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A simple generalisation of the Hill estimator

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  • Fátima Brilhante, M.
  • Ivette Gomes, M.
  • Pestana, Dinis
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    Abstract

    The classical Hill estimator of a positive extreme value index (EVI) can be regarded as the logarithm of the geometric mean, or equivalently the logarithm of the mean of order p=0, of a set of adequate statistics. A simple generalisation of the Hill estimator is now proposed, considering a more general mean of order p≥0 of the same statistics. Apart from the derivation of the asymptotic behaviour of this new class of EVI-estimators, an asymptotic comparison, at optimal levels, of the members of such class and other known EVI-estimators is undertaken. An algorithm for an adaptive estimation of the tuning parameters under play is also provided. A large-scale simulation study and an application to simulated and real data are developed.

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    Bibliographic Info

    Article provided by Elsevier in its journal Computational Statistics & Data Analysis.

    Volume (Year): 57 (2013)
    Issue (Month): 1 ()
    Pages: 518-535

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    Handle: RePEc:eee:csdana:v:57:y:2013:i:1:p:518-535

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    Web page: http://www.elsevier.com/locate/csda

    Related research

    Keywords: Bias estimation; Bootstrap methodology; Heavy tails; Optimal levels; Semi-parametric estimation; Statistics of extremes;

    References

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    1. Einmahl, J. & Dekkers, A. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Open Access publications from Tilburg University urn:nbn:nl:ui:12-125712, Tilburg University.
    2. MacDonald, A. & Scarrott, C.J. & Lee, D. & Darlow, B. & Reale, M. & Russell, G., 2011. "A flexible extreme value mixture model," Computational Statistics & Data Analysis, Elsevier, vol. 55(6), pages 2137-2157, June.
    3. Gomes, M. Ivette & Neves, Cláudia, 2008. "Asymptotic comparison of the mixed moment and classical extreme value index estimators," Statistics & Probability Letters, Elsevier, vol. 78(6), pages 643-653, April.
    4. Vandewalle, B. & Beirlant, J., 2006. "On univariate extreme value statistics and the estimation of reinsurance premiums," Insurance: Mathematics and Economics, Elsevier, vol. 38(3), pages 441-459, June.
    5. Dekkers, A. L. M. & Dehaan, L., 1993. "Optimal Choice of Sample Fraction in Extreme-Value Estimation," Journal of Multivariate Analysis, Elsevier, vol. 47(2), pages 173-195, November.
    6. 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.
    7. Beran, Jan & Schell, Dieter, 2012. "On robust tail index estimation," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3430-3443.
    8. M. Ivette Gomes & Cristina Miranda & Clara Viseu, 2007. "Reduced-bias tail index estimation and the jackknife methodology," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 61(2), pages 243-270.
    9. Gomes, M. Ivette & Pestana, Dinis, 2007. "A Sturdy Reduced-Bias Extreme Quantile (VaR) Estimator," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 280-292, March.
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