Robust Λ$\Lambda$‐Quantiles and Extremal Distributions

ABSTRACT In this paper, we investigate the robust models for ‐quantiles with partial information regarding the loss distribution, where ‐quantiles extend the classical quantiles by replacing the fixed probability level with a probability/loss function . We find that, under some assumptions, the robust ‐quantiles equal the ‐quantiles of the extremal distributions. This finding allows us to obtain the robust ‐quantiles by applying the results of robust quantiles in the literature. Our results are applied to uncertainty sets characterized by the following three different constraints, respectively: moment constraints, probability distance constraints via the Wasserstein metric, and marginal constraints in risk aggregation. We obtain some explicit expressions for robust ‐quantiles by deriving the extremal distributions for each uncertainty set. These results are applied to optimal portfolio selection under model uncertainty.