In this paper, we discuss the worst-case distortion riskmetrics for general distributions when only partial information (mean and variance) is known. This result is applicable to a general class of distortion risk measures and variability measures. Furthermore, we also consider the worst-case weighted entropy for general distributions when only partial information is available. Specifically, we provide some applications for entropies, weighted entropies and risk measures. The commonly used entropies include Gini functional, cumulative residual entropy, tail-Gini functional, cumulative Tsallis past entropy, extended Gini coefficient, among others. The risk measures contain some premium principles and shortfalls based on entropy. The shortfalls include the Gini shortfall, extended Gini shortfall, shortfall of cumulative residual entropy and shortfall of cumulative residual Tsallis entropy with order α.

中文翻译：

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最坏情况下的扭曲风险计量学和具有部分信息的加权熵


在本文中，我们讨论了当仅知道部分信息（均值和方差）时，一般分布的最坏情况失真风险计量学。此结果适用于一般类别的失真风险度量和可变性度量。此外，我们还考虑了当只有部分信息可用时，一般分布的最坏情况加权熵。具体来说，我们提供了一些关于熵、加权熵和风险度量的应用。常用的熵包括基尼泛函、累积残差熵、尾基尼泛函、累积 Tsallis 过去熵、扩展基尼系数等。风险衡量标准包含一些溢价原则和基于熵的不足。这些不足包括基尼缺口、扩展基尼缺口、累积残差熵缺口和有序α累积残差 Tsallis 熵缺口。