  Let $(X_1,Y_1),\ldots,(X_n,Y_n)$ 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 $\bigvee_{i=1}^nX_i$ and $\bigvee_{i=1}^nY_i$ 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.
