Abstract
This paper deals with the semi-parametric approach to the problem of statistical choice of extreme domains of attraction. Relying on concepts of regular variation theory, it investigates the asymptotic properties of Hasofer and Wang’s test statistic based on the k upper extremes taken from a sample of size n, when k behaves as an intermediate sequence k n rather than remaining fixed while the sample size increases. In the process a Greenwood type test statistic is proposed which turns out to be useful in discriminating heavy-tailed distributions. The finite sample behavior of both testing procedures is evaluated in the light of a simulation study. The testing procedures are then applied to three real data sets.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balkema AA, de Haan L (1974) Residual life time at great age. Ann Probab 2:792–804
Beirlant J, Goegebeur Y, Segers J, Teugels J (2004) Statistics of extremes: theory and applications. Wiley, London
Castillo E, Galambos J, Sarabia JM (1989) The selection of the domain of attraction of an extreme value distribution from a set of data. In: Hüsler J, Reiss R-D (eds) Lecture notes in statistics, vol 51. Springer, Berlin, pp 181–190
Castillo E, Hadi AS (1997) Fitting the generalized Pareto distribution to data. J Am Stat Assoc 92:1609–1620
Castillo E, Hadi AS, Balakrishnan N, Sarabia JM (2005) Extreme value and related models with applications in engineering and science. Wiley, Hoboken
de Haan L (1984) Slow variation and characterization of domains of attraction. In: de Oliveira JT (ed) Statistical extremes and applications. Reidel, Dordrecht, pp 31–48
de Haan L, Stadtmüller U (1996) Generalized regular variation of second order. J Aust Math Soc Ser A 61:381–395
de Zea Bermudez P, Amaral Turkman MA (2001) A predictive approach to tail probability estimation. Extremes 4:295–314
de Zea Bermudez P, Amaral Turkman MA (2003) Bayesian approach to parameter estimation of the generalized Pareto distribution. Test 12(1):259–277
Draisma G, de Haan L, Peng L, Pereira TT (1999) A bootstrap-based method to achieve optimality in estimation the extreme-value index. Extremes 2(4):367–404
Fraga Alves MI, Gomes MI (1996) Statistical choice of extreme value domains of attraction—a comparative analysis. Commun Stat Theory Methods 4(25):789–811
Gnedenko BV (1943) Sur la distribution limite du terme maximum d’une série aléatoire. Ann Math 44:423–453
Greenwood M (1946) The statistical study of infectious diseases. J Roy Stat Soc Ser A 109:85–109
Hasofer AM, Wang Z (1992) A test for extreme value domain of attraction. J Am Stat Assoc 87:171–177
Hsing T (1997) A case study of ozone pollution with XTREMES. In: Reiss R-D, Thomas M (eds) Statistical analysis of extreme values. Birkhäuser, Basel, pp 251–256
Marohn F (1998a) An adaptive test for Gumbel domain of attraction. Scand J Stat 25:311–324
Marohn F (1998b) Testing the Gumbel hypothesis via the POT-method. Extremes 1(2):191–213
Matthys G, Beirlant J (2001) Extreme quantile estimation for heavy-tailed distributions. Working paper, University Center of Statistics, Katholieke University Leuven. In: www.kuleuven.ac.be/ucs/research/publi.htm
Moran PAP (1947) The random division of an interval. J Roy Stat Soc Ser B 40:213–216
Neves C, Picek J, Fraga Alves MI (2006) Contribution of the maximum to the sum of excesses for testing max-domains of attraction. J Stat Plan Inference 136(4):1281–1301
Pickands J III (1975) Statistical inference using extreme order statistics. Ann Stat 3:119–131
Reiss R-D, Thomas M (2001) Statistical analysis of extreme values, with applications to insurance, finance, hydrology and other fields, 2nd edn. Birkhäuser, Basel
Shapiro SS, Wilk MB (1965) An analysis of variance test for normality (complete samples). Biometrika 52:591–611
Wang JZ (1995) Selection of the k largest order statistics for the domain of attraction of the Gumbel distribution. J Am Stat Assoc 90:1055–1061
Wang JZ, Cooke P, Li S (1996) Determination of domains of attraction based on a sequence of maxima. Aust J Stat 38:173–181
Weissman I (1978) Estimation of parameters and large quantiles based on the k largest observations. J Am Stat Assoc 73:812–815
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by FCT/POCTI/FEDER.
Rights and permissions
About this article
Cite this article
Neves, C., Fraga Alves, M.I. Semi-parametric approach to the Hasofer–Wang and Greenwood statistics in extremes. TEST 16, 297–313 (2007). https://doi.org/10.1007/s11749-006-0010-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11749-006-0010-1