Abstract
In this paper, we build a two-step estimator \(\hat{\gamma}_{\rm STEP}\), which satisfies \(\sqrt{k}(\hat{\gamma}_{\rm STEP}-\hat{\gamma}_{ML})\stackrel{P}{\rightarrow} 0\), where \(\hat{\gamma}_{ML}\) is the well-known maximum likelihood estimator of the extreme value index. Since the two-step estimator \(\hat{\gamma}_{\rm STEP}\) can be calculated easily as a function of the observations, it is much simpler to use in practice. By properly choosing the first step estimator, such as the Pickands estimator, we can even get a shift and scale invariant estimator with the above property.
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Balkema, A.A., de Haan, L.: Residual life time at great age. Ann. Probab. 2, 792–804 (1974)
Beirlant, J., Vynckier, P., Teugels, J.L.: Tail index estimation, pareto quantile plots, and regression diagnostics. J. Am. Stat. Assoc. 91, 1659–1667 (1996)
de Haan, L., Peng, L.: Comparison of tail index estimators. Stat. Neerl. 52(1), 60–70 (1998)
de Haan, L., Stadtmüller, U.: Generalized regular variation of second order. J. Aust. Math. Soc. Ser. A 61, 381–395 (1996)
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)
Drees, H.: On smooth statistical tail functionals. 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)
Gnedenko, B.: Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44, 423–453 (1943)
Grimshaw, S.D.: Computing maximum likelihood estimates for the generalized pareto distribution. Technometrics 35, 185–191 (1993)
Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)
Hosking, J., Wallis, J.: Parameter and quantile estimation for the generalized pareto distribution. Technometrics 29, 339–349 (1987)
Pickands III, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975)
Smith, R.L.: Estimating tails of probability distributions. Ann. Stat. 15, 1174–1207 (1987)
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The author thanks Laurens de Haan for motivating this work and giving helpful comments. The author also thanks two anonymous referees for their useful comments.
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Zhou, C. A two-step estimator of the extreme value index. Extremes 11, 281–302 (2008). https://doi.org/10.1007/s10687-008-0058-2
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DOI: https://doi.org/10.1007/s10687-008-0058-2