We obtain a functional analogue of the quantile function for probability measures admitting a continuous Lebesgue density on $\mathbb{R}^d$, and use it to characterize the class of non-trivial limit distributions of radially recentered and rescaled multivariate exceedances in geometric extremes. A new class of multivariate distributions is identified, termed radially stable generalized Pareto distributions, and is shown to admit certain stability properties that permit extrapolation to extremal sets along any direction in $\mathbb{R}^d$. Based on the limit Poisson point process likelihood of the radially renormalized point process of exceedances, we develop parsimonious statistical models that exploit theoretical links between structural star-bodies and are amenable to Bayesian inference. The star-bodies determine the mean measure of the limit Poisson process through a hierarchical structure. 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 the 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：针对$\mathbb{R}^d$上具有连续勒贝格密度的概率测度，我们获得了分位数函数的泛函对应形式，并将其用于刻画几何极值中径向中心重标化多元超阈值非平凡极限分布类。提出了一类新的多元分布族，称为径向稳定广义帕累托分布，并证明其具有特定稳定性性质，允许沿$\mathbb{R}^d$任意方向外推至极端集合。基于径向重归一化超阈值点过程的极限泊松点过程似然，我们开发了简约统计模型，该模型利用结构星形体之间的理论联系且适用于贝叶斯推断。星形体通过分层结构决定极限泊松过程的均值测度。我们的框架通过适当纳入几何超阈值角度方向信息强化统计推断，并在$d=2$和$d=3$维度下实现高效计算。此外，该框架自然引出回报水平集概念——以平均重现区间表示的规范分位集合，即一维回报水平的几何类比。通过模拟研究（展示稀有事件概率的优越预测性能）及两个案例（河流流量极值和海洋学极值）验证了方法有效性。