A location invariant moment-type estimator II

Ling, Cheng-Xiu; Peng, Zuoxiang; Nadarajah, Saralees

doi:10.1090/S0094-9000-09-00756-X

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
DOI: https://doi.org/10.1090/S0094-9000-09-00756-X
Published electronically: January 21, 2009
MathSciNet review: 2432781
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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.

1. Shihong Cheng and Jiazhu Pan, Asymptotic expansions of estimators for the tail index with applications, Scand. J. Statist. 25 (1998), no. 4, 717–728. MR 1666796, https://doi.org/10.1111/1467-9469.00131
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. Laurens de Haan and Ulrich Stadtmüller, Generalized regular variation of second order, J. Austral. Math. Soc. Ser. A 61 (1996), no. 3, 381–395. MR 1420345
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), no. 4, 1833–1855. MR 1026315, https://doi.org/10.1214/aos/1176347397
5. Holger Drees, A general class of estimators of the extreme value index, J. Statist. Plann. Inference 66 (1998), no. 1, 95–112. MR 1616999, https://doi.org/10.1016/S0378-3758(97)00076-1
6. Zhaozhi Fan, Estimation problems for distributions with heavy tails, J. Statist. Plann. Inference 123 (2004), no. 1, 13–40. MR 2058119, https://doi.org/10.1016/S0378-3758(03)00142-3
7. M. I. Fraga Alves, A location invariant Hill-type estimator, Extremes 4 (2001), no. 3, 199–217 (2002). MR 1907061, https://doi.org/10.1023/A:1015226104400
8. M. Ivette Gomes and M. João Martins, Generalizations of the Hill estimator-asymptotic versus finite sample behaviour, J. Statist. Plann. Inference 93 (2001), no. 1-2, 161–180. MR 1822394, https://doi.org/10.1016/S0378-3758(00)00201-9
9. M. Ivette Gomes and Orlando Oliveira, Censoring estimators of a positive tail index, Statist. Probab. Lett. 65 (2003), no. 3, 147–159. MR 2018025, https://doi.org/10.1016/j.spl.2003.07.011
10. Peter Hall, On estimating the endpoint of a distribution, Ann. Statist. 10 (1982), no. 2, 556–568. MR 653530
11. Bruce M. Hill, A simple general approach to inference about the tail of a distribution, Ann. Statist. 3 (1975), no. 5, 1163–1174. MR 0378204
12. Jiazhu Pan, Some results on estimation of the tail index of a distribution, Chinese Ann. Math. Ser. B 19 (1998), no. 2, 239–248. A Chinese summary appears in Chinese Ann. Math. Ser. A 19 (1998), no. 2, 284. MR 1655939
13. L. Peng, Asymptotically unbiased estimators for the extreme-value index, Statist. Probab. Lett. 38 (1998), no. 2, 107–115. MR 1627906, https://doi.org/10.1016/S0167-7152(97)00160-0
14. Zuo Xiang Peng, An extension of a Pickands-type estimator, Acta Math. Sinica (Chin. Ser.) 40 (1997), no. 5, 759–762 (Chinese, with English and Chinese summaries). MR 1612631
15. V. V. Petrov, Sums of independent random variables, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. MR 0388499
16. James Pickands III, Statistical inference using extreme order statistics, Ann. Statist. 3 (1975), 119–131. MR 0423667
17. Y. Qi and S. Cheng, Convergence of Pickands-type estimators, Chinese Sci. Bull. 37 (1992), 1409-1413.
18. Johan Segers, Abelian and Tauberian theorems on the bias of the Hill estimator, Scand. J. Statist. 29 (2002), no. 3, 461–483. MR 1925570, https://doi.org/10.1111/1467-9469.00301
19. Johan Segers, Generalized Pickands estimators for the extreme value index, J. Statist. Plann. Inference 128 (2005), no. 2, 381–396. MR 2102765, https://doi.org/10.1016/j.jspi.2003.11.004
20. Richard L. Smith, Maximum likelihood estimation in a class of nonregular cases, Biometrika 72 (1985), no. 1, 67–90. MR 790201, https://doi.org/10.1093/biomet/72.1.67
21. Richard L. Smith, Estimating tails of probability distributions, Ann. Statist. 15 (1987), no. 3, 1174–1207. MR 902252, https://doi.org/10.1214/aos/1176350499
22. Richard L. Smith and Ishay Weissman, Maximum likelihood estimation of the lower tail of a probability distribution, J. Roy. Statist. Soc. Ser. B 47 (1985), no. 2, 285–298. MR 816094
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. Xiao Qian Wang and Shi Hong Cheng, General regular variation of 𝑛-th order and the 2nd order Edgeworth expansion of the extreme value distribution. I, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 5, 1121–1130. MR 2176324, https://doi.org/10.1007/s10114-004-0486-0
25. Seokhoon Yun, On a generalized Pickands estimator of the extreme value index, J. Statist. Plann. Inference 102 (2002), no. 2, 389–409. Silver jubilee issue. MR 1896495, https://doi.org/10.1016/S0378-3758(01)00100-8