# Tail dependence in independence

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- Eurandom
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## Abstract

C:\PhD\eta\Drees\paper\eta-paper\eta13b.dvi TAIL DEPENDENCE IN INDEPENDENCE GERRIT DRAISMA� HOLGER DREES� ANA FERREIRA� AND LAURENS DE HAAN Abstract� We propose a new estimator of the parameter �� introduced by Ledford and Tawn ������� governing dependence in bivariate distributions with asymptotically independent componentwise maxima� We prove asymptotic normality of this estimator and two other estimators proposed in the quoted paper� For the latter we develop a weighted approximation result for a two dimensional rank process� We compare the estimators and a related test for asymptotic independence in a simulation study� Also we show consistency of the resulting estimator for failure probabilities in this set up� Our estimator for � is inspired by the work of Peng ������� Our less strict second order conditions are satis ed by the normal distribution� �� Introduction Suppose a region is protected by a river dam against �ooding� The water level is regularly observed at two stations� yielding a sample �X i � Y i �� � � i � n� If there is no other protection within the region� the whole area will be �ooded if the water level exceeds the height of the dam at one of both points� Hence the probability of a �ooding at a particular date is of the form PrfX i � u or Y i � vg������ We assume that �if necessary� after a suitable declustering� the vectors �X i � Y i � are independent and identically distributed with distribution function F � say� If the heights u and v of the dam are large� then multivariate extreme value theory provides a framework which allows a systematic estimation of the probability ������ For this� assume that there exist normalizing constants a n � c n � � and b n � d n � R such that lim n�� F n �a n x � b n � c n y � d n � lim n�� Pr n W n i�� X i � b n a n � x� W n i�� Y i � d n c n � y o G�x� y� ��� � for all but

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