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Modeling spatial extremes using normal mean-variance mixtures


Classical models for multivariate or spatial extremes are mainly based upon the asymptotically justified max-stable or generalized Pareto processes. These models are suitable when asymptotic dependence is present, i.e., the joint tail decays at the same rate as the marginal tail. However, recent environmental data applications suggest that asymptotic independence is equally important and, unfortunately, existing spatial models in this setting that are both flexible and can be fitted efficiently are scarce. Here, we propose a new spatial copula model based on the generalized hyperbolic distribution, which is a specific normal mean-variance mixture and is very popular in financial modeling. The tail properties of this distribution have been studied in the literature, but with contradictory results. It turns out that the proofs from the literature contain mistakes. We here give a corrected theoretical description of its tail dependence structure and then exploit the model to analyze a simulated dataset from the inverted Brown–Resnick process, hindcast significant wave height data in the North Sea, and wind gust data in the state of Oklahoma, USA. We demonstrate that our proposed model is flexible enough to capture the dependence structure not only in the tail but also in the bulk.

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Data availability statement

The hindcast wave height data is kindly provided by Philip Jonathan from Shell Research, and the wind gust data is freely available from

Code availability

The code can be provided upon request and will be made publicly available later.


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We are grateful to a referee and an Associate Editor for careful reading and constructive comments that improved our paper. We gratefully acknowledge Philip Jonathan of Shell Research for providing the wave height data analysed in Section 4.1.


This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Awards No. OSR-CRG2017-3434 and OSR-CRG2020-4394.

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Zhang, Z., Huser, R., Opitz, T. et al. Modeling spatial extremes using normal mean-variance mixtures. Extremes 25, 175–197 (2022).

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  • Asymptotic independence
  • Copula model
  • Generalized hyperbolic distribution
  • Normal mean-variance mixtures
  • Spatial extremes

AMS 2000 Subject Classifications

  • 60G70
  • 60E07
  • 62H20