BEST ATTAINABLE RATES OF CONVERGENCE FOR ESTIMATORS OF THE STABLE TAIL DEPENDENCE FUNCTION

It is well known that a bivariate distribution belongs to the domain o f attraction of an extreme value distribution G if and only if the mar ginals belong to the domain of attraction of the univariate marginal e xtreme value distributions and the dependence Function converges to th e stable tail dependence function of G. Hall and Welsh (1984, Ann. Sta tist. 12: 1079-1084) and Drees (1997b, Ann. Statist., to appear) addre ssed the problem of finding optimal rates of convergence for estimator s of the extreme value index of an univariate distribution. The presen t paper deals with the corresponding problem for the stable tail depen dence function. First an upper bound on the rate of convergence for es timators of the stable tail dependence function is established. Then i t is shown that this bound is sharp by proving that it is attained by the tail empirical dependence function. Finally, we determine the limi t distribution of this estimator if the dependence Function satisfies a certain second-order condition. (C) 1998 Academic Press.