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Testing for a $$\delta $$ δ -neighborhood of a generalized Pareto copula

Author

Listed:
  • Stefan Aulbach

    (University of Würzburg)

  • Michael Falk

    (University of Würzburg)

  • Timo Fuller

    (University of Würzburg)

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 $$\chi ^2$$ χ 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.

Suggested Citation

  • Stefan Aulbach & Michael Falk & Timo Fuller, 2019. "Testing for a $$\delta $$ δ -neighborhood of a generalized Pareto copula," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(3), pages 599-626, June.
    • Handle: RePEc:spr:aistmt:v:71:y:2019:i:3:d:10.1007_s10463-018-0657-x
    DOI: 10.1007/s10463-018-0657-x
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    References listed on IDEAS

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    1. Kojadinovic, Jean D. & Segers, Johan & Yan, Yun, 2011. "Large-sample tests of extreme-value dependence for multivariate copulas," LIDAM Discussion Papers ISBA 2011012, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Bucher, Axel & Segers, Johan & Volgushev, Stanislav, 2014. "When uniform weak convergence fails: empirical processes for dependence functions via epi- and hypographs," LIDAM Reprints ISBA 2014018, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Falk, Michael & Hofmann, Martin, 2012. "Sojourn times and the fragility index," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 1110-1128.
    4. Einmahl, J.H.J. & de Haan, L.F.M. & Li, D., 2006. "Weighted approximations of tail copula processes with applications to testing the bivariate extreme value condition," Other publications TiSEM 18b65ac3-ba79-4bff-ad53-2, Tilburg University, School of Economics and Management.
    5. Stefan Aulbach & Verena Bayer & Michael Falk, 2012. "A multivariate piecing-together approach with an application to operational loss data," Papers 1205.1617, arXiv.org.
    6. Kojadinovic, Ivan & Segers, Johan & Yan, Jun, 2011. "Large-sample tests of extreme-value dependence for multivariate copulas," LIDAM Reprints ISBA 2011025, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. Hofmann, Martin, 2013. "On the hitting probability of max-stable processes," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2516-2521.
    8. Charpentier, Arthur & Segers, Johan, 2009. "Tails of multivariate Archimedean copulas," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1521-1537, August.
    9. Dominik Kortschak & Hansjörg Albrecher, 2009. "Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 11(3), pages 279-306, September.
    10. Deheuvels, Paul, 1983. "Point processes and multivariate extreme values," Journal of Multivariate Analysis, Elsevier, vol. 13(2), pages 257-272, June.
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