A few generalizations of Kendall’s tau. Part I: Construction

Abstract

Complimenting our earlier work on generalizations of popular concordance measures in the sense of Scarsini for a pair of continuous random variables (X, Y) (such measures can be understood as functions of the bivariate copula C associated with (X, Y)), we focus on generalizations of Kendall’s τ. In Part I, we give two forms of such measures and also provide general bounds for their values, which are sharp in certain cases and depend on the values of Spearman’s ρ and the original Kendall’s τ. Part II is devoted to the intrinsic meaning of presented Kendall’s τ generalizations, their degree as polynomial-type concordance measures, and computational aspects.