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A simple generalisation of the Hill estimator


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


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.

Suggested Citation

  • Fátima Brilhante, M. & Ivette Gomes, M. & Pestana, Dinis, 2013. "A simple generalisation of the Hill estimator," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 518-535.
  • Handle: RePEc:eee:csdana:v:57:y:2013:i:1:p:518-535
    DOI: 10.1016/j.csda.2012.07.019

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    References listed on IDEAS

    1. Beran, Jan & Schell, Dieter, 2012. "On robust tail index estimation," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3430-3443.
    2. 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.
    3. 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.
    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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
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    Cited by:

    1. Gomes, M. Ivette & Henriques-Rodrigues, Lígia, 2016. "Competitive estimation of the extreme value index," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 128-135.
    2. Gomes, M. Ivette & Brilhante, M. Fátima & Caeiro, Frederico & Pestana, Dinis, 2015. "A new partially reduced-bias mean-of-order p class of extreme value index estimators," Computational Statistics & Data Analysis, Elsevier, vol. 82(C), pages 223-237.
    3. 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.
    4. Vygantas Paulauskas & Marijus Vaičiulis, 2017. "A class of new tail index estimators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 461-487, April.


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