Expansions of multivariate Pickands densities and testing the tail dependence

Multivariate extreme value distribution functions (EVDs) with standard reverse exponential margins and the pertaining multivariate generalized Pareto distribution functions (GPDs) can be parametrized in terms of their Pickands dependence function D with D=1 representing tail independence. Otherwise, one has to deal with tail dependence. Besides GPDs we include in our statistical model certain distribution functions (dfs) which deviate from the GPDs whereby EVDs serve as special cases. Our aim is to test tail dependence against rates of tail independence based on the radial component. For that purpose we study expansions and introduce a second order condition for the density (called Pickands density) of the joint distribution of the angular and radial component with the Pickands densities under GPDs as leading terms. A uniformly most powerful test procedure is established based on asymptotic distributions of radial components. It is argued that there is no loss of information if the angular component is omitted in the testing problem.