Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beirlant, J., G. Dierckx, Y. Goegebeur, and G. Matthys. 1999. Tail index estimation and an exponential regression model. Extremes, 2, 177–200.
Beirlant J., Y. Goegebeur, J. Segers, and J. Teugels. 2004. Statistics of extremes. Theory and applications. Hoboken, NJ: Wiley.
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.
Caeiro, F., and M. I. Gomes. 2006. A new class of estimators of a “scale” second order parameter. Extremes, 9, 193–211.
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.
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.
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.
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.
Caeiro, F., M. I. Gomes, and D. D. Pestana. 2005. Direct reduction of bias of the classical Hill estimator. Revstat, 3(2), 111–136.
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.
Ciuperca, G., and C. Mercadier. 2010. Semi-parametric estimation for heavy tailed distributions. Extremes, 13(1), 55–87.
de Haan, L., and L. Peng. 1998. Comparison of extreme value index estimators. Stat. Neerland., 52, 60–70.
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.
Feuerverger, A., and P. Hall. 1999. Estimating a tail exponent by modelling departure from a Pareto distribution. Ann. Stat., 27, 760–781.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Hall, P. 1982. On some simple estimates of an exponent of regular variation. J. R. Stat. Soc. B, 44, 37–42.
Hill, B. M. 1975. A simple general approach to inference about the tail of a distribution. Ann. Stat., 3, 1163–1174.
R Development Core Team. 2014. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Caeiro, F., Gomes, M.I. Revisiting the Maximum Likelihood Estimation of a Positive Extreme Value Index. J Stat Theory Pract 9, 200–218 (2015). https://doi.org/10.1080/15598608.2014.909754
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1080/15598608.2014.909754