Estimating failure probabilities

TL;DR

This paper develops an estimator for failure probabilities in risk management using bivariate extreme value theory, analyzing its asymptotic properties and practical implications.

Contribution

It introduces a new estimator for failure probabilities within the bivariate extreme value framework and examines its asymptotic behavior and confidence intervals.

Findings

Estimation error depends mainly on marginal distribution analysis accuracy.

The estimator's asymptotic properties are derived under natural conditions.

Discussion includes extensions to higher dimensions and practical issues.

Abstract

In risk management, often the probability must be estimated that a random vector falls into an extreme failure set. In the framework of bivariate extreme value theory, we construct an estimator for such failure probabilities and analyze its asymptotic properties under natural conditions. It turns out that the estimation error is mainly determined by the accuracy of the statistical analysis of the marginal distributions if the extreme value approximation to the dependence structure is at least as accurate as the generalized Pareto approximation to the marginal distributions. Moreover, we establish confidence intervals and briefly discuss generalizations to higher dimensions and issues arising in practical applications as well.