Asymptotically Distribution-Free Goodness-of-Fit Testing for Tail Copulas

Let (X1, Y1),…., (Xn, Yn) be an i.i.d. sample from a bivariate distribution function that lies in the max-domain of attraction of
an extreme value distribution. The asymptotic joint distribution of the standardized component-wise maxima
√n i=1 Xi and √n i=1 Yi is then characterized by the marginal extreme value indices and the tail copula R. We propose a procedure for constructing asymptotically distribution-free goodness-of-fit tests for the tail copula R. The procedure is based on a transformation of a suitable empirical process derived from a semi-parametric estimator of R. The transformed empirical
process converges weakly to a standard Wiener process, paving the way for a multitude of asymptotically distribution-free goodness-of-fit tests. We also extend our results to the m-variate (m > 2) case. In a simulation study we show that the limit theorems provide good approximations for finite samples and that tests based on the transformed empirical process have high power.