Open Access
February 2020 Tail expectile process and risk assessment
Abdelaati Daouia, Stéphane Girard, Gilles Stupfler
Bernoulli 26(1): 531-556 (February 2020). DOI: 10.3150/19-BEJ1137


Expectiles define a least squares analogue of quantiles. They are determined by tail expectations rather than tail probabilities. For this reason and many other theoretical and practical merits, expectiles have recently received a lot of attention, especially in actuarial and financial risk management. Their estimation, however, typically requires to consider non-explicit asymmetric least squares estimates rather than the traditional order statistics used for quantile estimation. This makes the study of the tail expectile process a lot harder than that of the standard tail quantile process. Under the challenging model of heavy-tailed distributions, we derive joint weighted Gaussian approximations of the tail empirical expectile and quantile processes. We then use this powerful result to introduce and study new estimators of extreme expectiles and the standard quantile-based expected shortfall, as well as a novel expectile-based form of expected shortfall. Our estimators are built on general weighted combinations of both top order statistics and asymmetric least squares estimates. Some numerical simulations and applications to actuarial and financial data are provided.


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Abdelaati Daouia. Stéphane Girard. Gilles Stupfler. "Tail expectile process and risk assessment." Bernoulli 26 (1) 531 - 556, February 2020.


Received: 1 August 2018; Revised: 1 May 2019; Published: February 2020
First available in Project Euclid: 26 November 2019

zbMATH: 07140508
MathSciNet: MR4036043
Digital Object Identifier: 10.3150/19-BEJ1137

Keywords: Asymmetric least squares , coherent risk measures , expected shortfall , expectile , extrapolation , Extremes , heavy tails , tail index

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 1 • February 2020
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