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

• Published in: Computational statistics & data analysis Vol. 71; pp. 542 - 567
• Main Authors: Sabourin, Anne, Naveau, Philippe
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2014; Elsevier
• Subjects: Algorithms; Applications; Bayesian theory; Computation; Markov chain; Mathematics; Methodology; Mixture models; monitoring; Multivariate extremes; Reversible-jump algorithm; Semi parametric Bayesian inference; Statistics; Statistics Theory; Semi parametric Bayesian inference; Multivariate extremes; Reversible-jump algorithm; Mixture models
• ISSN: 0167-9473; 1872-7352
• DOI: 10.1016/j.csda.2013.04.021

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.