Statistical inference for radially-stable generalized Pareto distributions and return level-sets in geometric extremes

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.