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

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  • 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
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    References listed on IDEAS

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    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.
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    Cited by:

    1. 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.
    2. 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.
    3. 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).
    4. 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.
    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. 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.
    7. 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.
    8. 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.
    9. 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.

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