Estimating the parameters of max-stable parametric models poses significant challenges, particularly when some parameters lie on the boundary of the parameter space. This situation arises when a subset of variables exhibits extreme values simultaneously, while the remaining variables do not -- a phenomenon commonly referred to as an extreme direction. A novel estimator is proposed for the parameters of a general parametric mixture model, incorporating a threshold exceedances approach based on a pseudo-norm penalization. The latter plays a crucial role in accurately identifying parameters at the boundary of the parameter space. Additionally, the estimator comes with a data-driven algorithm to detect groups of variables corresponding to extreme directions. The performance of the estimator is assessed in terms of both parameter estimation and the identification of extreme directions through extensive simulation studies. Finally, the method is applied to two real-world datasets: discharge measurements at stations along the Danube river, and financial portfolio losses from stocks listed on the NYSE, AMEX, and NASDAQ. In both applications, the sets of variables that can become large simultaneously are identified.

翻译：估计最大稳定参数模型的参数面临重大挑战，特别是当部分参数位于参数空间边界时。这种情况出现在变量子集同时呈现极值而其余变量不呈现极值的情形——这种现象通常被称为极值方向。针对一般参数混合模型的参数，提出了一种新的估计量，该估计量结合了基于伪范数惩罚的阈值超越方法。后者在准确识别参数空间边界的参数中起着关键作用。此外，该估计量附带一种数据驱动算法，用于检测对应于极值方向的变量分组。通过广泛的模拟研究，从参数估计和极值方向识别两个角度评估了该估计量的性能。最后，将该方法应用于两个真实数据集：多瑙河沿岸观测站的流量测量数据，以及纽约证券交易所、美国证券交易所和纳斯达克上市股票的金融投资组合损失数据。在这两个应用中，均识别出可能同时变大的变量集合。