Global and Tail Dependence: A Differential Geometry Approach

Measures of tail dependence between random variables aim to numericallyquantify the degree of association between their extreme realizations. Existingtail dependence coefficients (TDCs) are based on an asymptotic analysis ofrelevant conditional probabilities, and do not provide a complete framework inwhich to compare extreme dependence between two random variables. In fact, formany important classes of bivariate distributions, these coefficients take onnon-informative boundary values. We propose a new approach by first consideringglobal measures based on the surface area of the conditional cumulativeprobability in copula space, normalized with respect to departures fromindependence and scaled by the difference between the two boundary copulas ofco-monotonicity and counter-monotonicity. The measures could be approached bycumulating probability on either the lower left or upper right domain of thecopula space, and offer the novel perspective of being able to differentiateasymmetric dependence with respect to direction of conditioning. The resultingTDCs produce a smoother and more refined taxonomy of tail dependence. Theempirical performance of the measures is examined in a simulated data context,and illustrated through a case study examining tail dependence between stockindices.