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The spectral density function plays a key role in fitting the tail of multivariate extremal data and so in estimating probabilities of rare events. This function satisfies moment constraints but unlike the univariate extreme value distributions has no simple parametric form. Parameterized subfamilies of spectral densities have been suggested for use in applications, and nonparametric estimation procedures have been proposed, but semiparametric models for multivariate extremes have hitherto received little attention. We show that mixtures of Dirichlet distributions satisfying the moment constraints are weakly dense in the class of all nonparametric spectral densities, and discuss frequentist and Bayesian inference in this class based on the EM algorithm and reversible jump Markov chain Monte Carlo simulation. We illustrate the ideas using simulated and real data.
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large' observations under the assumption that they follow the spectral distribution. There has been little attention on studying the impact of this approximation on inference, and it turns out that it can yield significantly biased estimates. We provide a characterization of the angular distribution of excesses corresponding to the distribution of pseudo-angles of
large'observations that improves direct inference on the spectral distribution in the bivariate setting.
Extremal dependence is at the heart of extreme value modelling and numerous measures to quantify it have been proposed in the literature. In many applications, datasets seem to exhibit asymmetry in the dependence between the variables. Many parametric multivariate extreme-value models can accommodate asymmetry in the sense that the spectral density can be asymmetric, resulting in a non-exchangeable dependence structure. There has been little attention paid to quantifying asymmetry at extreme levels, which can be useful for diagnosis and model checking. We propose a coefficient of extremal asymmetry that quantifies the asymmetry at extreme levels for pairs of variables. We also propose two non-parametric estimators of the coefficient of extremal asymmetry and compare their properties through numerical simulation. The two estimators have diametrically opposed bias-variance trade-offs. The estimator based on maximum empirical likelihood performs well and is nearly unbiased.