Extreme M-quantiles as risk measures
Résumé
Quantiles are basic tools in extreme-value theory in general, and in actuarial and financial mathematics in particular. The alternative class of expectiles has recently been receiving a lot of attention in actuarial science, econometrics and statistical finance. Both of these notions can be embedded in a more general class of M-quantiles by means of $L_p$ optimization. These generalized $L_p$ quantiles can in some sense, for $p$ between 1 and 2, interpolate between ordinary quantiles and expectiles. We investigate here their estimation from the perspective of extreme values in the class of heavy tailed distributions. We construct estimators of intermediate and extreme $L_p$ quantiles and establish their asymptotic normality in a dependence framework motivated by financial and actuarial applications. We also investigate the potential of extreme $L_p$ quantiles as a tool for estimating the usual quantiles and expectiles themselves. We show the usefulness of extreme $L_p$ quantiles and elaborate the choice of $p$ through applications to some simulated and financial real data.