Corrected inference about the extreme Expected Shortfall in the general max-domain of attraction

The use of the Expected Shortfall as a solution for various deficiencies of quantiles has gained substantial traction in the field of risk assessment over the last 20 years. Existing approaches to its inference at extreme levels remain limited to distributions that are both heavy-tailed and have a finite second tail moment. This constitutes a strong 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 Expected Shortfall estimation above extreme quantiles in the class of distributions with finite first tail moment, regardless of whether the underlying extreme value index is positive, negative, or zero. By relying on the moment estimators of the scale and shape extreme value parameters, we construct corrected 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.