Abstract
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.
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Acknowledgements
We would like to thank the anonymous referees and the Associate Editor for providing suggestions towards substantially improving the paper. We are extremely grateful to a referee for pointing out the potential use of implicit function theorem towards providing a neat proof of Theorem 2.
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J. Blanchet gratefully acknowledges support from NSF, grants 1436700, 1915967, 1820942, 1838676, Norges and as well as DARPA award N6. K. Murthy gratefully acknowledges support from MOE grant SRG ESD 2018 134.
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Blanchet, J., He, F. & Murthy, K. On distributionally robust extreme value analysis. Extremes 23, 317–347 (2020). https://doi.org/10.1007/s10687-019-00371-1
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DOI: https://doi.org/10.1007/s10687-019-00371-1
Keywords
- Distributional robustness
- Generalized extreme value distributions
- KL-divergence
- Rényi divergence
- Quantile estimation