Spearman's ρ is larger than kendall's τ for positively dependent random variables

Abstract

Spearman's ρ and Kendall's τ are classical nonparametric measures of the degree of dependence bvetween two real-valued random variables X and Y. It is shown here that ρ(X, Y)≥τ(XX, Y)≥0 whenever one of X or Y is simultaneously left-tail decreasing and right-tail increasing in the other variable. A similar argument implies that ρ(X, Y) ≤τ(X, Y)≤0 whenever one of the two variables is simulataneously left-tail increasing and right-tail decreasing in the other. These hypothese, which are rather mild, are verified in a large number of stochastic models for bivariate continuous populations. several illustrative examples are provided. These findings offer a rationale for the oft reported tendency of spearman's correlation coefficient to be more extreme (that is, further away from zero) than Kendall's coefficient of concordance in moderate to large samples.

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