A geometric representation for multivariate extremes, based on the shapes of scaled sample clouds in light-tailed margins and their so-called limit sets, has recently been shown to connect several existing extremal dependence concepts. However, these results are purely probabilistic, and the geometric approach itself has not been fully exploited for statistical inference. We outline a method for parametric estimation of the limit set shape, which includes a useful non/semi-parametric estimate as a pre-processing step. More fundamentally, our approach provides a new class of asymptotically-motivated statistical models for the tails of multivariate distributions, and such models can accommodate any combination of simultaneous or non-simultaneous extremes through appropriate parametric forms for the limit set shape. Extrapolation further into the tail of the distribution is possible via simulation from the fitted model. A simulation study confirms that our methodology is very competitive with existing approaches, and can successfully allow estimation of small probabilities in regions where other methods struggle. We apply the methodology to two environmental datasets, with diagnostics demonstrating a good fit.

翻译：多元极值的几何表示，基于轻尾边际中缩放样本云及其所谓极限集的形状，最近被证明能连接几种现有的极值相依性概念。然而，这些结果纯粹是概率性的，且几何方法本身尚未被充分开发用于统计推断。我们提出了一种极限集形状的参数估计方法，其中包含一个有用的非参数/半参数估计作为预处理步骤。更根本的是，我们的方法为多元分布尾部提供了一类新的渐近驱动统计模型，此类模型可通过极限集形状的适当参数形式适应同时或非同时极值的任意组合。通过拟合模型的模拟，可进一步外推至分布更深层的尾部。模拟研究表明，我们的方法与现有方法相比极具竞争力，并能成功估计其他方法难以处理的区域中的小概率。我们将该方法应用于两个环境数据集，诊断结果显示出良好的拟合度。