We propose center-outward superquantile and expected shortfall functions, with applications to multivariate risk measurements, extending the standard notion of value at risk and conditional value at risk from the real line to $\mathbb{R}^d$. Our new concepts are built upon the recent definition of Monge-Kantorovich quantiles based on the theory of optimal transport, and they provide a natural way to characterize multivariate tail probabilities and central areas of point clouds. They preserve the univariate interpretation of a typical observation that lies beyond or ahead a quantile, but in a meaningful multivariate way. We show that they characterize random vectors and their convergence in distribution, which underlines their importance. Our new concepts are illustrated on both simulated and real datasets.

翻译：我们提出了中心-外围超分位数和期望短缺函数，并将其应用于多变量风险度量，从而将风险价值和条件风险价值的标准概念从实数线推广到 $\mathbb{R}^d$ 空间。我们的新概念建立在基于最优传输理论的 Monge-Kantorovich 分位数的最新定义之上，它们提供了一种自然的方式来刻画多变量尾部概率和点云的中心区域。这些概念保留了单变量情形下对位于分位数之外或之前的典型观测的解释，但以一种有意义的多变量方式进行。我们证明了它们能够刻画随机向量及其依分布收敛性，这凸显了其重要性。我们通过模拟和真实数据集对新概念进行了说明。