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Simple models for multivariate regular variation and the Hüsler–Reiß Pareto distribution

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  • Ho, Zhen Wai Olivier
  • Dombry, Clément

Abstract

We revisit multivariate extreme-value theory modeling by emphasizing multivariate regular variation and a multivariate version of Breiman’s Lemma. This allows us to recover in a simple framework the most popular multivariate extreme-value distributions, such as the logistic, negative logistic, Dirichlet, extremal- t and Hüsler–Reiß models. We then focus on the Hüsler–Reiß Pareto model and its surprising exponential family property. After a thorough study of this exponential family structure, we focus on maximum likelihood estimation: we prove the existence of asymptotically normal maximum likelihood estimators and provide simulation experiments assessing their finite-sample properties. We also consider the generalized Hüsler–Reiß Pareto model with different tail indices and a likelihood ratio test for discriminating constant tail index versus varying tail indices.

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  • Ho, Zhen Wai Olivier & Dombry, Clément, 2019. "Simple models for multivariate regular variation and the Hüsler–Reiß Pareto distribution," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 525-550.
    • Handle: RePEc:eee:jmvana:v:173:y:2019:i:c:p:525-550
    DOI: 10.1016/j.jmva.2019.04.008
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    References listed on IDEAS

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    1. Rootzen, Holger & Segers, Johan & Wadsworth, Jennifer, 2018. "Multivariate peaks over thresholds models," LIDAM Reprints ISBA 2018005, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Krupskii, Pavel & Joe, Harry & Lee, David & Genton, Marc G., 2018. "Extreme-value limit of the convolution of exponential and multivariate normal distributions: Link to the Hüsler–Reiß distribution," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 80-95.
    3. Belzile, Léo R. & Nešlehová, Johanna G., 2017. "Extremal attractors of Liouville copulas," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 68-92.
    4. R. Huser & A. C. Davison, 2013. "Composite likelihood estimation for the Brown--Resnick process," Biometrika, Biometrika Trust, vol. 100(2), pages 511-518.
    5. Davis, Richard A. & Mikosch, Thomas, 2008. "Extreme value theory for space-time processes with heavy-tailed distributions," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 560-584, April.
    6. Jennifer L. Wadsworth & Jonathan A. Tawn, 2014. "Efficient inference for spatial extreme value processes associated to log-Gaussian random functions," Biometrika, Biometrika Trust, vol. 101(1), pages 1-15.
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    Cited by:

    1. Hansjörg Albrecher & Martin Bladt & Mogens Bladt, 2021. "Multivariate matrix Mittag–Leffler distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(2), pages 369-394, April.
    2. Mourahib, Anas & Kiriliouk, Anna & Segers, Johan, 2023. "Multivariate generalized Pareto distributions along extreme directions," LIDAM Discussion Papers ISBA 2023034, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Hentschel, Manuel & Engelke, Sebastian & Segers, Johan, 2022. "Statistical Inference for Hüsler–Reiss Graphical Models Through Matrix Completions," LIDAM Discussion Papers ISBA 2022032, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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