  In risk management, often the probability must be estimated that a random
vector falls into an extreme failure set. In the framework of bivariate extreme
value theory, we construct an estimator for such failure probabilities and
analyze its asymptotic properties under natural conditions. It turns out that
the estimation error is mainly determined by the accuracy of the statistical
analysis of the marginal distributions if the extreme value approximation to
the dependence structure is at least as accurate as the generalized Pareto
approximation to the marginal distributions. Moreover, we establish confidence
intervals and briefly discuss generalizations to higher dimensions and issues
arising in practical applications as well.
