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Bayesian Dirichlet mixture model for multivariate extremes: A re-parametrization

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  • Sabourin, Anne
  • Naveau, Philippe

Abstract

The probabilistic framework of extreme value theory is well-known: the dependence structure of large events is characterized by an angular measure on the positive orthant of the unit sphere. The family of these angular measures is non-parametric by nature. Nonetheless, any angular measure may be approached arbitrarily well by a mixture of Dirichlet distributions. The semi-parametric Dirichlet mixture model for angular measures is theoretically valid in arbitrary dimension, but the original parametrization is subject to a moment constraint making Bayesian inference very challenging in dimension greater than three. A new unconstrained parametrization is proposed. This allows for a natural prior specification as well as a simple implementation of a reversible-jump MCMC. Posterior consistency and ergodicity of the Markov chain are verified and the algorithm is tested up to dimension five. In this non identifiable setting, convergence monitoring is performed by integrating the sampled angular densities against Dirichlet test functions.

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  • 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.
    • Handle: RePEc:eee:csdana:v:71:y:2014:i:c:p:542-567
    DOI: 10.1016/j.csda.2013.04.021
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    1. Goix, Nicolas & Sabourin, Anne & Clémençon, Stephan, 2017. "Sparse representation of multivariate extremes with applications to anomaly detection," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 12-31.
    2. 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.
    3. Maël Chiapino & Stephan Clémençon & Vincent Feuillard & Anne Sabourin, 2020. "A multivariate extreme value theory approach to anomaly clustering and visualization," Computational Statistics, Springer, vol. 35(2), pages 607-628, June.
    4. 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.
    5. J. L. Wadsworth & J. A. Tawn & A. C. Davison & D. M. Elton, 2017. "Modelling across extremal dependence classes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 149-175, January.
    6. Vettori, Sabrina & Huser, Raphael & Segers, Johan & Genton, Marc, 2017. "Bayesian Clustering and Dimension Reduction in Multivariate Extremes," LIDAM Discussion Papers ISBA 2017017, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. 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.

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