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An asymptotically unbiased minimum density power divergence estimator for the Pareto-tail index

Author

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  • Dierckx, Goedele
  • Goegebeur, Yuri
  • Guillou, Armelle

Abstract

We introduce a robust and asymptotically unbiased estimator for the tail index of Pareto-type distributions. The estimator is obtained by fitting the extended Pareto distribution to the relative excesses over a high threshold with the minimum density power divergence criterion. Consistency and asymptotic normality of the estimator is established under a second order condition on the distribution underlying the data, and for intermediate sequences of upper order statistics. The finite sample properties of the proposed estimator and some alternatives from the extreme value literature are evaluated by a small simulation experiment involving both uncontaminated and contaminated samples.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:121:y:2013:i:c:p:70-86
    DOI: 10.1016/j.jmva.2013.06.011
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    References listed on IDEAS

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    1. Holger Drees, 1998. "On Smooth Statistical Tail Functionals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 187-210, March.
    2. Vandewalle, B. & Beirlant, J. & Christmann, A. & Hubert, M., 2007. "A robust estimator for the tail index of Pareto-type distributions," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6252-6268, August.
    3. Kim, Moosup & Lee, Sangyeol, 2008. "Estimation of a tail index based on minimum density power divergence," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2453-2471, November.
    4. 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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    Cited by:

    1. Christophe Dutang & Yuri Goegebeur & Armelle Guillou, 2016. "Robust and Bias-Corrected Estimation of the Probability of Extreme Failure Sets," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 78(1), pages 52-86, February.
    2. Yuri Goegebeur & Armelle Guillou & Théo Rietsch, 2015. "Robust conditional Weibull-type estimation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(3), pages 479-514, June.
    3. Goegebeur, Yuri & Guillou, Armelle & Verster, Andréhette, 2014. "Robust and asymptotically unbiased estimation of extreme quantiles for heavy tailed distributions," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 108-114.
    4. Christophe Dutang & Yuri Goegebeur & Armelle Guillou, 2016. "Robust and bias-corrected estimation of the probability of extreme failure sets," Post-Print hal-01616187, HAL.
    5. Minkah, Richard & de Wet, Tertius & Ghosh, Abhik, 2022. "Robust Extreme Quantile Estimation for Pareto-Type tails through an Exponential Regression Model," AfricArxiv hf7vk, Center for Open Science.
    6. Goedele Dierckx & Yuri Goegebeur & Armelle Guillou, 2021. "Local Robust Estimation of Pareto-Type Tails with Random Right Censoring," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 70-108, February.
    7. Goegebeur, Yuri & Guillou, Armelle & Ho, Nguyen Khanh Le & Qin, Jing, 2020. "Robust nonparametric estimation of the conditional tail dependence coefficient," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    8. 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.
    9. M. Ivette Gomes & Armelle Guillou, 2015. "Extreme Value Theory and Statistics of Univariate Extremes: A Review," International Statistical Review, International Statistical Institute, vol. 83(2), pages 263-292, August.
    10. Goedele Dierckx & Yuri Goegebeur & Armelle Guillou, 2014. "Local robust and asymptotically unbiased estimation of conditional Pareto-type tails," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 330-355, June.

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