A few generalizations of Kendall’s tau. Part II: Intrinsic meaning, properties, and computational aspects - Manstavičius, Martynas (Nonlinear Analysis: Modelling and Control, 2025)
Continuing our investigation on generalizations of Kendall’s τ, started in Part I of the paper, here we elaborate on the intrinsic meaning and degree of such polynomial-type concordance measures, as well as present many examples of their computation. In particular, we interpret generalized Kendall’s τφ as the difference between the average capacities of concordance and discordance, and, for power-type distortion functions φ, we obtain polynomial-type concordance measures of various degree, which could stimulate further research of their characterization as achieved for degree-one polynomial-type concordance measures by Taylor, Edwards, and Mikusiński.
Keywords: Kendall’s tau; Scarsini axioms; bivariate copula; concordance measure; convex capacity; supermodular