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Limit distributions of upper order statistics for families of multivariate distributions

Authors:	Email author" target="_blank">Mario?V?Wüthrich Email author

Institution:	(1) Department of Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland

Abstract:	We consider a portfolio of dependent exchangeable random variables $X_1, \ldots, X_n$ , where the dependence structure is generated by a mixture model (Archimedean copulas belong to this class of models). Define the ordered sample $X_{1,n} \ge X_{2,n} \ge \ldots \ge X_{n,n}$ . We prove results of the following type: fix $k \in \mathbb{N}$ and choose ${\left( {c_{n} ,d_{n} } \right)}_{{n \in \mathbb{N}}}$ appropriately, then $(c_n^{-1}(X_{1,n}-d_n), \ldots, c_n^{-1}(X_{k,n}-d_n))$ converges in distribution to a random vector $(Y^{(1)},\cdots,Y^{(k)})$ as $n\to \infty$ , for which we can explicitly give the distribution.

Keywords:	Mixture models Archimedean copula Dependent random variables Upper order statistics Extreme value theory Near-maximum
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