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Selecting the optimal sample fraction in univariate extreme value estimation

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  • Drees, Holger
  • Kaufmann, Edgar

Abstract

In 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.

Suggested Citation

  • Drees, Holger & Kaufmann, Edgar, 1998. "Selecting the optimal sample fraction in univariate extreme value estimation," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 149-172, July.
    • Handle: RePEc:eee:spapps:v:75:y:1998:i:2:p:149-172
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    References listed on IDEAS

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    1. Holger Drees, 1998. "On Smooth Statistical Tail Functionals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 187-210, March.
    2. Dekkers, A. L. M. & Dehaan, L., 1993. "Optimal Choice of Sample Fraction in Extreme-Value Estimation," Journal of Multivariate Analysis, Elsevier, vol. 47(2), pages 173-195, November.
    3. Einmahl, J. H.J. & Dekkers, A. L.M. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Other publications TiSEM 81970cb3-5b7a-4cad-9bf6-2, Tilburg University, School of Economics and Management.
    4. 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.
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