Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function
Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function
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DOI：
10.1006/jmva.1997.1708
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发表时间：
1998
影响因子：
1.6
通讯作者：
H. Drees;Xing-Fang Huang
中科院分区：
文献类型：
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作者：
H. Drees;Xing-Fang Huang
众所周知，二元分布属于极值分布G的吸引域当且仅当边值属于一元边际极值分布的吸引域，且依赖函数收敛于G.Hall和Welsh(1984，Ann)的稳定尾部依赖函数。Statist.12,1079？1084)和Drees(1997b，Ann.统计学家，即将出现)解决了为单变量分布的极值指数的估计量寻找最优收敛速度的问题。本文讨论了稳定尾部相关函数的相应问题。首先，建立了稳定尾相关函数估计量收敛速度的一个上界。然后通过证明这个界是由尾部经验依赖函数得到的，从而证明了这个界是尖锐的。最后，当依赖函数满足一定的二阶条件时，我们确定了该估计量的极限分布。
It is well known that a bivariate distribution belongs to the domain of attraction of an extreme value distribution G if and only if the marginals belong to the domain of attraction of the univariate marginal extreme value distributions and the dependence function converges to the stable tail dependence function of G. Hall and Welsh (1984,Ann. Statist.12, 1079?1084) and Drees (1997b,Ann. Statist., to appear) addressed the problem of finding optimal rates of convergence for estimators of the extreme value index of an univariate distribution. The present paper deals with the corresponding problem for the stable tail dependence function. First an upper bound on the rate of convergence for estimators of the stable tail dependence function is established. Then it is shown that this bound is sharp by proving that it is attained by the tail empirical dependence function. Finally, we determine the limit distribution of this estimator if the dependence function satisfies a certain second-order condition.