We use a functional analogue of the quantile function for probability measures admitting a continuous Lebesgue density on $\mathbb{R}^d$ to characterise the class of non-trivial limit distributions of radially recentered and rescaled multivariate exceedances. A new class of multivariate distributions is identified, termed radially-stable generalised Pareto distributions, and is shown to admit certain stability properties that permit extrapolation to extremal sets along any direction in cones such as $\mathbb{R}^d$ and $\mathbb{R}_+^d$. Leveraging the limit Poisson point process likelihood of the point process of radially renormalised exceedances, we develop parsimonious statistical models that exploit theoretical links between structural star-bodies and are amenable to Bayesian inference. Our framework sharpens statistical inference by suitably including additional information from the angular directions of the geometric exceedances and facilitates efficient computations in dimensions $d=2$ and $d=3$. Additionally, it naturally leads to the notion of return level-set, which is a canonical quantile set expressed in terms of its average recurrence interval, and a geometric analogue of the uni-dimensional return level. We illustrate our methods with a simulation study showing superior predictive performance of probabilities of rare events, and with two case studies, one associated with river flow extremes, and the other with oceanographic extremes.

翻译：我们利用允许在$\mathbb{R}^d$上存在连续勒贝格密度的概率测度的分位函数的泛函类似物，刻画了径向重定位与重缩放多元超阈值序列的非平凡极限分布类。识别出一类新的多元分布，称为径向稳定广义帕累托分布，并证明其具有特定的稳定性性质，允许沿锥形区域（如$\mathbb{R}^d$和$\mathbb{R}_+^d$）的任意方向外推至极值集。通过利用径向重归一化超阈值点过程的极限泊松点过程似然，我们构建了简约的统计模型，该模型利用了结构星体之间的理论联系，并适用于贝叶斯推断。我们的框架通过恰当纳入几何超阈值角方向的额外信息增强了统计推断，并在维度$d=2$和$d=3$下实现了高效计算。此外，该框架自然引出返回水平集的概念——一种以其平均复发间隔表示的标准分位集，即一维返回水平的几何类比。我们通过一项模拟研究（展示了对罕见事件概率的卓越预测性能）以及两项案例研究（分别涉及河流流量极值和海洋学极值）来验证所提方法。