一个等同措施也不意味着一个变量是另一个变量的可衡量函数。然而,两种属性都是具有说服力依赖性衡量尺度的自然属性。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,该措施的特征是独立性和功能依赖性。我们的方法使用了由Hardy和Litlewood提出的单项重新排列概念,该概念适用于广泛的措施类别。特别是,我们可以确定一个重新排列的Spearman's $\rho$和一个等值的变量,只要两个变量都是独立的,而且只有一个变量是独立的,并且只有一个变量是能衡量其可靠性的可衡量的,那么价值是1美元。我们的方法使用一个简单的模型来说明它们是否具有可测量的极限性。</s>