Abstract The correlation ratio has been used to measure how much the behavior of one variable can be predicted by the other variable. In this paper, we derive a new expression of the correlation ratio based on copulas. We represent the copula correlation ratio in terms of Spearman’s rho of the ∗ -product of two copulas. Our expression provides a new way to obtain the copula correlation ratio, which is especially useful when a copula is closed under the ∗ -product operation. Moreover, we propose a Kendall’s tau copula correlation ratio that has not been considered in the literature. We apply the new expressions to investigate the theoretical properties of the copula correlation ratios, including difference and discontinuity. For multivariate copulas, we propose to define the copula correlation ratio matrices, and show their invariance property.

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关于copula相关比及其推广

摘要 相关比已被用来衡量一个变量的行为可以被另一个变量预测多少。在本文中，我们基于 copula 推导出一种新的相关比表达式。我们用两个 copula 的 * 乘积的 Spearman rho 表示 copula 相关比。我们的表达式提供了一种获得 copula 相关比的新方法，这在 ∗ -product 操作下 copula 闭合时特别有用。此外，我们提出了文献中未考虑过的 Kendall tau copula 相关比。我们应用新表达式来研究 copula 相关比的理论特性，包括差异和不连续性。对于多元 copula，我们建议定义 copula 相关比率矩阵，并展示它们的不变性。