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.

中文翻译：

---

随机数量观察的多元极值

经典的多元极值理论关注多元随机样本中极值的建模，建议使用最大稳定分布。在这项工作中，经典理论被扩展到考虑聚合数据的情况，例如随机数量观察的最大值。我们推导出一个关于聚集数据分布的吸引子的极限定理，它归结为一个新的最大稳定分布族。我们还连接了经典最大稳定分布的极值依赖结构和我们新的最大稳定分布系列的极值依赖结构。使用反演方法，我们从聚合数据的极值相关性的初步估计量开始，推导出不可观察数据的极值相关性的半参数复合估计量。此外，