论文标题
重新排列的依赖措施
Rearranged dependence measures
论文作者
论文摘要
对于两个随机变量$ x $和$ y $（例如Pearson's和Spearman的相关性，Kendall的$τ$和Gini's $γ$）的大多数流行依赖措施消失了。但是，消失的依赖度量也不一定意味着独立性，也不等于1的量度暗示一个变量是另一个变量的一个可测量函数。但是，这两种特性都是令人信服的依赖度量的自然特性。在本文中，我们提出了一种将给定依赖度量转换为新的依赖性措施的一般方法，该方法准确地表征了独立性和功能依赖性。我们的方法使用了Hardy和Littlewood引入的单调重排的概念，并且适用于广泛的措施。特别是，我们能够定义一个重新排列的Spearman的$ρ$和重新排列的Kendall的$τ$，当时确实达到了$ 0 $，并且仅当两个变量都是独立的，并且仅当一个变量$ 1 $，并且仅当一个变量是另一个变量是另一个变量的函数。我们还提出了重新排列的依赖度量的简单估计器，证明了它们的一致性，并通过模拟研究和数据示例说明了其有限样本属性。
Most of the popular dependence measures for two random variables $X$ and $Y$ (such as Pearson's and Spearman's correlation, Kendall's $τ$ and Gini's $γ$) 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 $ρ$ and a rearranged Kendall's $τ$ 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.
