The goal of this paper is to develop distributionally robust optimization (DRO) estimators, specifically for multidimensional Extreme Value Theory (EVT) statistics. EVT supports using semi-parametric models called max-stable distributions built from spatial Poisson point processes. While powerful, these models are only asymptotically valid for large samples. However, since extreme data is by definition scarce, the potential for model misspecification error is inherent to these applications, thus DRO estimators are natural. In order to mitigate over-conservative estimates while enhancing out-of-sample performance, we study DRO estimators informed by semi-parametric max-stable constraints in the space of point processes. We study both tractable convex formulations for some problems of interest (e.g. CVaR) and more general neural network based estimators. Both approaches are validated using synthetically generated data, recovering prescribed characteristics, and verifying the efficacy of the proposed techniques. Additionally, the proposed method is applied to a real data set of financial returns for comparison to a previous analysis. We established the proposed model as a novel formulation in the multivariate EVT domain, and innovative with respect to performance when compared to relevant alternate proposals.

翻译：本文旨在开发分布鲁棒优化（DRO）估计器，专门用于多维极值理论（EVT）统计量。极值理论支持使用基于空间泊松点过程构建的、称为最大稳定分布的半参数模型。尽管这些模型功能强大，但它们仅对大样本具有渐近有效性。然而，由于极端数据本质上稀缺，模型设定误差的可能性在这些应用中固有存在，因此DRO估计器成为自然选择。为了在提升样本外性能的同时缓解估计过度保守的问题，我们研究了在点过程空间中由半参数最大稳定约束信息引导的DRO估计器。我们既研究了针对某些特定问题（如CVaR）的可处理凸规划形式，也探讨了更通用的基于神经网络的估计器。两种方法均通过合成生成的数据进行了验证，成功恢复了预设特征，并证实了所提技术的有效性。此外，所提方法被应用于金融收益的实际数据集，以与先前分析进行比较。我们将所提模型确立为多元极值理论领域的一种新颖建模框架，并在与相关替代方案比较时，在性能方面展现出创新性。