# Data and uncertainty in extreme risks - a nonlinear expectations approach

## TL;DR

This paper explores how nonlinear expectations, especially Data-robust expectation, can improve the estimation of extreme risk measures like VaR and expected shortfall, particularly addressing challenges in heavy-tailed Pareto distributions.

## Contribution

It introduces a regularization method within the nonlinear expectations framework to better estimate tail quantities in heavy-tailed data, providing a qualitative criterion for reliable risk estimation.

## Key findings

- Regularization is necessary for nonlinear expectation-based tail estimation in Pareto distributions.
- The paper establishes a qualitative condition for the reliability of tail risk measures.
- It extends the nonlinear expectations approach to heavy-tailed extreme value contexts.

## Abstract

Estimation of tail quantities, such as expected shortfall or Value at Risk, is a difficult problem. We show how the theory of nonlinear expectations, in particular the Data-robust expectation introduced in [5], can assist in the quantification of statistical uncertainty for these problems. However, when we are in a heavy-tailed context (in particular when our data are described by a Pareto distribution, as is common in much of extreme value theory), the theory of [5] is insufficient, and requires an additional regularization step which we introduce. By asking whether this regularization is possible, we obtain a qualitative requirement for reliable estimation of tail quantities and risk measures, in a Pareto setting.