Ordering and inequalities for mixtures on risk aggregation
研究了当边际分布通过分布混合和分位数混合变得相似时，聚合集的排序关系，发现更同质的边际导致更大的聚合集和更严重的模型不确定性，并给出了风险度量最坏情况值的不等式。
Abstract 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.