The Annals of Applied Probability

Robust bounds in multivariate extremes

Sebastian Engelke and Jevgenijs Ivanovs

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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.

Article information

Ann. Appl. Probab., Volume 27, Number 6 (2017), 3706-3734.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes 62G32: Statistics of extreme values; tail inference 62G35: Robustness

Extremal dependence Pickands’ function model misspecification stress test robust bounds convex optimization


Engelke, Sebastian; Ivanovs, Jevgenijs. Robust bounds in multivariate extremes. Ann. Appl. Probab. 27 (2017), no. 6, 3706--3734. doi:10.1214/17-AAP1294.

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