Kendall and Spearman Rank Correlations for Skew-Elliptical Copulas

Abstract: In this paper, we derive explicit formulas of Kendall's tau and Spearman's rho rank correlations for two general classes of skew-elliptical copulas: normal location-scale mixture copulas and skew-normal scale mixture copulas, which encompass some widely used skew-$t$ and skew-normal copulas. These formulas establish mappings from copula parameters to rank correlation coefficients, potentially facilitating robust rank-based estimation of skew-elliptical copula models. Additionally, we investigate and compare the impacts of asymmetry parameters on the properties of both rank correlations within these two classes of skew-elliptical models. Notably, the introduction of asymmetry in normal location-scale mixture copulas restricts the attainable range of the rank correlations from $[-1,1]$ -- as observed under elliptical symmetry -- to a strict subset of $[-1,1]$. In contrast, the entire interval $[-1,1]$ remains attainable for skew-normal scale mixture copulas.