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

Author

Listed:
  • Mikael Escobar-Bach

    (University of Southern Denmark)

  • Yuri Goegebeur

    (University of Southern Denmark)

  • Armelle Guillou

    (Avancée, UMR 7501, Université de Strasbourg et CNRS)

  • Alexandre You

    (Direction Technique)

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: $$\alpha $$ α , which controls the trade-off between efficiency and robustness of the estimator, and a second-order parameter $$\tau $$ τ , which can be replaced by a fixed value or by an estimate. In case where $$\tau $$ τ has been replaced by the true value or by an external consistent estimator, our robust estimator is asymptotically unbiased, whereas in case where $$\tau $$ τ is mis-specified, one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 $$(\alpha , \tau )$$ ( α , τ ) and is compared with alternative estimators from the extreme value literature.

Suggested Citation

  • Mikael Escobar-Bach & Yuri Goegebeur & Armelle Guillou & Alexandre You, 2017. "Bias-corrected and robust estimation of the bivariate stable tail dependence function," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(2), pages 284-307, June.
  • Handle: RePEc:spr:testjl:v:26:y:2017:i:2:d:10.1007_s11749-016-0511-5
    DOI: 10.1007/s11749-016-0511-5
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    References listed on IDEAS

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    1. Rafael Schmidt & Ulrich Stadtmüller, 2006. "Non‐parametric Estimation of Tail Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 307-335, June.
    2. Beirlant, J. & Dierckx, G. & Guillou, A., 2011. "Bias-reduced estimators for bivariate tail modelling," Insurance: Mathematics and Economics, Elsevier, vol. 49(1), pages 18-26, July.
    3. Drees, Holger & Huang, Xin, 1998. "Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function," Journal of Multivariate Analysis, Elsevier, vol. 64(1), pages 25-47, January.
    4. Einmahl, John H.J. & de Haan, Laurens & Sinha, Ashoke Kumar, 1997. "Estimating the spectral measure of an extreme value distribution," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 143-171, October.
    5. Beirlant, J. & Vandewalle, B., 2002. "Some comments on the estimation of a dependence index in bivariate extreme value statistics," Statistics & Probability Letters, Elsevier, vol. 60(3), pages 265-278, December.
    6. Yuri Goegebeur & Armelle Guillou, 2013. "Asymptotically Unbiased Estimation of the Coefficient of Tail Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(1), pages 174-189, March.
    7. Beirlant, Jan & Escobar-Bach, Mikael & Goegebeur, Yuri & Guillou, Armelle, 2016. "Bias-corrected estimation of stable tail dependence function," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 453-466.
    8. Dierckx, Goedele & Goegebeur, Yuri & Guillou, Armelle, 2013. "An asymptotically unbiased minimum density power divergence estimator for the Pareto-tail index," Journal of Multivariate Analysis, Elsevier, vol. 121(C), pages 70-86.
    9. Anthony W. Ledford & Jonathan A. Tawn, 1997. "Modelling Dependence within Joint Tail Regions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(2), pages 475-499.
    10. M. Ivette Gomes & Laurens De Haan & Lígia Henriques Rodrigues, 2008. "Tail index estimation for heavy‐tailed models: accommodation of bias in weighted log‐excesses," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 31-52, February.
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    1. Härdle, Wolfgang Karl & Ling, Chengxiu, 2018. "How Sensitive are Tail-related Risk Measures in a Contamination Neighbourhood?," IRTG 1792 Discussion Papers 2018-010, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    2. Goegebeur, Yuri & Guillou, Armelle & Qin, Jing, 2019. "Robust estimation of the Pickands dependence function under random right censoring," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 101-114.
    3. Goegebeur, Yuri & Guillou, Armelle & Qin, Jing, 2024. "Dependent conditional tail expectation for extreme levels," Stochastic Processes and their Applications, Elsevier, vol. 171(C).
    4. Yuri Goegebeur & Armelle Guillou & Jing Qin, 2023. "Robust estimation of the conditional stable tail dependence function," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(2), pages 201-231, April.

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