一种新的相关系数及其正交分解和相关独立性检验
A new correlation coefficient, its orthogonal decomposition and associated tests of independence
摘要：
两个实随机变量$X$和$Y$的普通相关系数$ρ$的一个可能的缺点是零相关并不意味着独立。在本文中，我们引入了一个新的相关系数$ρ^*$，它假定值在0和1之间，等于零，如果两个变量是独立的，等于一，如果两个变量是线性相关的。系数$ρ^*$和$ρ^2$被证明是密切相关的代数，它们符合分布在$2\times 2$列联表。我们推导出一个正交分解的$ρ^*$作为一个积极的加权和的平方之间的某些边际特征函数的普通相关。估计$ρ^*$及其组件的相关性和它们的渐近分布进行了讨论，我们开发了可视化的工具，用于评估的性质可能的关联在一个双变量数据集。本文包括考虑等级（等级）版本的$ρ^*$以及使用$ρ^*$进行列联表分析。作为特例，得到了Cram{é} r-vonMises检验在$K$有序样本上的一个新推广.
英文摘要：
A possible drawback of the ordinary correlation coefficient $ρ$ for two real random variables $X$ and $Y$ is that zero correlation does not imply independence. In this paper we introduce a new correlation coefficient $ρ^*$ which assumes values between zero and one, equalling zero iff the two variables are independent and equalling one iff the two variables are linearly related. The coefficients $ρ^*$ and $ρ^2$ are shown to be closely related algebraically, and they coincide for distributions on a $2\times 2$ contingency table. We derive an orthogonal decomposition of $ρ^*$ as a positively weighted sum of squared ordinary correlations between certain marginal eigenfunctions. Estimation of $ρ^*$ and its component correlations and their asymptotic distributions are discussed, and we develop visual tools for assessing the nature of a possible association in a bivariate data set. The paper includes consideration of grade (rank) versions of $ρ^*$ as well as the use of $ρ^*$ for contingency table analysis. As a special case a new generalization of the Cram{é}r-von Mises test to $K$ ordered samples is obtained.
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