# Extreme M-quantiles as risk measures

1 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-01667201
Contributor : Stephane Girard <>
Submitted on : Tuesday, December 19, 2017 - 10:51:43 AM
Last modification on : Thursday, March 26, 2020 - 8:49:32 PM

### Identifiers

• HAL Id : hal-01667201, version 1

### Citation

Stéphane Girard, Abdelaati Daouia, Gilles Stupfler. Extreme M-quantiles as risk measures. 10th International Conference of the ERCIM WG on Computing and Statistics, Dec 2017, London, United Kingdom. ⟨hal-01667201⟩

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