Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions

Jonathan El Methni (1) , Gilles Stupfler (2)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 GREQAM - Groupement de Recherche en Économie Quantitative d'Aix-Marseille

Among the many possible ways to study the right tail of a real-valued random variable, a particularly general one is given by considering the family of its Wang distortion risk measures. This class of risk measures encompasses various interesting indicators, such as the widely used Value-at-Risk and Tail Value-at-Risk, which are especially popular in actuarial science, for instance. In this paper, we first build simple extreme analogues of Wang distortion risk measures. Special cases of the risk measures of interest include the extreme Value-at-Risk as well as the recently introduced extreme Conditional Tail Moment. We then introduce adapted estimators when the random variable of interest has a heavy-tailed distribution and we prove their asymptotic normality. The finite sample performance of our estimators is assessed on a simulation study and we showcase our technique on a set of real data.

Keywords: asymptotic normality; conditional tail moment; distortion risk measure; extreme-value statistics; heavy-tailed distribution

Domains: Mathematics [math]

Jonathan El Methni, Gilles Stupfler. Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions. 2015. ⟨hal-01145417v1⟩