IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v75y1998i2p149-172.html
   My bibliography  Save this article

Selecting the optimal sample fraction in univariate extreme value estimation

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(98)00017-9
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wager, Stefan, 2014. "Subsampling extremes: From block maxima to smooth tail estimation," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 335-353.
    2. Geluk, J. L. & Peng, Liang, 2000. "An adaptive optimal estimate of the tail index for MA(l) time series," Statistics & Probability Letters, Elsevier, vol. 46(3), pages 217-227, February.
    3. Frederico Caeiro & M. Ivette Gomes & Björn Vandewalle, 2014. "Semi-Parametric Probability-Weighted Moments Estimation Revisited," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 1-29, March.
    4. Hsieh, Ping-Hung, 2002. "An exploratory first step in teletraffic data modeling: evaluation of long-run performance of parameter estimators," Computational Statistics & Data Analysis, Elsevier, vol. 40(2), pages 263-283, August.
    5. Danielsson, J. & de Haan, L. & Peng, L. & de Vries, C. G., 2001. "Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation," Journal of Multivariate Analysis, Elsevier, vol. 76(2), pages 226-248, February.
    6. Barunik, Jozef & Vacha, Lukas, 2010. "Monte Carlo-based tail exponent estimator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4863-4874.
    7. Fátima Brilhante, M. & Ivette Gomes, M. & Pestana, Dinis, 2013. "A simple generalisation of the Hill estimator," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 518-535.
    8. Igor Fedotenkov, 2020. "A Review of More than One Hundred Pareto-Tail Index Estimators," Statistica, Department of Statistics, University of Bologna, vol. 80(3), pages 245-299.
    9. Marco Rocco, 2011. "Extreme value theory for finance: a survey," Questioni di Economia e Finanza (Occasional Papers) 99, Bank of Italy, Economic Research and International Relations Area.
    10. Christian Schluter, 2021. "On Zipf’s law and the bias of Zipf regressions," Empirical Economics, Springer, vol. 61(2), pages 529-548, August.
    11. Ana Ferreira & Casper G. de Vries, 2004. "Optimal Confidence Intervals for the Tail Index and High Quantiles," Tinbergen Institute Discussion Papers 04-090/2, Tinbergen Institute.
    12. Matheus Henrique Junqueira Saldanha & Adriano Kamimura Suzuki, 2023. "On dealing with the unknown population minimum in parametric inference," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 107(3), pages 509-535, September.
    13. Chao Huang & Jin-Guan Lin & Yan-Yan Ren, 2012. "Statistical Inferences for Generalized Pareto Distribution Based on Interior Penalty Function Algorithm and Bootstrap Methods and Applications in Analyzing Stock Data," Computational Economics, Springer;Society for Computational Economics, vol. 39(2), pages 173-193, February.
    14. de Haan, Laurens & Canto e Castro, Luisa, 2006. "A class of distribution functions with less bias in extreme value estimation," Statistics & Probability Letters, Elsevier, vol. 76(15), pages 1617-1624, September.
    15. Goedele Dierckx & Yuri Goegebeur & Armelle Guillou, 2021. "Local Robust Estimation of Pareto-Type Tails with Random Right Censoring," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 70-108, February.
    16. Einmahl, J.H.J. & Lin, T., 2003. "Asymptotic Normality of Extreme Value Estimators on C[0,1]," Discussion Paper 2003-132, Tilburg University, Center for Economic Research.
    17. Caers, Jef & Dyck, Jozef Van, 1998. "Nonparametric tail estimation using a double bootstrap method," Computational Statistics & Data Analysis, Elsevier, vol. 29(2), pages 191-211, December.
    18. Necir, Abdelhakim & Meraghni, Djamel, 2009. "Empirical estimation of the proportional hazard premium for heavy-tailed claim amounts," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 49-58, August.
    19. Małgorzata Just & Krzysztof Echaust, 2021. "An Optimal Tail Selection in Risk Measurement," Risks, MDPI, vol. 9(4), pages 1-16, April.
    20. Krzysztof Echaust & Małgorzata Just, 2020. "Value at Risk Estimation Using the GARCH-EVT Approach with Optimal Tail Selection," Mathematics, MDPI, vol. 8(1), pages 1-24, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:75:y:1998:i:2:p:149-172. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.