Distortion risk measure (DRM) plays a crucial role in management science and finance particularly actuarial science. Various DRMs have been introduced but little is discussed on which DRM at hand should be chosen to address a decision maker's (DM's) risk preference. This paper aims to fill out the gap. Specifically, we consider a situation where the true distortion function is unknown either because it is difficult to identify/elicit and/or because the DM's risk preference is ambiguous. We introduce a preference robust distortion risk measure (PRDRM), which is based on the worst-case distortion function from an ambiguity set of distortion functions to mitigate the impact arising from the ambiguity. The ambiguity set is constructed under well-known general principles such as concavity and inverse S-shapedness of distortion functions (overweighting on events from impossible to possible or possible to certainty and underweighting on those from possible to more possible) as well as new user-specific information such as sensitivity to tail losses, confidence intervals to some lotteries, and preferences to certain lotteries over others. To calculate the proposed PRDRM, we use the convex and/or concave envelope of a set of points to characterize the curvature of the distortion function and derive a tractable reformulation of the PRDRM when the underlying random loss is discretely distributed. Moreover, we show that the worst-case distortion function is a nondecreasing piecewise linear function and can be determined by solving a linear programming problem. Finally, we apply the proposed PRDRM to a risk capital allocation problem and carry out some numerical tests to examine the efficiency of the PRDRM model.

中文翻译：

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偏好鲁棒失真风险测度及其应用

失真风险度量（DRM）在管理科学和金融尤其是精算科学中起着至关重要的作用。已经引入了各种 DRM，但很少讨论应该选择手头的哪个 DRM 来解决决策者 (DM) 的风险偏好。本文旨在填补这一空白。具体来说，我们考虑真实失真函数未知的情况，因为它难以识别/引出和/或因为 DM 的风险偏好不明确。我们引入了一种偏好鲁棒失真风险度量（PRDRM），它基于一组模糊失真函数中的最坏情况失真函数，以减轻模糊产生的影响。歧义集是根据众所周知的一般原则构建的，例如畸变函数的凹性和反 S 形（对从不可能到可能或可能到确定的事件加权，对从可能到更可能的事件加权不足）以及新的用户-具体信息，例如对尾部损失的敏感性、对某些彩票的置信区间以及对某些彩票的偏好。为了计算所提出的 PRDRM，我们使用一组点的凸包络和/或凹包络来表征失真函数的曲率，并在基础随机损失离散分布时推导出 PRDRM 的易于处理的重构。而且，我们表明，最坏情况下的失真函数是一个非递减分段线性函数，可以通过求解线性规划问题来确定。最后，我们将提出的 PRDRM 应用于风险资本配置问题，并进行一些数值测试来检验 PRDRM 模型的效率。