Estimation of conditional extreme risk measures from heavy-tailed elliptical random vectors

TL;DR

This paper develops new estimators for conditional extreme risk measures in heavy-tailed elliptical distributions, analyzing their asymptotic properties and demonstrating their effectiveness through simulations and a financial data example.

Contribution

It introduces novel estimators for extremal parameters and conditional risk measures, with proven asymptotic properties, for heavy-tailed elliptical random vectors.

Findings

Estimators are consistent and asymptotically normal.

Simulation results confirm estimator efficiency.

Application to financial data illustrates practical usefulness.

Abstract

In this work, we focus on some conditional extreme risk measures estimation for elliptical random vectors. In a previous paper, we proposed a methodology to approximate extreme quantiles, based on two extremal parameters. We thus propose some estimators for these parameters, and study their asymptotic properties in the case of heavy-tailed distributions. Thereafter, from these parameters, we construct extreme conditional quantiles estimators, and give their consistency properties. Using recent results on the asymptotic relationship between quantiles and other risk measures, we deduce estimators for extreme conditional Lp-quantiles and Haezendonck-Goovaerts risk measures. In order to test the efficiency of our estimators, we propose a simulation study. A financial data example is also proposed.