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Asymptotically unbiased estimators for the extreme-value index

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  • Peng, L.

Abstract

Estimators of the extreme-value index are based on a set of upper order statistics. When the number of upper-order statistics used in the estimation of the extreme-value index is small, the variance of the estimator will be large. On the other hand, the use of a large number of upper statistics will introduce a big bias. There are several papers concerning how to balance the variance component and the bias component. In this paper, we give an unbiased estimator even if one uses a large number of upper-order statistics.

Suggested Citation

  • Peng, L., 1998. "Asymptotically unbiased estimators for the extreme-value index," Statistics & Probability Letters, Elsevier, vol. 38(2), pages 107-115, June.
    • Handle: RePEc:eee:stapro:v:38:y:1998:i:2:p:107-115
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    References listed on IDEAS

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    1. Laurens F.M. de Haan & Liang Peng & T.T. Pereira, 1997. "A Bootstrap-based Method to Achieve Optimality in Estimating the Extreme-value Index," Tinbergen Institute Discussion Papers 97-099/4, Tinbergen Institute.
    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. 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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    1. Cai, J., 2012. "Estimation concerning risk under extreme value conditions," Other publications TiSEM a92b089f-bc4c-41c2-b297-c, Tilburg University, School of Economics and Management.
    2. Wendy Shinyie & Noriszura Ismail & Abdul Jemain, 2014. "Semi-parametric Estimation Based on Second Order Parameter for Selecting Optimal Threshold of Extreme Rainfall Events," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 28(11), pages 3489-3514, September.
    3. Roman Matkovskyy, 2020. "A measurement of affluence and poverty interdependence across countries: Evidence from the application of tail copula," Bulletin of Economic Research, Wiley Blackwell, vol. 72(4), pages 404-416, October.
    4. Araújo Santos, Paulo & Fraga Alves, Isabel & Hammoudeh, Shawkat, 2013. "High quantiles estimation with Quasi-PORT and DPOT: An application to value-at-risk for financial variables," The North American Journal of Economics and Finance, Elsevier, vol. 26(C), pages 487-496.
    5. Bucher, Axel & Segers, Johan, 2015. "Maximum likelihood estimation for the Frechet distribution based on block maxima extracted from a time series," LIDAM Discussion Papers ISBA 2015023, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. 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.
    7. Wolfgang Glänzel & Henk F. Moed, 2013. "Opinion paper: thoughts and facts on bibliometric indicators," Scientometrics, Springer;Akadémiai Kiadó, vol. 96(1), pages 381-394, July.
    8. El Methni, Jonathan & Stupfler, Gilles, 2018. "Improved estimators of extreme Wang distortion risk measures for very heavy-tailed distributions," Econometrics and Statistics, Elsevier, vol. 6(C), pages 129-148.
    9. Maarten R C van Oordt & Chen Zhou, 2019. "Estimating Systematic Risk under Extremely Adverse Market Conditions," Journal of Financial Econometrics, Oxford University Press, vol. 17(3), pages 432-461.
    10. Beran, Jan & Schell, Dieter, 2012. "On robust tail index estimation," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3430-3443.
    11. Wager, Stefan, 2014. "Subsampling extremes: From block maxima to smooth tail estimation," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 335-353.
    12. Tsourti, Zoi & Panaretos, John, 2003. "Extreme Value Index Estimators and Smoothing Alternatives: A Critical Review," MPRA Paper 6390, University Library of Munich, Germany.
    13. Liu, Qing & Peng, Liang & Wang, Xing, 2017. "Haezendonck–Goovaerts risk measure with a heavy tailed loss," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 28-47.
    14. Djamel Meraghni & Abdelhakim Necir & Louiza Soltane, 2025. "Nelson-Aalen Tail Product-limit Process and Extreme Value Index Estimation Under Random Censorship," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 87(2), pages 526-574, August.
    15. Djamel Meraghni & Abdelhakim Necir, 2007. "Estimating the Scale Parameter of a Lévy-stable Distribution via the Extreme Value Approach," Methodology and Computing in Applied Probability, Springer, vol. 9(4), pages 557-572, December.
    16. Gomes, M. Ivette & Brilhante, M. Fátima & Caeiro, Frederico & Pestana, Dinis, 2015. "A new partially reduced-bias mean-of-order p class of extreme value index estimators," Computational Statistics & Data Analysis, Elsevier, vol. 82(C), pages 223-237.
    17. M. Ivette Gomes & Armelle Guillou, 2015. "Extreme Value Theory and Statistics of Univariate Extremes: A Review," International Statistical Review, International Statistical Institute, vol. 83(2), pages 263-292, August.
    18. Cui, Hengxin & Tan, Ken Seng & Yang, Fan & Zhou, Chen, 2022. "Asymptotic analysis of portfolio diversification," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 302-325.

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