Abstract
Motivated by Fraga Alves (Extremes 4:199–217, 2001)’s work, a new class of location invariant Hill-type estimators for the tail index of a heavy tailed distribution is proposed in the paper. Its asymptotic behavior is derived, and the optimal choice of the sample fraction is discussed by mean squared error. Asymptotic comparisons and simulation studies are presented to show that the new estimator performs well compared to the known ones.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beirlant, J., Dierckx, G., Guillou, A., Stărică, C.: On exponential representions of log-spacings of extreme order statistics. Extremes 5, 157–180 (2002)
Caeiro, F., Gomes, M.I.: A class of asymptotically unbiansed semi-parametric estimators of the tail index. Test 11, 345–364 (2002)
de Haan, L., Ferreira, A.: Extreme Value Theory. Springer, New York (2006)
de Haan, L., Peng, L.: Comparison of tail index estimators. Stat. Neerl. 52, 60–70 (1998)
Dekkers, A.L.M., Einmahl, J.H.J., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Stat. 4, 1833–1855 (1989)
Fraga Alves, M.I.: A location invariant Hill-type estimator. Extremes 4, 199–217 (2001)
Fraga Alves, M.I., Gomes, M.I., de Haan, L.: A new class of sem-parametric estimators of the second order parameter. Port. Math. 60, 193–213 (2003)
Fraga Alves, M.I., Gomes, M.I., de Haan, L., Neves, C.: Mixed moment estimator and location invariant alternatives. Extremes (2008). doi:10.1007/s10687-008-0073-3
Gomes, M. I, de Haan, L., Henriques Rodrigues, L.: Tail index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses. J. R. Stat. Soc., Ser. B 70, 31–53 (2008)
Gomes, M.I., de Haan, L., Peng, L.: Semi-parametric estimation of second order parameter in statistics of extremes. Extremes 5, 387–414 (2002)
Gomes, M.I., Henriques Rodrigues, L.: Tail index estimation for heavy tails: accommodation of bias in the excesses over a high threshold. Extremes (2008). doi:10.1007/s10687-008-0059-1
Gomes, M.I., Martins, M.J.: Generalizations of the Hill estimator-asymptotic versus finite sample behavior. J. Stat. Plan. Inference 93, 161–180 (2001)
Gomes, M.I., Martins, M.J.: “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. Extremes 5, 5–31 (2002)
Gomes, M.I., Martins, M.J.: Bias reduction and explicit estimation of the tail index. J. Stat. Plan. Inference 124, 361–378 (2004)
Gomes, M.I., Martins, M.J., Neves, M.: Alternatives to a semi-parametric estimator of parameters of rare events—the Jackknife methodology. Extremes 3, 207–229 (2000)
Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)
Li, J., Peng, Z., Nadarajah, S.: A class of unbiased location invariant Hill-type estimators for heavy tailed distributions. Electron. J. Stat. 2, 829–847 (2008)
Ling, C., Peng, Z., Nadarajah, S.: A location invariant moment-type estimator (I). Theory Probab. Math. Stat. 76, 22–30 (2007a)
Ling, C., Peng, Z., Nadarajah, S.: A location invariant moment-type estimator (II). Theory Probab. Math. Stat. 77, 167–178 (2007b)
Peng, L.: Asymptotically unbiased estimator for the extreme-value index. Stat. Probab. Lett. 2, 107–115 (1998)
Peng, L., Qi, Y.: A new calibration method of constructing empirical likelihood-based confidence intervals for the tail index. Aust. N. Z. J. Stat. 48, 59–66 (2006a)
Peng, L., Qi, Y.: Confidence regions for high quantiles of a heavy tailed distribution. Ann. Stat. 4, 1964–1986 (2006b)
Qi, Y.: On the tail index of a heavy tailed distribution. Ann. Inst. Stat. Math. (2008). doi:10.1007/s10463-008-0176-2
Rodriguez, R.N.: A guide to the Burr type XII distributions. Biometrics 64, 129–134 (1977)
Segers, J.: Residual estimators. J. Stat. Plan. Inference 98, 15–27 (2001)
Tadikamalla, P.R.: A look at the Burr and related distributions. Int. Stat. Rev. 48, 337–344 (1980)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, J., Peng, Z. & Nadarajah, S. Asymptotic normality of location invariant heavy tail index estimator. Extremes 13, 269–290 (2010). https://doi.org/10.1007/s10687-009-0088-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-009-0088-4