Selecting the optimal sample fraction in univariate extreme value estimation
AbstractIn general, estimators of the extreme value index of i.i.d. random variables crucially depend on the sample fraction that is used for estimation. In case of the well-known Hill estimator the optimal number knopt of largest order statistics was given by Hall and Welsh (1985) as a function of some parameters of the unknown distribution function F, which was assumed to admit a certain expansion. Moreover, an estimator of knopt was proposed that is consistent if a second-order parameter [rho] of F belongs to a bounded interval. In contrast, we introduce a sequential procedure that yields a consistent estimator of knopt in the full model without requiring prior information about [rho]. Then it is demonstrated that even in a more general setup the resulting adaptive Hill estimator is asymptotically as efficient as the Hill estimator based on the optimal number of order statistics. Finally, it is shown by Monte Carlo simulations that also for moderate sample sizes the procedure shows a reasonable performance, which can be improved further if [rho] is restricted to bounded intervals.
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Bibliographic InfoArticle provided by Elsevier in its journal Stochastic Processes and their Applications.
Volume (Year): 75 (1998)
Issue (Month): 2 (July)
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Web page: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description
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- Hall, Peter, 1990. "Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 177-203, February.
- Einmahl, J. & Dekkers, A. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Open Access publications from Tilburg University urn:nbn:nl:ui:12-125712, Tilburg University.
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