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：在风险变量向量建模中，极端情景往往备受关注。阈值超越峰值方法的核心思想是：渐近意义上，超过高阈值向量的超额部分服从多元广义帕累托分布。然而，现有文献主要聚焦于所有风险变量同时取极大值的情形。在实际应用中，该假设常难以满足，尤其是在高维场景下。针对这一局限性，我们研究不同风险变量组可能呈现联合极端性而其他变量组未呈现联合极端性的情景。这些可区分的变量组源自相应最大稳定分布蕴含的角度测度，故称为"极端方向"。我们在多元广义帕累托分布框架下探究此类极端方向，重点关注其相对于适当控制测度的概率密度函数。此外，我们提出一种随机构造方法，使得任意预指定的风险变量组构成该分布的极端方向。该构造采用平滑最大线性模型形式，能够容纳所有可能的最大稳定相依结构。最后，我们为多元广义帕累托分布引入通用模拟算法，并为Logistic族和Hüsler-Reiss族的扩展形式提供具体实现方案，使其能够承载任意极端方向。