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

TL;DR

This paper derives explicit formulas for Kendall's tau and Spearman's rho for two classes of asymmetric copulas, revealing how asymmetry affects the range of attainable rank correlations and providing insights into their dependence structures.

Contribution

The paper introduces explicit formulas linking copula parameters to rank correlation coefficients for normal and skew-normal scale mixture copulas, including their asymmetry effects.

Findings

Asymmetry in normal scale mixture copulas restricts rank correlation range.

Skew-normal scale mixture copulas can attain the full [-1,1] correlation interval.

Derived formulas facilitate understanding of dependence structures in asymmetric copulas.

Abstract

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.