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Divergence based robust estimation of the tail index through an exponential regression model

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  • Abhik Ghosh

    (Indian Statistical Institute)

Abstract

The extreme value theory is very popular in applied sciences including finance, economics, hydrology and many other disciplines. In univariate extreme value theory, we model the data by a suitable distribution from the general max-domain of attraction characterized by its tail index; there are three broad classes of tails—the Pareto type, the Weibull type and the Gumbel type. The simplest and most common estimator of the tail index is the Hill estimator that works only for Pareto type tails and has a high bias; it is also highly non-robust in presence of outliers with respect to the assumed model. There have been some recent attempts to produce asymptotically unbiased or robust alternative to the Hill estimator; however all the robust alternatives work for any one type of tail. This paper proposes a new general estimator of the tail index that is both robust and has smaller bias under all the three tail types compared to the existing robust estimators. This essentially produces a robust generalization of the estimator proposed by Matthys and Beirlant (Stat Sin 13:853–880, 2003) under the same model approximation through a suitable exponential regression framework using the density power divergence. The robustness properties of the estimator are derived in the paper along with an extensive simulation study. A method for bias correction is also proposed with application to some real data examples.

Suggested Citation

  • Abhik Ghosh, 2017. "Divergence based robust estimation of the tail index through an exponential regression model," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(2), pages 181-213, June.
  • Handle: RePEc:spr:stmapp:v:26:y:2017:i:2:d:10.1007_s10260-016-0364-9
    DOI: 10.1007/s10260-016-0364-9
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    References listed on IDEAS

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    1. 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.
    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.
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    4. Einmahl, J. H.J. & Dekkers, A. L.M. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Other publications TiSEM 81970cb3-5b7a-4cad-9bf6-2, Tilburg University, School of Economics and Management.
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    Cited by:

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    2. 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).
    3. 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.
    4. 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".

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