×

Estimation of the extreme-value index and generalized quantile plots. (English) Zbl 1123.62034

Summary: In extreme-value analysis, a central topic is the adaptive estimation of the extreme-value index \(\gamma\). Hitherto, most of the attention in this area has been devoted to the case \(\gamma>0\), that is, when \(\overline{F}\) is a regularly varying function with index \(-1/\gamma\). In addition to the well-known B. M. Hill estimator [Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)], many other estimators are currently available. Among the most important are the kernel-type estimators and the weighted least-squares slope estimators based on the Pareto quantile plot or the Zipf plot, as reviewed by S. Csörgő and L. Viharos [Estimating the tail index. B. Szyszkowicz (ed.), Asymptotic Methods Probab. Stat., 833–881 (1998; Zbl 1042.62543)]. Using an exponential regression model (ERM) for spacings between successive extreme order statistics, both J. Beirlant et al. [Extremes 2, No. 2, 177–200 (!999; Zbl 0947.62034)] and A. Feuerverger and P. Hall [Ann. Stat. 27, No. 2, 760–781 (1999: Zbl 0942.62059)] introduced bias-reduced estimators.
For the case where \(\gamma\) is real, Hill’s estimator has been generalized to a moment-type estimator by A. L. M. Dekkers et al. [ibid. 17, No. 4, 1833–1855 (1989; Zbl 0701.62029)]. Alternatively, J. Beirlant et al. [J. Am. Stat. Assoc. 91, No. 436, 1659–1667 (1996; Zbl 0881.62077)] introduced a Hill-type estimator that is based on the generalized quantile plot. Another popular estimation method follows from maximum likelihood estimation applied to the generalizations of the Pareto distribution. In the present paper, slope estimators for \(\gamma>0\) are generalized to the case where \(\gamma\) is real-valued. This is accomplished by replacing the Zipf plot by a generalized quantile plot. We make an asymptotic comparison of our estimator with the moment estimator and with the maximum likelihood estimator. A case study illustrates our findings. Finally, we offer a regression model that generalizes the ERM in that it allows the construction of bias-reduced estimators. Moreover, the model provides an adaptive selection rule for the number of extremes needed in several of the existing estimators.

MSC:

