Kendall's tau and Spearman's rho for normal location-scale and skew-normal scale mixture copulas

We derive explicit formulas for Kendall’s tau and Spearman’s rho for two broad classes of asymmetric copulas: normal location-scale mixture copulas and skew-normal scale mixture copulas. These classes encompass widely used specifications, including the normal scale mixture, skew-normal, and various skew-$t$ copulas, as special cases. The derived formulas establish functional mappings from copula parameters to rank correlation coefficients, and we investigate and compare how asymmetry parameters influence rank correlation properties and drive departures from the elliptically symmetric case within these two classes. A notable finding is that the introduction of asymmetry in normal location-scale mixture copulas restricts the attainable range of rank correlations from the standard [-1,1] interval, which is 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.

💡 Research Summary

This paper derives explicit closed‑form expressions for Kendall’s τ and Spearman’s ρ for two broad families of asymmetric copulas: normal location‑scale mixture (NLSM) copulas and skew‑normal scale mixture (SNSM) copulas. Both families are built on stochastic representations involving a multivariate normal component mixed over a positive scalar random variable, but they differ in how asymmetry is introduced. In NLSM the asymmetry enters through a location‑mixing vector β (the “skewness” vector) in the representation X = μ + Wβ + √W Z, whereas in SNSM asymmetry is introduced via the skew‑normal shape vector α in X = μ + √W Z with Z∼SN(0,Σ,α).

The authors start from the general copula‑based definitions of τ and ρ_S, namely
τ = 4∬_{