On the properties of some nonparametric concordance measures in the discrete case

It is shown here that Kendall's τ and Spearman's ρ are monotone with respect to the concordance ordering of pairs of discrete as well as continuous random variables. This extends and completes results of [Tchen, A.H., 1980, The Annals of Probability, 8, 814–827.] It is also shown that various relationships between Kendall's τ and Spearman's ρ mentioned in [Nelsen, R.B., 1999, An Introduction to Copulas. Lecture Notes in Statistics no. 139 (New York: Springer).] remain valid for discrete variables. In particular, a result of [Capéraà, P. and Genest, C., 1993, Journal of Nonparametric Statistics, 2, 183–194.] is extended to the case of discrete random pairs. Finally, an analytic expression is given for the most extreme values of Kendall's τ and Spearman's ρ associated with discrete uniform variates.