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，极端情景通常具有特殊意义。超阈值方法的核心思想在于，从渐近角度看，超过高维阈值向量的超额部分服从多元广义帕累托分布。然而，现有文献主要关注所有风险变量同时取较大值的情况。在实际中，这一假设往往难以满足，尤其在高维场景下。针对这一局限性，我们研究了不同风险变量组可能联合出现极端值而其他组未出现极端值的情况。这些可区分的组来源于相应最大稳定分布固有的角测度，因此称为极端方向。我们在多元广义帕累托分布框架下探索了此类极端方向，重点关注其相对于合适主导测度的概率密度函数。此外，我们提出了一种随机构造方法，允许任意预先指定的风险变量组构成分布的极端方向。该构造采用平滑最大线性模型的形式，并能容纳所有可能的最大稳定依赖结构。同时，我们引入了一种专为多元广义帕累托分布设计的通用模拟算法，并为能够承载任意极端方向的logistic扩展族与Hüsler-Reiss扩展族提供了具体实现。