标题： 关于Kendall和Spearman相关系数的非退化性 显示英文标题

标题： On the Non-degeneracy of Kendall's and Spearman's Correlation Coefficients

摘要： 霍夫丁证明了，肯德尔和斯皮尔曼关于两个连续随机变量 \(X\) 和 \(Y\) 的非参数相关性度量各自渐近服从正态分布，其渐近方差具有形式 \(\sigma^2/n\)——前提是非退化条件 \(\sigma^2 > 0\) 成立，其中 \(\sigma^2\) 是一个由 \(X\) 和 \(Y\) 的联合分布（记作 \(\mu\)）确定的表达式（始终非负）。 给出了关于 \(\mu\) 的支撑集（记作 \(S\)）使得 \(\sigma^2 > 0\) 成立的充分条件，这些条件适用于两种相关性统计量。 其中一个条件是：存在一个矩形，其所有顶点都在 \(S\) 中，边平行于 \(X\) 和 \(Y\) 轴，并且该矩形内部也有一个点属于 \(S\)。 另一个充分条件是 \(S\) 的勒贝格测度非零。 显示英文摘要

摘要： Hoeffding proved that Kendall's and Spearman's nonparametric measures of correlation between two continuous random variables X and Y are each asymptotically normal with an asymptotic variance of the form sigma^2/n -- provided the non-degeneracy condition sigma^2>0 holds, where sigma^2 is a certain (always nonnegative) expression which is determined by the joint distribution (say mu) of X and Y. Sufficient conditions for sigma^2>0 in terms of the support set (say S) of mu are given, the same for both correlation statistics. One of them is that there exist a rectangle with all its vertices in S, sides parallel to the X and Y axes, and an interior point also in S. Another sufficient condition is that the Lebesgue measure of S be nonzero.