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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.

Suggested Citation

  • 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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    References listed on IDEAS

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    1. Janet E. Heffernan & Jonathan A. Tawn, 2004. "A conditional approach for multivariate extreme values (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(3), pages 497-546.
    2. Einmahl, J.H.J. & Segers, J.J.J., 2008. "Maximum Empirical Likelihood Estimation of the Spectral Measure of an Extreme Value Distribution," Discussion Paper 2008-42, Tilburg University, Center for Economic Research.
    3. Einmahl, J.H.J. & de Haan, L.F.M. & Piterbarg, V.I., 2001. "Nonparametric estimation of the spectral measure of an extreme value distribution," Other publications TiSEM c3485b9b-a0bd-456f-9baa-0, Tilburg University, School of Economics and Management.
    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. Roberts, G. O. & Smith, A. F. M., 1994. "Simple conditions for the convergence of the Gibbs sampler and Metropolis-Hastings algorithms," Stochastic Processes and their Applications, Elsevier, vol. 49(2), pages 207-216, February.
    6. 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.
    7. Simon Guillotte & François Perron & Johan Segers, 2011. "Non‐parametric Bayesian inference on bivariate extremes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(3), pages 377-406, June.
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    1. repec:eee:jmvana:v:161:y:2017:i:c:p:12-31 is not listed on IDEAS
    2. repec:eee:stapro:v:128:y:2017:i:c:p:60-66 is not listed on IDEAS
    3. 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.
    4. repec:bla:jorssb:v:79:y:2017:i:1:p:149-175 is not listed on IDEAS
    5. 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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