Tail risk measures are fully determined by the distribution of the underlying loss beyond its quantile at a certain level, with Value-at-Risk and Expected Shortfall being prime examples. They are induced by law-based risk measures, called their generators, evaluated on the tail distribution. This paper establishes joint identifiability and elicitability results of tail risk measures together with the corresponding quantile, provided that their generators are identifiable and elicitable, respectively. As an example, we establish the joint identifiability and elicitability of the tail expectile together with the quantile. The corresponding consistent scores constitute a novel class of weighted scores, nesting the known class of scores of Fissler and Ziegel for the Expected Shortfall together with the quantile. For statistical purposes, our results pave the way to easier model fitting for tail risk measures via regression and the generalized method of moments, but also model comparison and model validation in terms of established backtesting procedures.

翻译：尾部风险度量完全由底层损失在特定水平分位数之外的分布所决定，其中风险价值和预期短缺是典型示例。它们由基于定律的风险度量（称为其生成元）在尾部分布上的评估所导出。本文证明，只要尾部风险度量的生成元分别具有可识别性和可激发性，则尾部风险度量与相应分位数具有联合可识别性与可激发性。作为示例，我们建立了尾部期望损失与分位数的联合可识别性与可激发性。相应的一致性评分构成了一类新颖的加权评分，其中嵌套了Fissler和Ziegel针对预期短缺与分位数所建立的已知评分类别。在统计应用方面，我们的研究结果为通过回归和广义矩估计方法更简便地拟合尾部风险度量模型铺平了道路，同时也为基于既定回测程序的模型比较与验证提供了理论基础。