风险聚集下混合模型的序与不等式
Ordering and Inequalities for Mixtures on Risk Aggregation
摘要：
集结集是鲁棒风险集结研究中的一个重要对象，它代表了模型中由于未知的依赖关系而产生的不确定性。在本文中，我们研究了两个聚集集之间的序关系，其中的边缘集的相关的两个简单的操作：分布混合和分位数混合。直觉上，这些操作通过使边缘分布相似而使它们“均匀化”。从我们的研究结果中得出的一般结论是，更多的“同质”边缘导致更大的聚合集，从而更严重的模型不确定性，虽然分位数混合的情况要比分布混合复杂得多。我们继续研究的最坏情况下的风险度量值的风险聚合，这代表了保守的计算监管资本的不等式。在其他结果中，我们得到了一个顺序关系的风险值下的分位数混合的边际分布单调密度。数值结果显示了理论结果，并进一步启发了一些问题。最后，我们提供了相关不确定性下的投资组合多元化和合并p值在多个假设检验中的应用，并讨论了我们的结果与联合混合性的联系。
英文摘要：
Aggregation sets, which represent model uncertainty due to unknown dependence, are an important object in the study of robust risk aggregation. In this paper, we investigate ordering relations between two aggregation sets for which the sets of marginals are related by two simple operations: distribution mixtures and quantile mixtures. Intuitively, these operations ``homogenize" marginal distributions by making them similar. As a general conclusion from our results, more ``homogeneous" marginals lead to a larger aggregation set, and thus more severe model uncertainty, although the situation for quantile mixtures is much more complicated than that for distribution mixtures. We proceed to study inequalities on the worst-case values of risk measures in risk aggregation, which represent conservative calculation of regulatory capital. Among other results, we obtain an order relation on VaR under quantile mixture for marginal distributions with monotone densities. Numerical results are presented to visualize the theoretical results and further inspire some conjectures. Finally, we provide applications on portfolio diversification under dependence uncertainty and merging p-values in multiple hypothesis testing, and discuss the connection of our results to joint mixability.
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