Estimation of Expected Shortfall under Various Experimental Conditions

Our primary aim is to find an estimate of the expected shortfall in various situations: (1) Nonparametric situation, when the probability distribution of the incurred loss is unknown, only satisfying some general conditions. Then, following [3], the expected shortfall can be expressed through a minimization of a well known quantile criterion and its numerical estimate is based on the empirical quantile functionof the loss. (2) The distribution function of the loss is known, but the loss can be contaminated by an additive measurement error: Estimating the expected shortfallin such a case exploits the concept of pseudo-capacities elaborated in [11] and [6] and its numerical value is based on the empirical quantile function of the suitable capacity. (3) The loss distribution can be contaminated by the heavy right tail with Pareto index > 1. The problem of interest is in this case to evaluate the effect of the Pareto index on the resulting expected shortfall.