62G32 Statistics of extreme values; tail inference
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
62G08 Nonparametric regression and quantile regression
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] Balkema, A. and de Haan, L. (1974) Residual life at great age. Ann. Probab., 2, 792-804. · Zbl 0295.60014
[2] Beirlant, J., Vynckier, P. and Teugels, J.L. (1996a) Tail index estimation, Pareto quantile plots, and regression diagnostics. J. Amer. Statist. Assoc., 91, 1659-1667. JSTOR: · Zbl 0881.62077
[3] Beirlant, J., Vynckier, P. and Teugels, J.L. (1996b) Excess functions and estimation of the extreme value index. Bernoulli, 2, 293-318. · Zbl 0870.62019
[4] Beirlant, J., Dierckx, G., Goegebeur Y. and Matthys, G. (1999) Tail index estimation and an exponential regression model. Extremes, 2, 177-200. · Zbl 0947.62034
[5] Beirlant, J., Dierckx, G., Guillou, A. and Starica, C. (2002) On exponential representations of logspacings of extreme order statistics. Extremes, 5, 157-180. · Zbl 1036.62040
[6] Castillo, E. and Hadi, A.S. (1997) Fitting the generalized Pareto distribution to data. J. Amer. Statist. Assoc., 92, 1609-1620. JSTOR: · Zbl 0919.62014
[7] Coles, S.G. and Powell, E.A. (1996) Bayesian methods in extreme value modelling: a review and new developments. Internat. Statist. Rev., 64, 119-136. · Zbl 0853.62025
[8] Csörgö, S. and Viharos, L. (1998) Estimating the tail index. In B. Szyszkowicz (ed.), Asymptotic Methods in Probability and Statistics, pp. 833-881. Amsterdam: North-Holland. · Zbl 1042.62543
[9] Danielsson, J.L., de Haan, L., Peng, L. and de Vries, C.G. (2001) Using a bootstrap based method to choose the sample fraction in tail index estimation. J. Multivariate Anal., 76, 226-248. · Zbl 0976.62044
[10] Davison, A.C. and Smith, R.L. (1990) Models for exceedances over high thresholds. J. Roy. Statist. Soc. Ser. B, 52, 393-442. JSTOR: · Zbl 0706.62039
[11] de Haan, L. and Rootzén, H. (1993) On the estimation of high quantiles. J. Statist. Plann. Inference, 35, 1-13. · Zbl 0770.62026
[12] de Haan, L. and Stadtmüller, U. (1996) Generalized regular variation of second order. J. Austral. Math. Soc. Ser. A, 61, 381-395. · Zbl 0878.26002
[13] Dekkers, A.L.M., Einmahl, J.H.J. and de Haan, L. (1989) A moment estimator for the index of an extreme-value distribution. Ann. Statist., 17, 1833-1855. · Zbl 0701.62029
[14] Draisma, G., de Haan, L., Peng, L. and Pereira, T.T. (1999) A bootstrap based method to achieve optimality in estimating the extreme value index. Extremes, 2, 367-404. · Zbl 0972.62014
[15] Drees, H. (1998) On smooth statistical tail functionals. Scand. J. Statist., 25, 187-210. · Zbl 0923.62032
[16] Drees, H. and Kaufmann, E. (1998) Selecting the optimal sample fraction in univariate extreme value estimation. Stochastic Process. Appl., 75, 149-172. · Zbl 0926.62013
[17] Drees, H., de Haan, L. and Resnick, S. (2000) How to make a Hill plot. Ann. Statist., 28, 254-274. · Zbl 1106.62333
[18] Drees, H., Ferreira, A. and de Haan, L. (2004) On maximum likelihood estimation of the extreme value index. Ann. Appl. Probab., 14, 1179-1201. · Zbl 1102.62051
[19] Ferreira, A. (2002) Optimal asymptotic estimation of small exceedance probabilities. J. Statist. Plann. Inference, 53, 83-102. · Zbl 0988.62029
[20] Ferreira, A., de Haan, L. and Peng, L. (2003) On optimizing the estimation of high quantiles of a probability distribution. Statistics, 37, 401-434. · Zbl 1210.62052
[21] Feuerverger, A. and Hall, P. (1999) Estimating a tail exponent by modelling departure from a Pareto distribution. Ann. Statist., 27, 760-781. · Zbl 0942.62059
[22] Fraga Alves, M.I., de Haan, L. and Lin, T. (2003) Estimation of the parameter controlling the speed of convergence in extreme value theory. Math. Methods Statist., 12, 155-176.
[23] Groeneboom, P., Lopuhaä, H.P. and de Wolf, P.P. (2003) Kernel-type estimators for the extreme value index. Ann. Statist., 31, 1956-1995. · Zbl 1047.62046
[24] Hall, P. (1982) On some simple estimates of an exponent of regular variation. J. Roy. Statist. Soc. Ser. B, 44, 37-42. JSTOR: · Zbl 0521.62024
[25] Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Statist., 3, 1163-1174. · Zbl 0323.62033
[26] Hosking, J.R.M., Wallis, J.R. and Wood, E.F. (1985) Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics, 27, 251-261. JSTOR:
[27] Kratz, M. and Resnick, S. (1996) The qq-estimator and heavy tails. Comm. Statist. Stochastic Models, 12, 699-724. · Zbl 0887.62025
[28] Mason, D.M. and Turova, T.S. (1994) Weak convergence of the Hill estimator process. In J. Galambos, J. Lechner and E. Simiu (eds), Extreme Value Theory and Applications. Dordrecht: Kluwer Academic.
[29] Pickands, J. III (1975) Statistical inference using extreme order statistics. Ann. Statist., 3, 119-131. · Zbl 0312.62038
[30] Resnick, S. and Starica, C. (1997) Smoothing the Hill estimator. Adv. Appl. Probab., 29, 271-293. JSTOR: · Zbl 0873.60021
[31] Schultze, J. and Steinebach, J. (1996) On least squares estimates of an exponential tail coefficient. Statist. Decisions, 14, 353-372. · Zbl 0893.62023
[32] Smith, R.L. (1987) Estimating tails of probability distributions. Ann. Statist., 15, 1174-1207. · Zbl 0642.62022
[33] Smith, R.L. (1989) Extreme value analysis of environmental times series: an application to trend detection in ground-level ozone. Statist. Sci., 4, 367-393. · Zbl 0955.62646
[34] Vanroelen, G. (2003) Generalized regular variation, order statistics and real inversion formulas. Doctoral thesis, Katholieke Universiteit Leuven, Leuven, Belgium.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.