# An asymptotically unbiased moment estimator of a negative extreme value index

Frederico Caeiro; M. Ivette Gomes

Discussiones Mathematicae Probability and Statistics (2010)

- Volume: 30, Issue: 1, page 5-19
- ISSN: 1509-9423

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topFrederico Caeiro, and M. Ivette Gomes. "An asymptotically unbiased moment estimator of a negative extreme value index." Discussiones Mathematicae Probability and Statistics 30.1 (2010): 5-19. <http://eudml.org/doc/277012>.

@article{FredericoCaeiro2010,

abstract = {In this paper we consider a new class of consistent semi-parametric estimators of a negative extreme value index, based on the set of the k largest observations. This class of estimators depends on a control or tuning parameter, which enables us to have access to an estimator with a null second-order component of asymptotic bias, and with a rather interesting mean squared error, as a function of k. We study the consistency and asymptotic normality of the proposed estimators. Their finite sample behaviour is obtained through Monte Carlo simulation.},

author = {Frederico Caeiro, M. Ivette Gomes},

journal = {Discussiones Mathematicae Probability and Statistics},

keywords = {extreme value index; semi-parametric estimation; moment estimator},

language = {eng},

number = {1},

pages = {5-19},

title = {An asymptotically unbiased moment estimator of a negative extreme value index},

url = {http://eudml.org/doc/277012},

volume = {30},

year = {2010},

}

TY - JOUR

AU - Frederico Caeiro

AU - M. Ivette Gomes

TI - An asymptotically unbiased moment estimator of a negative extreme value index

JO - Discussiones Mathematicae Probability and Statistics

PY - 2010

VL - 30

IS - 1

SP - 5

EP - 19

AB - In this paper we consider a new class of consistent semi-parametric estimators of a negative extreme value index, based on the set of the k largest observations. This class of estimators depends on a control or tuning parameter, which enables us to have access to an estimator with a null second-order component of asymptotic bias, and with a rather interesting mean squared error, as a function of k. We study the consistency and asymptotic normality of the proposed estimators. Their finite sample behaviour is obtained through Monte Carlo simulation.

LA - eng

KW - extreme value index; semi-parametric estimation; moment estimator

UR - http://eudml.org/doc/277012

ER -

## References

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- [2] A.L.M. Dekkers, J.H.J. Einmahl and L. de Haan, A moment estimator for the index of an extreme-value distribution, The Annals of Statistics 17 (4) (1989), 1833-1855. Zbl0701.62029
- [3]G. Draisma, L. de Haan, L. Peng and T. Themido Pereira, A bootstrap-based method to achieve optimality in estimating the extreme value index, Extremes 2 (4) (1999), 367-404. Zbl0972.62014
- [4] M.I. Fraga Alves, Weiss-Hill estimator, Test 10 (2001), 203-224.
- [5] B.V. Gnedenko, Sur la distribution limite du terme maximum d'une série aléatoire, Ann. Math. 44 (1943), 423-453. Zbl0063.01643
- [6] M.I. Gomes, L. de Haan and L. Henriques Rodrigues, Tail Index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses, J. R. Stat. Soc. Ser. B 70 (1) (2008), 31-52. Zbl05563342
- [7] M.I. Gomes, M.J. Martins and M.M. Neves, Improving second order reduced-bias tail index estimation, Revstat 5 (2) (2007), 177-207. Zbl05217613
- [8] M.I. Gomes and C. Neves, Asymptotic comparison of the mixed moment and classical extreme value index estimators, Statistics & Probability Letters 78 (2008), 643-653. Zbl05268494
- [9] M.I. Gomes and O. Oliveira, The bootstrap methodology in Statistics of Extremes - choice of the optimal sample fraction, Extremes 4 (4) (2001), 331-358. Zbl1023.62048
- [10] L. de Haan, On Regular Variation and its Application to the Weak Convergence of Sample Extremes, Mathematical Centre Tract 32, Amesterdam 1970. Zbl0226.60039
- [11] L. de Haan and A. Ferreira, Extreme Value Theory: an Introduction, Springer, LLC New York 2006. Zbl1101.62002
- [12] B.M. Hill, A Simple General Approach to Inference About the Tail of a Distribution, The Annals of Statistics 3 (5) (1975), 1163-1174. Zbl0323.62033

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