Extreme risk measures estimation: A bridge between asymptotic theory and finite-sample inference

In an increasingly uncertain world, it is crucial to be able to measure the potential impact of certain risks (like a stock market shock or a natural disaster), which raises two important questions. Firstly, the choice of a suitable risk measure. If the classical αth quantile (0 < α < 1) has long been essential, it has more recently been subject to some criticism, which has allowed the emergence of other risk measures like αth expectile or αth Expected Shortfall. Secondly, in order to assess the magnitude of worst potential scenarios, the use of extreme risk measures (α→ 1) is necessary. Their estimation in practice is challenging, and constitute the common thread of the presentation, straddling theory and practice. Indeed, the theoretical properties of all the proposed estimators are provided and deeply analyzed, which allows to correct some pitfalls (strong bias and poor coverage probabilities) observed in a finite sample size setting, and propose very accurate quantile, expectile and Expected Shortfall estimators (and confidence intervals). Applications in insurance, finance and environmental sciences are also proposed.