Theory of Probability and Mathematical Statistics

Author(s): Cheng-Xiu Ling; Zuoxiang Peng; Saralees Nadarajah
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, vipusk 76 (2007).
Journal: Theor. Probability and Math. Statist. No. 76 (2008), 23-31.
MSC (2000): Primary 60F99
Posted: July 10, 2008
Retrieve article in: PDF DVI PostScript

Abstract: The moment's 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 to estimate 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 is made location invariant by a random shift. Its weak consistency and strong consistency are derived, in a semiparametric setup.

1.: S. Cheng and L. de Haan, Penultimate approximation for Hill's estimator, Scand. J. Statist. 28 (2001), 569-575. MR 1858418 (2002h:62156)
2.: 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)
3.: A. Cuntz, E. Haeusler, and J. Segers, Edgeworth expansions for the distribution function of the Hill estimator, Discussion paper from Tilburg University, Center for Economic Research (2003), no. 8.
4.: L. de Haan, Slow variation and characterization of domains of attraction, Statistical Extremes and Applications, Vimeiro, 1983, Reidel, Dordrecht, 1984, pp. 31-48. MR 784812 (87b:60036)
5.: L. de Haan and U. Stadtmüller, Generalized regular variation of second order, J. Aust. Math. Soc., A 61 (1996), 381-395. MR 1420345 (97g:26001)
6.: A. L. M. Dekkers, J. H. J. Einmahl, and L. de Haan, A moment estimator for the index of an extreme-value distribution, Ann. Stat. 17 (1989), 1833-1855. MR 1026315 (91i:62033)
7.: H. Drees Refined Pickands estimators of the extreme value index, Ann. Statist. 23 (1995), 2059-2080. MR 1389865 (97d:62069)
8.: H. Drees, A general class of estimators of the extreme value index, J. Statist. Plann. Inference 66 (1998), 95-112. MR 1616999 (99c:62085)
9.: H. Drees and E. Kaufmann, Selecting the optimal sample fraction in univariate extreme value index, Stochastic Process. Appl. 75 (1998), 149-172. MR 1632189 (99h:62034)
10.: Z. Fan, Estimation problems for distributions with heavy tails, J. Statist. Plann. Inference 123 (2004), 13-40. MR 2058119 (2005c:62059)
11.: M. I. Fraga Alves, A location invariant Hill-type estimator, Extremes 4 (2001), 199-217. MR 1907061 (2003d:62129)
12.: J. L. Geluk and L. de Haan, Regular variation, extensions, and Tauberian theorems, CWI Tract 40 (1987). MR 906871 (89a:26002)
13.: M. I. Gomes and M. J. Martins, Generalizations of the Hill estimator-asymptotic versus finite sample behavior, J. Statist. Plann. Inference 93 (2001), 161-180. MR 1822394
14.: M. I. Gomes and O. Oliveira, Censoring estimators of a positive tail index, Statist. Probab. Lett. 65 (2003), 147-159. MR 2018025 (2005d:62155)
15.: P. Hall, On estimating the endpoint of a distribution, Ann. Statist. 10 (1982), 556-568. MR 653530 (83f:62043)
16.: 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)
17.: J. Pan, Some results on estimation of the tail index of a distribution, Chinese Ann. Math. Ser. B 19 (1998), 239-248. MR 1655939 (2000a:62037)
18.: Z. Peng, Extension of Pickands' estimator, Math. Sinica 40 (1997), 759-762. MR 1612631 (99c:62111)
19.: L. Peng, Asymptotically unbiased estimators for the extreme value index, Statist. Probab. Lett. 38 (1998), 107-115. MR 1627906 (99e:62056)
20.: J. Pickands, Statistical inference using extreme order statistics, Ann. Statist. 3 (1975), 119-131. MR 0423667 (54:11642)
21.: Y. Qi and S. Cheng, Convergence of Pickands-type estimators, Chinese Sci. Bull. 37 (1992), 1409-1413.
22.: J. Segers, Abelian and Tauberian theorems for the bias of the Hill estimator, Scand. J. Statist. 29 (2001), 461-483. MR 1925570 (2003k:62159)
23.: J. Segers, Generalized Pickands estimators for the extreme value index, J. Statist. Plann. Inference 128 (2005), 381-396. MR 2102765
24.: R. L. Smith, Maximum likelihood estimation in a class of nonregular cases, Biometrika 72 (1985), 67-90. MR 790201 (86k:62053)
25.: R. L. Smith, Estimation of tails of probability distribution, Ann. Statist. 15 (1987), 1174-1207. MR 902252 (88j:62096)
26.: 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)
27.: Z. Tsourti and I. Panaretos, Extreme Value Index Estimators and Smoothing Alternatives: Review and Simulation Comparison, Technical Report, 2001.
28.: X. Wang and S. Cheng, General regular variation of th order and the second order Edgeworth expansion of the extreme value distribution. I, Acta Math. Sin., Engl. Ser. 21 (2005), no. 5, 1121-1130. MR 2176324 (2006i:60063)
29.: S. Yun, On a general Pickands estimator of extreme value index, J. Statist. Plann. Inference 102 (2002), 389-409. MR 1896495 (2003e:62050)

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

Cheng-Xiu Ling
Affiliation: Department of Mathematics, Southwest Normal University, Chongqing 400715, P. R. China
Email: pzx@swu.edu.cn

Zuoxiang Peng
Affiliation: Department of Mathematics, Southwest Normal University, Chongqing 400715, P. R. China

Saralees Nadarajah
Affiliation: Department of Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska 68583, USA
Email: snadaraj@unlserve.unl.edu

DOI: 10.1090/S0094-9000-08-00728-X
PII: S 0094-9000(08)00728-X
Keywords: Extreme value index, location invariant property, moment estimation, strong and weak consistencies, order statistics, regular varying functions
Received by editor(s): 29/NOV/2005
Posted: July 10, 2008
Copyright of article: Copyright 2008, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1547-7363(e) 0094-9000(p)

Previous number \| Table of contents \| Next number Recently posted articles \| Most recent number \| All numbers Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS