This paper offers a mathematical invention that shows how to convert integrated quantiles, which often appear in risk measures, into integrated cumulative distribution functions, which are technically more tractable from various perspectives. The invention helps to avoid a number of technical assumptions that have been traditionally imposed when working with quantities containing quantiles. In particular it helps to completely avoid the requirement of the existence of a probability density function. The developed results explain and illustrate the invention, whose byproducts include the assessment of model uncertainty and misspecification, and the derivation of statistical inference results.

翻译：本文提出了一项数学发明，展示了如何将风险度量中常见的积分分位数转化为从多个角度在技术上更易处理的积分累积分布函数。该发明有助于避免在处理包含分位数的量时传统上需要施加的若干技术假设，尤其是完全避免了对概率密度函数存在性的要求。所取得的结果解释并阐明了这一发明，其副产品包括模型不确定性与设定错误的评估，以及统计推断结果的推导。