To provide a comprehensive summary of the tail distribution, the expected shortfall is defined as the average over the tail above (or below) a certain quantile of the distribution. The expected shortfall regression captures the heterogeneous covariate-response relationship and describes the covariate effects on the tail of the response distribution. Based on a critical observation that the superquantile regression from the operations research literature does not coincide with the expected shortfall regression, we propose and validate a novel optimization-based approach for the linear expected shortfall regression, without additional assumptions on the conditional quantile models. While the proposed loss function is implicitly defined, we provide a prototype implementation of the proposed approach with some initial expected shortfall estimators based on binning techniques. With practically feasible initial estimators, we establish the consistency and the asymptotic normality of the proposed estimator. The proposed approach achieves heterogeneity-adaptive weights and therefore often offers efficiency gain over existing linear expected shortfall regression approaches in the literature, as demonstrated through simulation studies.

翻译：为全面描述尾部分布特征，期望损失被定义为分布中高于（或低于）特定分位数的尾部区域的平均值。期望损失回归能够捕捉协变量与响应变量之间的异质性关系，并描述协变量对响应分布尾部的影响。基于一个重要发现——运筹学文献中的超分位数回归与期望损失回归并不一致，本文提出并验证了一种基于优化的线性期望损失回归新方法，该方法无需对条件分位数模型附加任何假设。尽管所提出的损失函数是隐式定义的，我们仍基于分箱技术提供了该方法的原型实现及若干初始期望损失估计量。在采用实际可行的初始估计量条件下，我们证明了所提估计量的一致性与渐近正态性。该方法实现了异质性自适应加权，因此相较于文献中现有的线性期望损失回归方法，通常能获得效率提升，这一结论已通过模拟研究得到验证。