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

Abstract

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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.

How to cite

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Frederico 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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  1. [1] F. Caeiro, M.I. Gomes and D.D. Pestana, Direct reduction of bias of the classical Hill estimator, Revstat 3 (2) (2005), 113-136. Zbl1108.62049
  2. [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. [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. [4] M.I. Fraga Alves, Weiss-Hill estimator, Test 10 (2001), 203-224. 
  5. [5] B.V. Gnedenko, Sur la distribution limite du terme maximum d'une série aléatoire, Ann. Math. 44 (1943), 423-453. Zbl0063.01643
  6. [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. [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. [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. [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. [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. [11] L. de Haan and A. Ferreira, Extreme Value Theory: an Introduction, Springer, LLC New York 2006. Zbl1101.62002
  12. [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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