Estimation of Expected Shortfall under Various Experimental Conditions

TL;DR

This paper investigates methods to estimate expected shortfall across various scenarios, including unknown distributions, contaminated data, and heavy-tailed losses, providing practical approaches for each case.

Contribution

It introduces estimation techniques for expected shortfall under nonparametric, measurement error, and heavy-tail conditions, extending existing methods with new insights.

Findings

Expected shortfall can be estimated via empirical quantiles in nonparametric cases.

Pseudo-capacities are useful for estimating expected shortfall with contaminated data.

The Pareto index significantly influences the expected shortfall in heavy-tailed distributions.

Abstract

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.