  We study distributional robustness in the context of Extreme Value Theory
(EVT). We provide a data-driven method for estimating extreme quantiles in a
manner that is robust against incorrect model assumptions underlying the
application of the standard Extremal Types Theorem. Typical studies in
distributional robustness involve computing worst case estimates over a model
uncertainty region expressed in terms of the Kullback-Leibler discrepancy. We
go beyond standard distributional robustness in that we investigate different
forms of discrepancies, and prove rigorous results which are helpful for
understanding the role of a putative model uncertainty region in the context of
extreme quantile estimation. Finally, we illustrate our data-driven method in
various settings, including examples showing how standard EVT can significantly
underestimate quantiles of interest.
