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February 2019 Extreme M-quantiles as risk measures: From $L^{1}$ to $L^{p}$ optimization
Abdelaati Daouia, Stéphane Girard, Gilles Stupfler
Bernoulli 25(1): 264-309 (February 2019). DOI: 10.3150/17-BEJ987


The class of quantiles lies at the heart of extreme-value theory and is one of the basic tools in risk management. The alternative family of expectiles is based on squared rather than absolute error loss minimization. It has recently been receiving a lot of attention in actuarial science, econometrics and statistical finance. Both quantiles and expectiles can be embedded in a more general class of M-quantiles by means of $L^{p}$ optimization. These generalized $L^{p}$-quantiles steer an advantageous middle course between ordinary quantiles and expectiles without sacrificing their virtues too much for $1<p<2$. In this paper, we investigate their estimation from the perspective of extreme values in the class of heavy-tailed distributions. We construct estimators of the intermediate $L^{p}$-quantiles and establish their asymptotic normality in a dependence framework motivated by financial and actuarial applications, before extrapolating these estimates to the very far tails. 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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Abdelaati Daouia. Stéphane Girard. Gilles Stupfler. "Extreme M-quantiles as risk measures: From $L^{1}$ to $L^{p}$ optimization." Bernoulli 25 (1) 264 - 309, February 2019.


Received: 1 November 2016; Revised: 1 May 2017; Published: February 2019
First available in Project Euclid: 12 December 2018

zbMATH: 07007208
MathSciNet: MR3892320
Digital Object Identifier: 10.3150/17-BEJ987

Keywords: $L^{p}$ optimization , asymptotic normality , dependent observations , expectiles , extrapolation , Extreme values , heavy tails , Mixing , quantiles , tail risk

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 1 • February 2019
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