On exact regions between measures of concordance and Chatterjee’s rank correlation for lower semilinear copulas

• Exact regions between measures of concordance for lower semilinear (LSL) copulas are determined. • A proof of the conjecture on the tau-rho region of the LSL family is provided. • A closed-form expression for Chatterjee’s rank correlation is presented. • The relationship between Chatterjee’s rank correlation and popular measures of concordance for LSL copulas is investigated. We explore how the classical concordance measures–Kendall’s τ , Spearman’s rank correlation ρ , and Spearman’s footrule ϕ –relate to Chatterjee’s rank correlation ξ when restricted to lower semilinear copulas. First, we provide a complete characterization of the attainable τ − ρ region for this class, thus resolving the conjecture of Maislinger and Trutschnig [1]. Building on this result, we then derive the exact τ − ϕ and ϕ − ρ regions, obtain a closed-form relationship between ξ and τ , and establish the exact τ − ξ region. In particular, we prove that ξ never exceeds τ , ρ , or ϕ . Our results clarify the relationship between undirected and directed dependence measures and reveal novel insights into the dependence structures that result from lower semilinear copulas.