Flexible Statistical Inference for Multivariate Extremes

The main aim of this thesis is to study extreme value theory in a multivariate framework and to propose new methods for modeling and estimating dependence between several variables at extreme levels. We first give a brief overview of the main concepts of extreme value theory in the univariate framework, focusing on recent extensions of peak-over-threshold models that help automate threshold selection. We compare the performance of these models in different contexts and assess their effectiveness for high quantile estimation and extrapolation outside the range of observed values. In the multivariate case, the asymptotic dependence structure between extremes is entirely characterized by angular variables projected onto the unit simplex. Under moderate conditions, the only constraint on the probability distribution functions of these angular variables is that their marginal means be equal. We make two main contributions to this area. First, we present a non-parametric test to compare the extremal dependence of two bivariate random variables through their angular distribution functions, which we estimate by maximising the empirical likelihood. This allows us to enforce the constraints while avoiding parametric assumptions and results in increased power. The empirical likelihood approach is further exploited to define a valid non-parametric estimator of the Pickands dependence function. The second main contribution is the introduction of tilted mixtures of Dirichlet distributions. This new class of functions is dense in the space of angular distributions and well-defined in all dimensions. We adopt a Bayesian approach and propose a fast and easy-to-use reversible jump Markov chain Monte Carlo procedure which allows us to estimate the number of components in the mixture from the data, making the model semi-parametric. Furthermore, the mixture captures heterogeneity in the extremal dependence structure and allows the probabilistic clustering of observations. We illustrate the virtues of our proposal using simulations and applications to air pollution and financial data.