Journal of Statistical Theory and Practice

, Volume 9, Issue 1, pp 200–218 | Cite as

Revisiting the Maximum Likelihood Estimation of a Positive Extreme Value Index

  • Frederico CaeiroEmail author
  • M. Ivette Gomes


In this article, we revisit Feuerverger and Hall’s maximum likelihood estimation of the extreme value index. Based on those estimators we propose new estimators that have the smallest possible asymptotic variance, equal to the asymptotic variance of the Hill estimator. The full asymptotic distributional properties of the estimators are derived under a general third-order framework for heavy tails. Applications to a real data set and to simulated data are also presented.


Bias estimation Heavy tails Semiparametric estimation Statistics of extremes 

AMS Subject Classification

62G05 62G20 62G32 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Beirlant, J., G. Dierckx, Y. Goegebeur, and G. Matthys. 1999. Tail index estimation and an exponential regression model. Extremes, 2, 177–200.MathSciNetCrossRefGoogle Scholar
  2. Beirlant J., Y. Goegebeur, J. Segers, and J. Teugels. 2004. Statistics of extremes. Theory and applications. Hoboken, NJ: Wiley.CrossRefGoogle Scholar
  3. Beirlant, J., F. Caeiro, and M. I. Gomes. 2012. An overview and open research topics in the field of statistics of univariate extremes. Revstat, 10(1), 1–31.MathSciNetzbMATHGoogle Scholar
  4. Caeiro, F., and M. I. Gomes. 2006. A new class of estimators of a “scale” second order parameter. Extremes, 9, 193–211.MathSciNetCrossRefGoogle Scholar
  5. Caeiro, F., and M. I. Gomes. 2011. Asymptotic comparison at optimal levels of reduced-bias extreme value index estimators. Stat. Neerland., 65(4), 462–488.MathSciNetCrossRefGoogle Scholar
  6. Caeiro, F., and M. I. Gomes. 2012. A reduced bias estimator of a ‘scale’ second order parameter. In AIP Conf. Proc, ed. T. E. Simos, G. Psihoyios, C. Tsitouras, and Z. Anastassi, 1479, 1114–1117.Google Scholar
  7. Caeiro, F., and M. I. Gomes. 2013. Asymptotic comparison at optimal levels of minimum-variance reduced-bias tail index estimators. In Advances in regression, survival analysis, extreme values, Markov processes and other statistical applications, Studies in Theoretical and Applied Statistics, ed. J. Lita da Silva, F. Caeiro, I. Natário, and C.A. Braumann, 83–91. Berlin, Heidelberg: Springer-Verlag.CrossRefGoogle Scholar
  8. Caeiro, F., and M. I. Gomes. 2014. A semi-parametric estimator of a shape second order parameter. In New advances in statistical modeling and applications, Studies in Theoretical and Applied Statistics, ed. A. Pacheco, R. Santos, M. Rosário Oliveira, and C. D. Paulino, 137–144. Switzerland: Springer International.CrossRefGoogle Scholar
  9. Caeiro, F., M. I. Gomes, and D. D. Pestana. 2005. Direct reduction of bias of the classical Hill estimator. Revstat, 3(2), 111–136.MathSciNetzbMATHGoogle Scholar
  10. Caeiro, F., M. I. Gomes, and L. Henriques-Rodrigues. 2009. Reduced-bias tail index estimators under a third order framework. Commun. Stat. Theory Methods, 38(7), 1019–1040.MathSciNetCrossRefGoogle Scholar
  11. Ciuperca, G., and C. Mercadier. 2010. Semi-parametric estimation for heavy tailed distributions. Extremes, 13(1), 55–87.MathSciNetCrossRefGoogle Scholar
  12. de Haan, L., and L. Peng. 1998. Comparison of extreme value index estimators. Stat. Neerland., 52, 60–70.CrossRefGoogle Scholar
  13. Deme, E. H., L. Gardes, and S. Girard. 2013. On the estimation of the second order parameter for heavy-tailed distributions. Revstat, 11(3), 277–299.MathSciNetzbMATHGoogle Scholar
  14. Feuerverger, A., and P. Hall. 1999. Estimating a tail exponent by modelling departure from a Pareto distribution. Ann. Stat., 27, 760–781.MathSciNetCrossRefGoogle Scholar
  15. Fraga Alves, M. I., M. I. Gomes, and L. de Haan. 2003. A new class of semi-parametric estimators of the second order parameter. Port. Math., 60(2), 193–213.MathSciNetzbMATHGoogle Scholar
  16. Geluk, J., and L. de Haan. 1987. Regular variation, extensions and Tauberian theorems. CWI Tract 40, Center for Mathematics and Computer Science, Amsterdam, The Netherlands.zbMATHGoogle Scholar
  17. Goegebeur, Y., J. Beirlant, and T. de Wet. 2008. Linking Pareto-tail kernel goodness-of-fit statistics with tail index at optimal threshold and second order estimation. Revstat, 6(1), 51–69.MathSciNetzbMATHGoogle Scholar
  18. Gomes, M. I., L. Canto e Castro, M. I. Fraga Alves, and D. D. Pestana. 2008a. Statistics of extremes for IID data and breakthroughs in the estimation of the extreme value index: Laurens de Haan leading contributions. Extremes, 11(1), 3–34.MathSciNetCrossRefGoogle Scholar
  19. Gomes, M. I., L. de Haan, and L. Peng. 2002. Semi-parametric estimation of the second order parameter in statistics of extremes. Extremes, 5(4), 387–414.MathSciNetCrossRefGoogle Scholar
  20. Gomes, M. I., L. de Haan, and L. Henriques-Rodrigues. 2008b. Tail index estimation for heavy-tailed models: Accommodation of bias in weighted log-excesses. J. R. Stat. Soci., B70(1), 31–52.MathSciNetzbMATHGoogle Scholar
  21. Gomes, M. I., L. Henriques-Rodrigues, H. Pereira, and D. Pestana. 2010. Tail index and second order parameters’ semi-parametric estimation based on the log-excesses. J. Stat. Comput. Simul., 80(6), 653–666.MathSciNetCrossRefGoogle Scholar
  22. Gomes, M. I., and M. J. Martins. 2002. “Asymptotically unbiased” estimators of the extreme value index based on external estimation of the second order parameter. Extremes, 5(1), 5–31.MathSciNetCrossRefGoogle Scholar
  23. Gomes, M. I., M. J. Martins, and M. M. Neves. 2007. Improving second order reduced-bias extreme value index estimation. Revstat, 5(2), 177–207.MathSciNetzbMATHGoogle Scholar
  24. Hall, P. 1982. On some simple estimates of an exponent of regular variation. J. R. Stat. Soc. B, 44, 37–42.MathSciNetzbMATHGoogle Scholar
  25. Hill, B. M. 1975. A simple general approach to inference about the tail of a distribution. Ann. Stat., 3, 1163–1174.MathSciNetCrossRefGoogle Scholar
  26. R Development Core Team. 2014. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Google Scholar
  27. Reiss, R.-D., and M. Thomas. 2007. Statistical analysis of extreme values, with application to insurance, finance, hydrology and other fields, 3rd ed. Basel, Switzerland: Birkhäuser Verlag.zbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2015

Authors and Affiliations

  1. 1.Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade Nova de LisboaLisboa, CaparicaPortugal
  2. 2.Centro de Estatística e Aplicações, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal

Personalised recommendations