How to detect different tail behaviors of two risk random variables with the same mean is an important task. In this paper, motivated by Burzoni et al. (2022), a class of convex risk measures, referred to as adjusted higher-order Expected Shortfall (ES), is introduced and studied. The adjusted risk measure quantifies risk as the minimum amount of capital that has to be raised and injected into a financial position to ensure that its higher-order ES does not exceed a pre-specified threshold for every probability level. This new risk measure is intimately linked to dual higher-order increasing convex order by choosing the risk threshold to be the higher-order ES of a special benchmark random loss. The dual representation for (adjusted) higher-order Expected Shortfall is also given.

如何检测具有相同均值的两个风险随机变量的不同尾部行为是一项重要任务。在本文中，受到Burzoni 等人的启发。(2022)引入并研究了一类凸风险度量，称为调整后的高阶预期缺口（ES）。调整后的风险度量将风险量化为必须筹集并注入财务状况的最低资本金额，以确保其高阶 ES 不超过每个概率水平的预先指定的阈值。通过选择风险阈值作为特殊基准随机损失的高阶 ES，这种新的风险度量与双高阶递增凸阶密切相关。还给出了（调整后的）高阶预期缺口的双重表示。