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.

中文翻译：

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混合风险聚合的排序和不等式

聚合集代表了未知依赖导致的模型不确定性，是鲁棒风险聚合研究的重要对象。在本文中，我们研究了两个聚合集之间的排序关系，其中边缘集通过两个简单的操作相关：分布混合和分位数混合。直观地说，这些操作通过使边缘分布相似来“同质化”边缘分布。作为我们结果的一般结论，更多的“同质”边际导致更大的聚合集，因此更严重的模型不确定性，尽管分位数混合物的情况比分布混合物的情况复杂得多。我们继续研究风险聚合中风险度量的最坏情况值的不等式，这代表了监管资本的保守计算。在其他结果中，对于具有单调密度的边际分布，我们获得了分位数混合下 VaR 的顺序关系。给出了数值结果以可视化理论结果并进一步激发一些猜想。最后，我们提供了依赖不确定性下投资组合多样化的应用，并在多个假设检验中合并 p 值，并讨论了我们的结果与联合混合性的联系。