The tail dependograph

Abstract

All characterizations of non-degenerate multivariate tail dependence structures are both functional and infinite-dimensional. Taking advantage of the Hoeffding–Sobol decomposition, we derive new indices to measure and summarize the strength of dependence in a multivariate extreme value analysis. The tail superset importance coefficients provide a pairwise ordering of the asymptotic dependence structure. We then define the tail dependograph, which visually ranks the extremal dependence between the components of the random vector of interest. For the purpose of inference, a rank-based statistic is derived and its asymptotic behavior is stated. These new concepts are illustrated with both theoretical models and real data, showing that our methodology performs well in practice.

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Acknowledgments

The authors would like to thank Christian Genest (McGill University, Montréal, Canada) for fruitful discussions. The first author would also like to thank Roland Denis and Benoît Fabrèges (Institut Camille Jordan, Université de Lyon, France) whose discussions and workshop led to great progress in the codes associated with this paper. The authors would like to thank the editor and referees for their helpful comments.

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Correspondence to Cécile Mercadier.

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Mercadier, C., Roustant, O. The tail dependograph. Extremes 22, 343–372 (2019). https://doi.org/10.1007/s10687-019-00345-3

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Keywords

  • Global sensitivity analysis
  • Hoeffding–Sobol decomposition
  • Multivariate extreme value analysis
  • Pairwise index
  • Tail dependency graph

AMS 2000 Subject Classifications

  • Primary 62J10
  • 62G32
  • 62H20
  • Secondary 62J15
  • 62G20
  • 62-09