Risk measures are defined as functionals of the portfolio loss distribution, thus implicitly assuming the knowledge of such a distribution. However, in practical applications, the need for estimation arises and with it the need to study the effects of mis-specification errors, as well as estimation errors on the final conclusion. In this paper we focus on the qualitative robustness of a sequence of estimators for set-valued risk measures. These properties are studied in detail for two well-known examples of set-valued risk measures: the value-at-risk and the maximum average value-at-risk. Our results illustrate, in particular, that estimation of set-valued value-at-risk can be given in terms of random sets. Moreover, we observe that historical set-valued value-at-risk, while failing to be sub-additive, leads to a more robust procedure than alternatives such as the maximum likelihood average value at-risk.

中文翻译：

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设定值风险值的定性鲁棒性

风险度量被定义为投资组合损失分布的功能，因此隐含地假设了这种分布的知识。但是，在实际应用中，需要进行估计，并且需要研究错误指定错误的影响以及最终结论上的估计错误。在本文中，我们集中于针对集合值风险测度的一系列估计量的定性稳健性。针对两个著名的设定值风险度量示例对这些属性进行了详细研究：风险值和最大风险平均值。我们的结果特别说明，可以用随机集的形式给出风险中的集值估计值。此外，我们观察到历史设定值的风险值虽然没有成为次加性的，