Convex weak concordance measures and their constructions

Abstract

Considering the framework of weak concordance measures introduced by Liebscher in 2014, we propose and study convex weak concordance measures. This class of dependence measures contains as a proper subclass the class of all convex concordance measures, introduced and studied by Mesiar et al. in 2022, and thus it also covers the well-known concordance measures as Spearman's ρ, Gini's γ and Blomqvist's β. The class of all convex weak concordance measures also contains, for example, Spearman's footrule ϕ, which is not a concordance measure. In this paper, we first introduce basic convex weak concordance measures built in general by means of a single point (u,v)∈▽={(u,v)∈]0,1[2|u≥v} and its transpose (v,u) only. Then, based on basic convex weak concordance measures and probability measures on the Borel subsets of ▽, two rather general constructions of convex weak concordance measures are proposed, discussed and exemplified. Inspired by Edwards et al., probability measures-based constructions are generalized to Borel measures on B(]0,1[2)-based constructions also allowing some infinite measures to be considered. Finally, it is shown that the presented constructions also cover the mentioned standard (convex weak) concordance measures ρ, γ, β, ϕ and provide alternative formulas for them.