Open Access
December 2017 Robust bounds in multivariate extremes
Sebastian Engelke, Jevgenijs Ivanovs
Ann. Appl. Probab. 27(6): 3706-3734 (December 2017). DOI: 10.1214/17-AAP1294

Abstract

Extreme value theory provides an asymptotically justified framework for estimation of exceedance probabilities in regions where few or no observations are available. For multivariate tail estimation, the strength of extremal dependence is crucial and it is typically modeled by a parametric family of spectral distributions. In this work, we provide asymptotic bounds on exceedance probabilities that are robust against misspecification of the extremal dependence model. They arise from optimizing the statistic of interest over all dependence models within some neighborhood of the reference model. A certain relaxation of these bounds yields surprisingly simple and explicit expressions, which we propose to use in applications. We show the effectiveness of the robust approach compared to classical confidence bounds when the model is misspecified. The results are further applied to quantify the effect of model uncertainty on the Value-at-Risk of a financial portfolio.

Citation

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Sebastian Engelke. Jevgenijs Ivanovs. "Robust bounds in multivariate extremes." Ann. Appl. Probab. 27 (6) 3706 - 3734, December 2017. https://doi.org/10.1214/17-AAP1294

Information

Received: 1 July 2016; Revised: 1 February 2017; Published: December 2017
First available in Project Euclid: 15 December 2017

zbMATH: 06848277
MathSciNet: MR3737936
Digital Object Identifier: 10.1214/17-AAP1294

Subjects:
Primary: 60G70 , 62G32 , 62G35

Keywords: Convex optimization , Extremal dependence , model misspecification , Pickands’ function , robust bounds , stress test

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.27 • No. 6 • December 2017
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