This paper studies a basic notion of distributional shape known as orthounimodality (OU) and its use in shape-constrained distributionally robust optimization (DRO). As a key motivation, we argue how such type of DRO is well-suited to tackle multivariate extreme event estimation by giving statistically valid confidence bounds on target extremal probabilities. In particular, we explain how DRO can be used as a nonparametric alternative to conventional extreme value theory that extrapolates tails based on theoretical limiting distributions, which could face challenges in bias-variance control and other technical complications. We also explain how OU resolves the challenges in interpretability and robustness faced by existing distributional shape notions used in the DRO literature. Methodologically, we characterize the extreme points of the OU distribution class in terms of what we call OU sets and build a corresponding Choquet representation, which subsequently allows us to reduce OU-DRO into moment problems over infinite-dimensional random variables. We then develop, in the bivariate setting, a geometric approach to reduce such moment problems into finite dimension via a specially constructed variational problem designed to eliminate suboptimal solutions. Numerical results illustrate how our approach gives rise to valid and competitive confidence bounds for extremal probabilities.
翻译:本文研究一种基本分布形状的基本概念,即所谓的正单模式(OU)及其在受限制的分布稳健优化(DRO)中的应用。作为一个关键动机,我们争论这种类型的DRO如何适合应对多种变式极端事件估计,方法是在目标极端概率上给出统计上有效的信任界限。特别是,我们解释DRO如何作为传统极端价值理论的非参数替代物使用,这种理论根据理论限制分布来推断尾巴,这可能面临偏差控制和其他技术并发症的挑战。我们还解释OU如何解决DRO文献中所用现有分布形状概念在解释性和稳健性方面面临的挑战。从方法上讲,我们用我们所谓的OU设置和构建一个相应的Choquet代表来描述OU分配等级的极端点,从而使我们能够将O-DRO问题降低到与无限随机变量相比的时空问题。我们随后在双轨设置中开发一种几何方法,通过特别构建的变异性方法将这种时刻性问题降低到有限的空间层面,通过特别构建的变化方法来说明我们所设计的亚性变异性结果。