Right-invariant generalized metrics applied to rank correlation coefficients

In this paper we investigate properties of rank correlation coefficients that can be derived from right-invariant generalized metrics on the symmetric group. We prove some new inequalities between a number of generalized metrics, and we characterize the sample sizes for which several (right-invariant) rank correlation coefficients can equal zero. Using the Hausdorff generalized metric, we show how to construct circular rank correlation coefficients from regular rank correlation coefficients. In addition, we show how generalized triangle inequalities satisfied by generalized metrics on the symmetric group can be used to create new partial rank correlation coefficients (that measure the association between two variables controlling for the effect of a third one).