A novel positive dependence property and its impact on a popular class of concordance measures

Abstract

A novel positive dependence property is introduced, called positive measure inducing (PMI for short), being fulfilled by numerous copula classes, including Gaussian, Student $t$, Fréchet, Farlie–Gumbel–Morgenstern and Frank copulas; it is conjectured that even all positive quadrant dependent Archimedean copulas meet this property. From a geometric viewpoint, a PMI copula concentrates more mass near the main diagonal than in the opposite diagonal. A striking feature of PMI copulas is that they impose an ordering on a certain class of copula-induced measures of concordance, the latter originating in Edwards et al. (2004) and including Spearman’s rho $\rho$ and Gini’s gamma $\gamma$, leading to numerous new inequalities such as $3\gamma \ge 2\rho$. The measures of concordance within this class are estimated using (classical) empirical copulas and the intrinsic construction via empirical checkerboard copulas, and the estimators’ asymptotic behavior is determined. Building upon the presented inequalities, asymptotic tests are constructed having the potential of being used for detecting whether the underlying dependence structure of a given sample is PMI, which in turn can be used for excluding certain copula families from model building. The excellent performance of the tests is demonstrated in a simulation study and by means of a real-data example.