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A characterization of gumbel's family of extreme value distributions

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  • Genest, Christian
  • Rivest, Louis-Paul

Abstract

In this note, a family of multivariate extremal distributions proposed by Gumbel (1960) is characterized among those whose dependence function is an Archimedean copula. The domains of attraction of Gumbel's distributions are also determined within this class.

Suggested Citation

  • Genest, Christian & Rivest, Louis-Paul, 1989. "A characterization of gumbel's family of extreme value distributions," Statistics & Probability Letters, Elsevier, vol. 8(3), pages 207-211, August.
    • Handle: RePEc:eee:stapro:v:8:y:1989:i:3:p:207-211
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    Citations

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    Cited by:

    1. Li, Haijun, 2009. "Orthant tail dependence of multivariate extreme value distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 243-256, January.
    2. Dutfoy Anne & Parey Sylvie & Roche Nicolas, 2014. "Multivariate Extreme Value Theory - A Tutorial with Applications to Hydrology and Meteorology," Dependence Modeling, De Gruyter, vol. 2(1), pages 1-19, June.
    3. Okhrin Ostap & Okhrin Yarema & Schmid Wolfgang, 2013. "Properties of hierarchical Archimedean copulas," Statistics & Risk Modeling, De Gruyter, vol. 30(1), pages 21-54, March.
    4. Mai, Jan-Frederik, 2018. "Extreme-value copulas associated with the expected scaled maximum of independent random variables," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 50-61.
    5. Mazo, Gildas & Girard, Stéphane & Forbes, Florence, 2015. "A class of multivariate copulas based on products of bivariate copulas," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 363-376.
    6. Matthieu Garcin & Maxime L. D. Nicolas, 2021. "Nonparametric estimator of the tail dependence coefficient: balancing bias and variance," Papers 2111.11128, arXiv.org, revised Jul 2023.
    7. Ostap Okhrin, 2010. "Fitting high-dimensional Copulae to Data," SFB 649 Discussion Papers SFB649DP2010-022, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    8. Genest, Christian & Rivest, Louis-Paul, 2001. "On the multivariate probability integral transformation," Statistics & Probability Letters, Elsevier, vol. 53(4), pages 391-399, July.
    9. Li, Haijun & Wu, Peiling, 2013. "Extremal dependence of copulas: A tail density approach," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 99-111.
    10. Mainik Georg & Rüschendorf Ludger, 2012. "Ordering of multivariate risk models with respect to extreme portfolio losses," Statistics & Risk Modeling, De Gruyter, vol. 29(1), pages 73-106, March.
    11. Ressel, Paul, 2011. "A revision of Kimberling's results -- With an application to max-infinite divisibility of some Archimedean copulas," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 207-211, February.
    12. Charpentier, A. & Fougères, A.-L. & Genest, C. & Nešlehová, J.G., 2014. "Multivariate Archimax copulas," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 118-136.
    13. Hofert, Marius & Huser, Raphaël & Prasad, Avinash, 2018. "Hierarchical Archimax copulas," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 195-211.
    14. Ressel Paul, 2022. "Stable tail dependence functions – some basic properties," Dependence Modeling, De Gruyter, vol. 10(1), pages 225-235, January.
    15. Ressel, Paul, 2013. "Homogeneous distributions—And a spectral representation of classical mean values and stable tail dependence functions," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 246-256.
    16. Tankov, Peter, 2016. "Tails of weakly dependent random vectors," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 73-86.
    17. Yongzhao Chen & Ka Chun Cheung & Sheung Chi Phillip Yam & Fei Lung Yuen & Jia Zeng, 2023. "On the Diversification Effect in Solvency II for Extremely Dependent Risks," Risks, MDPI, vol. 11(8), pages 1-22, August.
    18. Enkelejd Hashorva & Didier Rullière, 2019. "Asymptotic Domination Of Sample Maxima," Working Papers hal-02277020, HAL.
    19. Hashorva, Enkelejd & Rullière, Didier, 2020. "Asymptotic domination of sample maxima," Statistics & Probability Letters, Elsevier, vol. 160(C).
    20. Yong Ma & Zhengjun Zhang & Weiguo Zhang & Weidong Xu, 2015. "Evaluating the Default Risk of Bond Portfolios with Extreme Value Theory," Computational Economics, Springer;Society for Computational Economics, vol. 45(4), pages 647-668, April.
    21. Segers, J.J.J., 2004. "Non-Parametric Inference for Bivariate Extreme-Value Copulas," Discussion Paper 2004-91, Tilburg University, Center for Economic Research.
    22. Capéraà, Philippe & Fougères, Anne-Laure & Genest, Christian, 2000. "Bivariate Distributions with Given Extreme Value Attractor," Journal of Multivariate Analysis, Elsevier, vol. 72(1), pages 30-49, January.
    23. Mohamed Achibi & Michel Broniatowski & Catherine Duveau & Alice Marboeuf, 2012. "Bivariate Cox models and copulas," Journal of Risk and Reliability, , vol. 226(5), pages 476-487, October.
    24. Bücher, Axel & Dette, Holger & Volgushev, Stanislav, 2012. "A test for Archimedeanity in bivariate copula models," Journal of Multivariate Analysis, Elsevier, vol. 110(C), pages 121-132.
    25. Segers, J.J.J., 2004. "Non-Parametric Inference for Bivariate Extreme-Value Copulas," Other publications TiSEM 3e837d24-e733-407c-bfaa-f, Tilburg University, School of Economics and Management.

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