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

TL;DR

This paper investigates the conditions under which Kendall's and Spearman's correlation coefficients are non-degenerate, providing criteria based on the support set of the joint distribution of the variables.

Contribution

It offers new sufficient conditions for the non-degeneracy of these correlation measures in terms of the support set of the joint distribution.

Findings

Conditions involving rectangles with vertices in the support set.

Nonzero Lebesgue measure of the support set implies non-degeneracy.

Both correlation measures share similar support-based criteria.

Abstract

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.