Orateur
Description
When assessing extreme risks, several risk measures depend on extreme value parameters that can be estimated via empirical mean excesses. A key mathematical challenge in studying these estimators is their reliance on high-order statistics above a random threshold. In this talk, we use simple yet novel derandomization arguments to derive the joint asymptotic distribution of these tail empirical excesses and Expected Shortfall with the underlying threshold level. This high-level result allows for a strong degree of heterogeneity in the data-generating process as well as serial dependence. In the case of independent observations with a heavy-tailed average distribution, we obtain asymptotic normality results for the Hill estimator of the extreme value index, the Weissman estimator of extreme quantiles, and two estimators of Expected Shortfall above an extreme level. These results hold under substantially weaker, yet easily verifiable and interpretable conditions than those in the recent literature. In particular, we establish explicit closed-form expressions for the asymptotic bias and variance of each estimator. Our assumptions hold in a wide range of models where existing results may not apply, including scenarios of contaminated samples. We illustrate the practical consequences of our results on simulated data and a real data application to financial risk management. If time permits, we also discuss extensions of this derandomization approach to multivariate data.