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, Volume 26, Issue 2, pp 284–307 | Cite as

Bias-corrected and robust estimation of the bivariate stable tail dependence function

  • Mikael Escobar-Bach
  • Yuri GoegebeurEmail author
  • Armelle Guillou
  • Alexandre You
Original Paper

Abstract

The stable tail dependence function gives a full characterisation of the extremal dependence between two or more random variables. In this paper, we propose an estimator for this function which is robust against outliers in the sample. The estimator is derived from a bivariate second-order tail model together with a proper transformation of the bivariate observations, and its asymptotic properties are studied under some suitable regularity conditions. Our estimation procedure depends on two parameters: \(\alpha \), which controls the trade-off between efficiency and robustness of the estimator, and a second-order parameter \(\tau \), which can be replaced by a fixed value or by an estimate. In case where \(\tau \) has been replaced by the true value or by an external consistent estimator, our robust estimator is asymptotically unbiased, whereas in case where \(\tau \) is mis-specified, one loses this property, but still our estimator performs quite well with respect to bias. The finite sample performance of our robust and bias-corrected estimator of the stable tail dependence function is examined on a simulation study involving uncontaminated and contaminated samples. In particular, its behavior is illustrated for different values of the pair \((\alpha , \tau )\) and is compared with alternative estimators from the extreme value literature.

Keywords

Multivariate extreme value statistics Stable tail dependence function Robustness Bias-correction 

Mathematics Subject Classification

62G05 62G20 62G32 

Notes

Acknowledgements

The authors thank the associate editor and the reviewers for their helpful comments which led to improvements of their paper.

Supplementary material

11749_2016_511_MOESM_ESM.pdf (719 kb)
Supplementary material 1 (pdf 719 KB)

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Copyright information

© Sociedad de Estadística e Investigación Operativa 2016

Authors and Affiliations

  • Mikael Escobar-Bach
    • 1
  • Yuri Goegebeur
    • 1
    Email author
  • Armelle Guillou
    • 2
  • Alexandre You
    • 3
  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark
  2. 2.Institut Recherche MathématiqueAvancée, UMR 7501, Université de Strasbourg et CNRSStrasbourg cedexFrance
  3. 3.Société Générale Insurance – SogessurDirection TechniqueParis La Défense 2France

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