IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v100y2009i4p794-815.html
   My bibliography  Save this article

Existence and consistency of the maximum likelihood estimator for the extreme value index

Author

Listed:
  • Zhou, Chen

Abstract

The paper is about the asymptotic properties of the maximum likelihood estimator for the extreme value index. Under the second order condition, Drees et al. [H. Drees, A. Ferreira, L. de Haan, On maximum likelihood estimation of the extreme value index, Ann. Appl. Probab. 14 (2004) 1179-1201] proved asymptotic normality for any solution of the likelihood equations (with shape parameter [gamma]>-1/2) that is not too far off the real value. But they did not prove that there is a solution of the equations satisfying the restrictions. In this paper, the existence is proved, even for [gamma]>-1. The proof just uses the domain of attraction condition (first order condition), not the second order condition. It is also proved that the estimator is consistent. When the second order condition is valid, following the current proof, the existence of a solution satisfying the restrictions in the above-cited reference is a direct consequence.

Suggested Citation

  • Zhou, Chen, 2009. "Existence and consistency of the maximum likelihood estimator for the extreme value index," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 794-815, April.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:4:p:794-815
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(08)00187-5
    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. 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.
    2. Einmahl, J. H. & Mason, D. M., 1988. "Strong limit theorems for weighted quantile processes," Other publications TiSEM 4bbe972d-b641-42a4-b2b8-0, Tilburg University, School of Economics and Management.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. Deyuan Li & Liang Peng & Yongcheng Qi, 2011. "Empirical likelihood confidence intervals for the endpoint of a distribution function," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 353-366, August.
    3. Mavis Pararai & Broderick O. Oluyede & Gayan Warahena-Liyanage, 2016. "The Beta Lindley-Poisson Distribution with Applications," Journal of Statistical and Econometric Methods, SCIENPRESS Ltd, vol. 5(4), pages 1-1.
    4. 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.
    5. Ahmed, Hanan, 2022. "Extreme value statistics using related variables," Other publications TiSEM 246f0f13-701c-4c0d-8e09-e, Tilburg University, School of Economics and Management.
    6. Zhou, Chen, 2010. "The extent of the maximum likelihood estimator for the extreme value index," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 971-983, April.
    7. Wolfgang Kössler & Janine Ott, 2019. "Two-sided variable inspection plans for arbitrary continuous populations with unknown distribution," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 103(3), pages 437-452, September.
    8. Oorschot, Jochem & Segers, Johan & Zhou, Chen, 2022. "Tail inference using extreme U-statistics," LIDAM Discussion Papers ISBA 2022014, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. Broderick O. Oluyede & Boikanyo Makubate & Adeniyi F. Fagbamigbe & Precious Mdlongwa, 2018. "A New Burr XII-Weibull-Logarithmic Distribution for Survival and Lifetime Data Analysis: Model, Theory and Applications," Stats, MDPI, vol. 1(1), pages 1-15, June.

    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. Hüsler, Jürg & Li, Deyuan & Müller, Samuel, 2006. "Weighted least squares estimation of the extreme value index," Statistics & Probability Letters, Elsevier, vol. 76(9), pages 920-930, May.
    2. de Valk, Cees, 2016. "A large deviations approach to the statistics of extreme events," Other publications TiSEM 117b3ba0-0e40-4277-b25e-d, Tilburg University, School of Economics and Management.
    3. 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.
    4. 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.
    5. 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.
    6. Phornchanok Cumperayot & Casper G. de Vries, 2006. "Large Swings in Currencies driven by Fundamentals," Tinbergen Institute Discussion Papers 06-086/2, Tinbergen Institute.
    7. Einmahl, J.H.J. & de Haan, L.F.M. & Krajina, A., 2009. "Estimating Extreme Bivariate Quantile Regions," Other publications TiSEM 007ce0a9-dd94-4301-ad62-1, Tilburg University, School of Economics and Management.
    8. Estate Khmaladze & Wolfgang Weil, 2008. "Local empirical processes near boundaries of convex bodies," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(4), pages 813-842, December.
    9. Einmahl, John H.J. & de Haan, Laurens & Sinha, Ashoke Kumar, 1997. "Estimating the spectral measure of an extreme value distribution," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 143-171, October.
    10. Xue-Zhong He & Youwei Li, 2017. "The adaptiveness in stock markets: testing the stylized facts in the DAX 30," Journal of Evolutionary Economics, Springer, vol. 27(5), pages 1071-1094, November.
    11. Einmahl, J.H.J. & Li, Jun & Liu, Regina, 2015. "Bridging Centrality and Extremity : Refining Empirical Data Depth using Extreme Value Statistics," Discussion Paper 2015-020, Tilburg University, Center for Economic Research.
    12. Ghosh, Souvik & Resnick, Sidney, 2010. "A discussion on mean excess plots," Stochastic Processes and their Applications, Elsevier, vol. 120(8), pages 1492-1517, August.
    13. Wager, Stefan, 2014. "Subsampling extremes: From block maxima to smooth tail estimation," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 335-353.
    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. Charnchai Leuwattanachotinan & Casper G. de Vries, 2015. "Extreme Linkages in Financial Markets: Macro Shocks and Systemic Risk," PIER Discussion Papers 2, Puey Ungphakorn Institute for Economic Research.
    16. Zhu, Sha & Dekker, Rommert & van Jaarsveld, Willem & Renjie, Rex Wang & Koning, Alex J., 2017. "An improved method for forecasting spare parts demand using extreme value theory," European Journal of Operational Research, Elsevier, vol. 261(1), pages 169-181.
    17. Daouia, Abdelaati & Laurent, Thibault & Noh, Hohsuk, 2017. "npbr: A Package for Nonparametric Boundary Regression in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 79(i09).
    18. Engle, Robert F. & Manganelli, Simone, 2001. "Value at risk models in finance," Working Paper Series 75, European Central Bank.
    19. 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.
    20. Christian Schluter, 2018. "Top Incomes, Heavy Tails, and Rank-Size Regressions," Econometrics, MDPI, vol. 6(1), pages 1-16, March.

    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:jmvana:v:100:y:2009:i:4:p:794-815. 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/622892/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.