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 $\RR^d$. 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：我们提出了中心向外超分位数和期望亏空函数，并应用于多元风险度量，将标准的风险价值与条件风险价值概念从实数轴扩展到$\RR^d$空间。这些新概念基于最优传输理论中近期定义的Monge-Kantorovich分位数，提供了一种自然的方式来刻画多元尾概率与点云中心区域。它们保留了单变量情形中典型观测值位于分位数之外或之前的解释意义，但以有意义的多元方式呈现。我们证明这些函数可刻画随机向量及其依分布收敛性，凸显了其重要性。通过模拟数据集和真实数据集验证了所提新概念的有效性。