Conventional methods for extreme event estimation rely on well-chosen parametric models asymptotically justified from extreme value theory (EVT). These methods, while powerful and theoretically grounded, could however encounter a difficult bias-variance tradeoff that exacerbates especially when data size is too small, deteriorating the reliability of the tail estimation. In this paper, we study a framework based on the recently surging literature of distributionally robust optimization. This approach can be viewed as a nonparametric alternative to conventional EVT, by imposing general shape belief on the tail instead of parametric assumption and using worst-case optimization as a resolution to handle the nonparametric uncertainty. We explain how this approach bypasses the bias-variance tradeoff in EVT. On the other hand, we face a conservativeness-variance tradeoff which we describe how to tackle. We also demonstrate computational tools for the involved optimization problems and compare our performance with conventional EVT across a range of numerical examples.
翻译:极端事件估计的常规方法依赖于从极端价值理论(EVT)中可以有合理的理由的精心选择的参数模型。这些方法虽然有强大和理论依据,但可能会遇到一个困难的偏差权衡,特别是在数据大小过小,尾数估计的可靠性下降的情况下,这种权衡会加剧。在本文中,我们研究一个基于最近快速增长的分布强力优化文献的框架。这个方法可以被视为常规 EVT的非参数替代方法,在尾部上强加一般形状的信念,而不是参数假设,并利用最坏的情况优化作为解决非参数不确定性的解决方案。我们解释这种方法如何绕过EVT中的偏差权衡。另一方面,我们面临一种保守的偏差权衡,我们描述如何解决。我们还展示了涉及优化问题的计算工具,并将我们与常规 EVT的绩效在一系列数字实例中进行比较。