Working with shuffles we establish a close link between Kendall's tau, the so-called length measure, and the surface area of bivariate copulas and derive some consequences. While it is well-known that Spearman's rho of a bivariate copula A is a rescaled version of the volume of the area under the graph of A, in this contribution we show that the other famous concordance measure, Kendall's tau, allows for a simple geometric interpretation as well - it is inextricably linked to the surface area of A.
翻译:肯德尔 Tau 系数、长度测度和二元 Copula 的表面积之间的关联及其对自相似支撑 Copula 的影响
翻译后的摘要:
利用洗牌方法,我们建立了肯德尔 Tau 系数、所谓的长度测度和二元 Copula 的表面积之间的紧密联系,并得出了一些结论。虽然人们早已知道二元 Copula A 的斯皮尔曼 rho 系数是 A 图形下面积的缩放版本,但在这篇文章中,我们展示了另一个著名的一致性衡量标准——肯德尔 Tau 系数——也具有简单的几何解释,它与 A 的表面积密不可分。