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The classical multivariate extreme-value theory concerns the modeling of extremes in a multivariate random sample, suggesting the use of max-stable distributions. In this work, the classical theory is extended to the case where aggregated data, such as maxima of a random number of observations, are considered. We derive a limit theorem concerning the attractors for the distributions of the aggregated data, which boil down to a new family of max-stable distributions. We also connect the extremal dependence structure of classical max-stable distributions and that of our new family of max-stable distributions. Using an inversion method, we derive a semiparametric composite-estimator for the extremal dependence of the unobservable data, starting from a preliminary estimator of the extremal dependence of the aggregated data. Furthermore, we develop the large-sample theory of the composite-estimator and illustrate its finite-sample performance via a simulation study.
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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.