May 2022 Joint inference on extreme expectiles for multivariate heavy-tailed distributions
Simone A. Padoan, Gilles Stupfler
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Bernoulli 28(2): 1021-1048 (May 2022). DOI: 10.3150/21-BEJ1375


Expectiles induce a law-invariant, coherent and elicitable risk measure that has received substantial attention in actuarial and financial risk management contexts. A number of recent papers have focused on the behaviour and estimation of extreme expectile-based risk measures and their potential for risk assessment was highlighted in financial and actuarial real data applications. Joint inference of several extreme expectiles has however been left untouched; in fact, even the inference about a marginal extreme expectile turns out to be a difficult problem in finite samples, even though an accurate idea of estimation uncertainty is crucial for the construction of confidence intervals in applications to risk management. We investigate the joint estimation of extreme marginal expectiles of a random vector with heavy-tailed marginal distributions, in a general extremal dependence model. We use these results to derive corrected confidence regions for extreme expectiles, as well as a test for the equality of tail expectiles. The methods are showcased in a finite-sample simulation study and on real financial data.


The authors acknowledge an anonymous Associate Editor and two anonymous reviewers for their very helpful comments that led to a much improved version of this paper. Part of this research was carried out when the authors were visiting each other at Bocconi University and the University of Nottingham where G. Stupfler was previously based. Support from the Nottingham PEF Fund is gratefully acknowledged. S.A. Padoan is supported by the Bocconi Institute for Data Science and Analytics (BIDSA), Italy. G. Stupfler is supported by the French National Research Agency under the grant ANR-19-CE40-0013/ExtremReg project. G. Stupfler also acknowledges support from an AXA Research Fund Award on “Mitigating risk in the wake of the COVID-19 pandemic”.


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Simone A. Padoan. Gilles Stupfler. "Joint inference on extreme expectiles for multivariate heavy-tailed distributions." Bernoulli 28 (2) 1021 - 1048, May 2022.


Received: 1 August 2020; Revised: 1 May 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

MathSciNet: MR4388928
zbMATH: 07526574
Digital Object Identifier: 10.3150/21-BEJ1375

Keywords: expectiles , Extremal dependence , heavy tails , joint convergence , joint inference , tail copula , testing

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 2 • May 2022
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