Inference on the Angular Distribution of Extremes

The spectral distribution plays a key role in the statistical modelling of multivariate extremes, as it defines the dependence structure of multivariate extreme-value distributions and characterizes the limiting distribution of the relative sizes of the components of large multivariate observations. No parametric family captures all possible types of multivariate dependence, and numerous parametric models have been proposed.

Inference on the spectral distribution is typically based on the pseudo-angles of 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.