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A mixture model for multivariate extremes

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  • M.‐O. Boldi
  • A. C. Davison

Abstract

Summary. The spectral density function plays a key role in fitting the tail of multivariate extre‐mal data and so in estimating probabilities of rare events. This function satisfies moment con‐straints but unlike the univariate extreme value distributions has no simple parametric form. Parameterized subfamilies of spectral densities have been suggested for use in applications, and non‐parametric estimation procedures have been proposed, but semiparametric models for multivariate extremes have hitherto received little attention. We show that mixtures of Dirichlet distributions satisfying the moment constraints are weakly dense in the class of all non‐parametric spectral densities, and discuss frequentist and Bayesian inference in this class based on the EM algorithm and reversible jump Markov chain Monte Carlo simulation. We illustrate the ideas using simulated and real data.

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  • M.‐O. Boldi & A. C. Davison, 2007. "A mixture model for multivariate extremes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(2), pages 217-229, April.
    • Handle: RePEc:bla:jorssb:v:69:y:2007:i:2:p:217-229
    DOI: 10.1111/j.1467-9868.2007.00585.x
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    Cited by:

    1. Einmahl, J.H.J. & Krajina, A. & Segers, J., 2011. "An M-Estimator for Tail Dependence in Arbitrary Dimensions," Discussion Paper 2011-013, Tilburg University, Center for Economic Research.
    2. Bao, Junshu & Hanson, Timothy E., 2016. "A mean-constrained finite mixture of normals model," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 93-99.
    3. Hanson, Timothy E. & de Carvalho, Miguel & Chen, Yuhui, 2017. "Bernstein polynomial angular densities of multivariate extreme value distributions," Statistics & Probability Letters, Elsevier, vol. 128(C), pages 60-66.
    4. Cooley, Daniel & Davis, Richard A. & Naveau, Philippe, 2010. "The pairwise beta distribution: A flexible parametric multivariate model for extremes," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2103-2117, October.
    5. Fries, Sébastien, 2018. "Conditional moments of noncausal alpha-stable processes and the prediction of bubble crash odds," MPRA Paper 97353, University Library of Munich, Germany, revised Nov 2019.
    6. Sabourin, Anne & Naveau, Philippe, 2014. "Bayesian Dirichlet mixture model for multivariate extremes: A re-parametrization," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 542-567.
    7. Khader Khadraoui & Pierre Ribereau, 2019. "Bayesian Inference with M-splines on Spectral Measure of Bivariate Extremes," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 765-788, September.
    8. Padoan, Simone A., 2011. "Multivariate extreme models based on underlying skew-t and skew-normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 977-991, May.
    9. Sabourin, Anne, 2015. "Semi-parametric modeling of excesses above high multivariate thresholds with censored data," Journal of Multivariate Analysis, Elsevier, vol. 136(C), pages 126-146.
    10. de Carvalho, Miguel & Oumow, Boris & Segers, Johan & WarchoÅ‚, MichaÅ‚, 2012. "A Euclidean likelihood estimator for bivariate tail dependence," LIDAM Discussion Papers ISBA 2012013, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Belzile, Léo R. & Nešlehová, Johanna G., 2017. "Extremal attractors of Liouville copulas," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 68-92.
    12. Hu, Shuang & Peng, Zuoxiang & Segers, Johan, 2022. "Modelling multivariate extreme value distributions via Markov trees," LIDAM Discussion Papers ISBA 2022021, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    13. Lee, J. & Fan, Y. & Sisson, S.A., 2015. "Bayesian threshold selection for extremal models using measures of surprise," Computational Statistics & Data Analysis, Elsevier, vol. 85(C), pages 84-99.
    14. E. Zanini & E. Eastoe & M. J. Jones & D. Randell & P. Jonathan, 2020. "Flexible covariate representations for extremes," Environmetrics, John Wiley & Sons, Ltd., vol. 31(5), August.
    15. Segers, Johan, 2012. "Max-Stable Models For Multivariate Extremes," LIDAM Discussion Papers ISBA 2012011, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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