When modeling a vector of risk variables, extreme scenarios are often of special interest. The peaks-over-thresholds method hinges on the notion that, asymptotically, the excesses over a vector of high thresholds follow a multivariate generalized Pareto distribution. However, existing literature has primarily concentrated on the setting when all risk variables are always large simultaneously. In reality, this assumption is often not met, especially in high dimensions. In response to this limitation, we study scenarios where distinct groups of risk variables may exhibit joint extremes while others do not. These discernible groups are derived from the angular measure inherent in the corresponding max-stable distribution, whence the term extreme direction. We explore such extreme directions within the framework of multivariate generalized Pareto distributions, with a focus on their probability density functions in relation to an appropriate dominating measure. Furthermore, we provide a stochastic construction that allows any prespecified set of risk groups to constitute the distribution's extreme directions. This construction takes the form of a smoothed max-linear model and accommodates the full spectrum of conceivable max-stable dependence structures. Additionally, we introduce a generic simulation algorithm tailored for multivariate generalized Pareto distributions, offering specific implementations for extensions of the logistic and H\"usler-Reiss families capable of carrying arbitrary extreme directions.

翻译：在对风险变量向量进行建模时，极端情景往往备受关注。阈值超越法基于以下渐近概念：超过高阈值的尾部余量服从多元广义帕累托分布。然而，现有文献主要集中在所有风险变量同时处于大数值状态的情形。现实情况中，这一假设往往难以成立，尤其是在高维场景下。针对这一局限性，本文研究了不同风险变量组可能呈现联合极端性而其他组则不呈现的情形。这些可区分的组源自相应最大稳定分布所固有的角度度量，故称为"极端方向"。我们在多元广义帕累托分布框架下探讨此类极端方向，重点关注其相对于适当主导度量的概率密度函数。此外，我们提出了一种随机构造方法，允许任意预设的风险变量组构成分布的极端方向。该构造以平滑最大线性模型的形式实现，能够涵盖所有可想象的最大稳定依赖结构。最后，我们引入了一种适用于多元广义帕累托分布的通用模拟算法，并为Logistic和Hüsler-Reiss分布族的扩展形式提供了具体实施方法，使其能够承载任意极端方向。