Strong Consistency of Nonparametric Bayesian Inferential Methods for Multivariate Max-Stable Distributions
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Predicting extreme events is important in many applications in risk analysis. The extreme-value theory suggests modelling extremes by max-stable distributions. The Bayesian approach provides a natural framework for statistical prediction. Marcon, Padoan and Antoniano [Electron. J. Stat.10 (2016) 3310--3337] proposed a nonparametric Bayesian estimation method for bivariate max-stable distributions, representing the main (infinite dimensional) parametrizations of the dependence structure with polynomials in Bernstein form. In this article, we describe a similar inferential method, but which alternatively models the dependence structure by splines. Then, for both approaches we establish the strong consistency of the posterior distributions, under the main parametrizations of the dependence structure. Next, we describe an inferential framework that extends the Bernstein polynomials based approach to max-stable distributions in arbitrary dimensions (greater than two) and we derive the posterior consistency results also in this case. Initially, the consistency results are obtained assuming that the data follow a max-stable distribution with known margins. However, the latter only provides an asymptotic model for sufficiently large sample sizes and its margins are known, potentially, apart from some unknown parameters. Then, we extend the consistency results to the case where the data come from a distribution that is in a neighbourhood of a max-stable distribution and to the case where the margins of the max-stable distribution are heavy-tailed with unknown tail indices.
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