Abstract
A new class of estimators of the extreme value index is developed. It has a simple form and is asymptotically very close to the maximum likelihood estimator for a wide class of heavy-tailed models. We also propose an alternative class of estimators, dependent on a tuning parameter p ∈ (0,1) and invariant for changes in both scale and/or location. Such a tuning parameter can help us to choose the number of top order statistics to be used in the estimation of extreme parameters.
Similar content being viewed by others
References
Araújo Santos, P., Fraga Alves, M.I., Gomes, M.I.: Peaks over random threshold methodology for tail index and quantile estimation. Revstat 4(3), 227–247 (2006)
Beirlant, J., Dierckx, G., Guillou, A.: Estimation of the extreme value index and generalized quantile plots. Bernoulli 11(6), 949–970 (2005)
Beirlant, J., Vynckier, P., Teugels, J.: Excess functions and estimation of the extreme-value index. Bernoulli 2, 293–318 (1996)
Billingsley, P.: Probability and Measure. Wiley, New York (1979)
Caeiro, F., Gomes, M.I., Pestana, D.D.: Direct reduction of bias of the classical Hill estimator. Revstat 3(2), 113–136 (2005)
de Haan, L.: On Regular Variation and its Application to Weak Convergence of Sample Extremes. Mathematisch Centrum, Amsterdam (1970)
de Haan, L.: Slow variation and characterization of domains of attraction. In: de Oliveira, T. (ed.) Statistical Extremes and Applications, pp. 31–48. D. Reidel, Dordrecht (1984)
de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering, New York (2006)
Dekkers, A.L.M., Einmahl, J.H.J., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Stat. 17, 1833–1855 (1989)
Draisma, G., de Haan, L., Peng, L., Ferreira, A.: A bootstrap-based method to achieve optimality in estimating the extreme value index. Extremes 2(4), 367–404 (1999)
Drees, H.: On smooth statistical tail functions. Scand. J. Statist. 25, 187–210 (1998)
Drees, H., Ferreira, A., de Haan, L.: On maximum likelihood estimation of the extreme value index. Ann. Appl. Probab. 14, 1179–1201 (2004)
Ferreira, A., de Haan, L., Peng, L.: On optimizing the estimation of high quantiles of a probability distribution. Statistics 37(5), 401–434 (2003)
Fisher, R.A., Tippett, L.H.C.: Limiting forms of the frequency of the largest or smallest member of a sample. Proc. Camb. Philos. Soc. 24, 180–190 (1928)
Fraga Alves, M.I., Gomes M.I., de Haan, L., Neves, C.: A note on second order conditions in extreme value theory: linking general and heavy tails conditions. Revstat 5(3), 285–305 (2007)
Geluk, J., de Haan, L.: Regular Variation, Extensions and Tauberian Theorems. CWI Tract 40, Center for Mathematics and Computer Science, Amsterdam, The Nethelands (1987)
Gomes, M.I., Pestana, D.: A sturdy reduced-bias extreme quantile (VaR) estimator. J. Am. Stat. Assoc. 102(477), 280–292 (2007)
Gomes, M.I., Martins, M.J., Neves, M.: Improving second order reduced bias extreme value index estimation. Revstat 5(2), 177–207 (2007)
Gomes, M.I., Fraga Alves, M.I., Araújo Santos, P.: PORT hill and moment estimators for heavy-tailed models. Commun. Stat., Simul. Comput. 37, 1281–1306 (2008a)
Gomes, M.I., de Haan, L., Henriques Rodrigues, L.: Tail index estimation through accommodation of bias in the weighted log-excesses. J. Royal Stat. Soc. B70(1), 31–52 (2008b)
Hill, B.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)
Hosking, J.R.M., Wallis, J.R.: Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29, 339–349 (1987)
Jenkinson, A.F.: The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Quart. J. Meteorol. Soc. 81, 158–171 (1955)
Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975)
Smirnov, N.: Limit distributions for terms of a variational series. Am. Math. Soc. Transl. Ser. I 11, 82–143 (1952)
Smith, R.: Estimating tails of probability distributions. Ann. Stat. 15, 1174–1207 (1987)
von Mises, R.: La distribution de la plus grande de n valeurs. Revue Math. Union Interbalcanique 1, 141–160 (1936). Reprinted in selected papers of Richard von Mises. Am. Soc. 2, 271–294 (1964)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by FCT / POCTI, POCI, PCDT and PPCDT / FEDER.
Rights and permissions
About this article
Cite this article
Fraga Alves, M.I., Gomes, M.I., de Haan, L. et al. Mixed moment estimator and location invariant alternatives. Extremes 12, 149–185 (2009). https://doi.org/10.1007/s10687-008-0073-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-008-0073-3