A unified theory of extreme Expected Shortfall inference

The use of the Expected Shortfall as a solution for various deficiencies of quan-tiles has gained substantial traction over the last 20 years. Its inference at extreme levels is a difficult problem in statistics, with existing approaches typically being limited to heavy-tailed distributions having a finite second tail moment. This constitutes a substantial restriction in areas like finance and environmental science, where the random variable of interest may have a much heavier tail or, at the opposite, may be light-tailed or short-tailed. Under a wider semiparametric extreme value framework, we develop comprehensive asymptotic theory for extreme Expected Shortfall estimation in the general class of distributions with finite first tail moment. By relying on the moment estimators of the scale and shape extreme value pa-rameters, we construct refined asymptotic confidence intervals whose finite-sample coverage is found to be close to the nominal level on simulated data. We illustrate the usefulness of our construction on two sets of financial loss returns and flood insurance claims data.