Expected shortfall (ES), also known as conditional value-at-risk, is a widely recognized risk measure that complements value-at-risk by capturing tail-related risks more effectively. Compared with quantile regression, which has been extensively developed and applied across disciplines, ES regression remains in its early stage, partly because the traditional empirical risk minimization framework is not directly applicable. In this paper, we develop a nonparametric framework for expected shortfall regression based on a two-step approach that treats the conditional quantile function as a nuisance parameter. Leveraging the representational power of deep neural networks, we construct a two-step ES estimator using feedforward ReLU networks, which can alleviate the curse of dimensionality when the underlying functions possess hierarchical composition structures. However, ES estimation is inherently sensitive to heavy-tailed response or error distributions. To address this challenge, we integrate a properly tuned Huber loss into the neural network training, yielding a robust deep ES estimator that is provably resistant to heavy-tailedness in a non-asymptotic sense and first-order insensitive to quantile estimation errors in the first stage. Comprehensive simulation studies and an empirical analysis of the effect of El Niño on extreme precipitation illustrate the accuracy and robustness of the proposed method.

翻译：期望损失（ES），亦称条件风险价值，是一种被广泛认可的风险度量指标，它通过更有效地捕捉尾部相关风险来补充风险价值。与已在各学科中得到广泛发展和应用的分位数回归相比，ES回归仍处于早期阶段，部分原因是传统的经验风险最小化框架无法直接适用。本文基于将条件分位数函数视为冗余参数的两步法，提出了一种非参数化的期望损失回归框架。利用深度神经网络的表征能力，我们使用前馈ReLU网络构建了两步ES估计器，当底层函数具有层次复合结构时，该估计器能够缓解维度灾难问题。然而，ES估计本质上对重尾响应或误差分布具有敏感性。为应对这一挑战，我们将经过适当调优的Huber损失函数整合到神经网络训练中，从而得到一种稳健的深度ES估计器。该估计器在非渐近意义上被证明对重尾性具有可证实的抵抗能力，并对第一阶段的分位数估计误差具有一阶不敏感性。综合模拟研究及关于厄尔尼诺现象对极端降水影响的实证分析，验证了所提方法的准确性与稳健性。