Testing for a \(\delta \)-neighborhood of a generalized Pareto copula

Article

Abstract

A multivariate distribution function F is in the max-domain of attraction of an extreme value distribution if and only if this is true for the copula corresponding to F and its univariate margins. Aulbach et al.  (Bernoulli 18(2), 455–475, 2012.  https://doi.org/10.3150/10-BEJ343) have shown that a copula satisfies the extreme value condition if and only if the copula is tail equivalent to a generalized Pareto copula (GPC). In this paper, we propose a \(\chi ^2\)-goodness-of-fit test in arbitrary dimension for testing whether a copula is in a certain neighborhood of a GPC. The test can be applied to stochastic processes as well to check whether the corresponding copula process is close to a generalized Pareto process. Since the p value of the proposed test is highly sensitive to a proper selection of a certain threshold, we also present graphical tools that make the decision, whether or not to reject the hypothesis, more comfortable.

Keywords

Multivariate max-domain of attraction Multivariate extreme value distribution Copula D-norm Generalized Pareto copula \(\chi ^2\)-goodness-of-fit test Max-stable processes Functional max-domain of attraction 

Notes

Acknowledgements

The authors are grateful to Kilani Ghoudi for his hint to compute the asymptotic distribution of the above test statistics using Imhof’s (1961) method. The authors thank an anonymous associate editor and anonymous referee for their constructive notes and comments, from which this final version of the paper has benefited a lot.

Supplementary material

10463_2018_657_MOESM1_ESM.pdf (745 kb)
Supplementary material 1 (pdf 745 KB)

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Copyright information

© The Institute of Statistical Mathematics, Tokyo 2018

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany

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