Expected shortfall and Harell-Davis estimators of value-at-risk

The most widely used estimator for the Value-at-Risk is thecorresponding order statistic. It relies on a single historic observation date,therefore it can exhibit high variability and provides little information aboutthe distribution of losses around the tail. In this paper we purpose to replacethis estimator of VaR by an appropriately chosen estimator of the ExpectedShortfall. We also consider the Harrel-Davis estimator of VaR and give somecomparative analysis among these estimators