The expected shortfall is defined as the average over the tail below (or above) a certain quantile of a probability distribution. The expected shortfall regression provides powerful tools for learning the relationship between a response variable and a set of covariates while exploring the heterogeneous effects of the covariates. In the health disparity research, for example, the lower/upper tail of the conditional distribution of a health-related outcome, given high-dimensional covariates, is often of importance. Under sparse models, we propose the lasso-penalized expected shortfall regression and establish non-asymptotic error bounds, depending explicitly on the sample size, dimension, and sparsity, for the proposed estimator. To perform statistical inference on a covariate of interest, we propose a debiased estimator and establish its asymptotic normality, from which asymptotically valid tests can be constructed. We illustrate the finite sample performance of the proposed method through numerical studies and a data application on health disparity.

翻译：期望短缺定义为概率分布中某个分位数下方（或上方）尾部区域的平均值。期望短缺回归为探索协变量的异质性效应、学习响应变量与一组协变量之间的关系提供了强大工具。例如，在健康差异研究中，给定高维协变量条件下健康相关结果的条件分布的下尾/上尾通常具有重要研究价值。针对稀疏模型，我们提出了LASSO惩罚期望短缺回归，并建立了所提出估计量的非渐近误差界，该误差界显式依赖于样本量、维度和稀疏性。为对感兴趣的协变量进行统计推断，我们提出了一种去偏估计量，并建立了其渐近正态性，从而可构造渐近有效的检验。通过数值实验和健康差异数据应用案例，我们验证了所提方法的有限样本性能。