# Tail models and the statistical limit of accuracy in risk assessment

## TL;DR

This paper investigates the statistical limits of accurately estimating high quantiles in risk management, focusing on tail models like the generalized Pareto distribution and analyzing the bias and variance of quantile estimators in finite samples.

## Contribution

It derives the finite sample distribution, bias, and variance of tail quantile estimators, providing insights into their accuracy limits in risk assessment scenarios.

## Key findings

- Finite sample distribution of quantile estimators derived
- Bias and variance of estimators quantified
- Impact analysis on unknown distribution quantiles conducted

## Abstract

In risk management, tail risks are of crucial importance. The assessment of risks should be carried out in accordance with the regulatory authority's requirement at high quantiles. In general, the underlying distribution function is unknown, the database is sparse, and therefore special tail models are used. Very often, the generalized Pareto distribution is employed as a basic model, and its parameters are determined with data from the tail area. With the determined tail model, statisticians then calculate the required high quantiles. In this context, we consider the possible accuracy of the calculation of the quantiles and determine the finite sample distribution function of the quantile estimator, depending on the confidence level and the parameters of the tail model, and then calculate the finite sample bias and the finite sample variance of the quantile estimator. Finally, we present an impact analysis on the quantiles of an unknown distribution function.