Tail expectile process and risk assessment

Daouia, Abdelaati; Girard, St�phane; Stupfler, Gilles

Abstract

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.

Citation

Daouia, A., Girard, S., & Stupfler, G. (2020). Tail expectile process and risk assessment. Bernoulli, 26(1), 531-556. https://doi.org/10.3150/19-BEJ1137

• Journal Article Type: Article
• Acceptance Date: Jun 28, 2019
• Online Publication Date: Nov 26, 2019
• Publication Date: 2020-02
• Deposit Date: Jul 1, 2019
• Publicly Available Date: Jul 1, 2019
• Journal: Bernoulli
• Print ISSN: 1350-7265
• Electronic ISSN: 1573-9759
• Publisher: Bernoulli Society for Mathematical Statistics and Probability
• Peer Reviewed: Peer Reviewed
• Volume: 26
• Issue: 1
• Pages: 531-556
• DOI: https://doi.org/10.3150/19-BEJ1137
• Keywords: Asymmetric least squares, Coherent risk measures, Expected shortfall, Expectile, Extrapolation, Extremes, Heavy tails, Tail index
• Public URL: https://nottingham-repository.worktribe.com/output/2247119
• Publisher URL: https://projecteuclid.org/euclid.bj/1574758837