Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



A location invariant moment-type estimator II

Authors: Cheng-Xiu Ling, Zuoxiang Peng and Saralees Nadarajah
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 77 (2007).
Journal: Theor. Probability and Math. Statist. 77 (2008), 177-189
MSC (2000): Primary 60F99
Published electronically: January 21, 2009
MathSciNet review: 2432781
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The moment estimator (Dekkers et al. (1989)) has been used in extreme value theory to estimate the tail index, but it is not location invariant. The location invariant Hill-type estimator (Fraga Alves (2001)) is only suitable for estimating positive indices. In this paper, a new moment-type estimator is studied, which is location invariant. This new estimator is based on the original moment-type estimator, but it is made location invariant by a random shift. Its asymptotic normality is derived, in a semiparametric setup.

References [Enhancements On Off] (What's this?)

  • 1. S. Cheng and J. Pan, Asymptotic expansions of estimators for the tail index with applications, Scand. J. Statist. 25 (1998), 717-728. MR 1666796 (2000d:62079)
  • 2. A. Cuntz, E. Haeusler, and J. Segers, Edgeworth expansions for the distribution function of the Hill estimator, Discussion Paper, vol. 8, Center for Economic Research, Tilburg University, 2003.
  • 3. L. de Haan and U. Statmüller, Generalized regular variation of second order, J. Austral. Math. Soc. Ser. A 61 (1996), 381-395. MR 1420345 (97g:26001)
  • 4. A. L. M. Dekkers, J. H. J. Einmahl, and L. De Haan, A moment estimator for the index of an extreme-value distribution, Ann. Statist. 17 (1989), 1833-1855. MR 1026315 (91i:62033)
  • 5. H. Drees, A general class of estimators of the extreme value index, J. Statist. Plann. Inference 66 (1998), 95-112. MR 1616999 (99c:62085)
  • 6. Z. Fan, Estimation problems for distributions with heavy tails, J. Statist. Plann. Inference 123 (2004), 13-40. MR 2058119 (2005c:62059)
  • 7. M. I. Fraga Alves, A location invariant Hill-type estimator, Extremes 3 (2001), 199-217. MR 1907061 (2003d:62129)
  • 8. M. I. Gomes and J. Martins, Generalizations of the Hill estimator-asymptotic versus finite sample behavior, J. Statist. Plann. Inference 93 (2001), 161-180. MR 1822394
  • 9. M. I. Gomes and O. Oliveira, Censoring estimators of a positive tail index, Statist. Probab. Lett. 65 (2003), 147-159. MR 2018025 (2005d:62155)
  • 10. P. Hall, On estimating the endpoint of a distribution, Ann. Statist. 3 (1982), 556-568. MR 653530 (83f:62043)
  • 11. B. M. Hill, A simple general approach to inference about the tail of a distribution, Ann. Statist. 3 (1975), 1163-1174. MR 0378204 (51:14373)
  • 12. J. Pan, Some results on estimation of the tail index of a distribution, Chin. Ann. Math. Ser. B 19 (1998), 239-248. MR 1655939 (2000a:62037)
  • 13. L. Peng, Asymptotically unbiased estimators for the extreme value index, Statist. Probab. Lett. 38 (1998), 107-115. MR 1627906 (99e:62056)
  • 14. Z. Peng, Extension of Pickands' estimator, Acta Math. Sinica (Chin. Ser.) 40 (1997), 759-762. MR 1612631 (99c:62111)
  • 15. V. V. Petrov, Sums of Independent Random Variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 82, Springer-Verlag, New York-Heidelberg, 1975. MR 0388499 (52:9335)
  • 16. J. Pickands, Statistical inference using extreme order statistics, Ann. Statist. 3 (1975), 119-131. MR 0423667 (54:11642)
  • 17. Y. Qi and S. Cheng, Convergence of Pickands-type estimators, Chinese Sci. Bull. 37 (1992), 1409-1413.
  • 18. J. Segers, Abelian and Tauberian theorems for the bias of the Hill estimator, Scand. J. Statist. 29 (2001), 461-483. MR 1925570 (2003k:62159)
  • 19. J. Segers, Generalized Pickands estimators for the extreme value index, J. Statist. Plann. Inference 128 (2005), 381-396. MR 2102765
  • 20. R. L. Smith, Maximum likelihood estimation in a class of nonregular cases, Biometrika 72 (1985), 67-90. MR 790201 (86k:62053)
  • 21. R. L. Smith, Estimation of tails of probability distribution, Ann. Statist. 15 (1987), 1174-1207. MR 902252 (88j:62096)
  • 22. R. L. Smith and I. Weissman, Maximum likelihood estimation of the lower tail of probability distribution, J. Roy. Statist. Soc. Ser. B 47 (1985), 285-298. MR 816094 (87b:62035)
  • 23. Z. Tsourti and I. Panaretos, Extreme value index estimators and smoothing alternatives: review and simulation comparison, Technical Report, vol. 149, Department of Statistics, Athens University of Economics and Business, 2001.
  • 24. X. Wang and S. Cheng, General regular variation of $ n$th order and the second order Edgeworth expansion of the extreme value distribution. I, Acta Math. Sin. (English Ser.) 21 (2005), no. 5, 1121-1130. MR 2176324 (2006i:60063)
  • 25. S. Yun, On a general Pickands estimator of extreme value index, J. Statist. Plann. Inference 102 (2002), 389-409. MR 1896495 (2003e:62050)

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60F99

Retrieve articles in all journals with MSC (2000): 60F99

Additional Information

Cheng-Xiu Ling
Affiliation: Department of Mathematics, Southwest Normal University, Chongqing 400715, People’s Republic of China

Zuoxiang Peng
Affiliation: Department of Mathematics, Southwest Normal University, Chongqing 400715, People’s Republic of China

Saralees Nadarajah
Affiliation: Department of Statistics, University of Nebraska–Lincoln, Lincoln, Nebraska 68583

Keywords: Extreme value index, location invariant property, moment estimation, asymptotic normality, order statistics, regular varying functions
Received by editor(s): November 29, 2005
Published electronically: January 21, 2009
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society