Abstract
Estimators of the extreme-value index are based on a set of upper order statistics. We present an adaptive method to choose the number of order statistics involved in an optimal way, balancing variance and bias components. Recently this has been achieved for the similar but some what less involved case of regularly varying tails (Drees and Kaufmann, 1997); Danielsson et al., 1996). The present paper follows the line of proof of the last mentioned paper.
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Bingham, N.H., Goldie, C.M., and Teugels, J.L., Regular Variation, Cambridge University Press, New York, 1987.
Cooil, B., “Limiting multivariate distributions of intermediate order statistics,” Ann. Probab. 13, 469–477, (1985).
Danielsson, J., de Haan, L., Peng, L., and de Vries, C.G., Using a bootstrap method to choose the sample fraction in the tail index estimation, Journal of Multivariate Analysis, to appear, 1996.
de Haan, L. and Peng, L., “Rates of convergence for bivariate extremes,” Journal of Multivariate Analysis 61(2), 195–230, (1997).
de Haan, L. and Resnick, S.I., “Second order regular variation and rates of convergence in extreme value theory,” Annals of Probability, 24, 97–124, (1996).
de Haan, L. and Stadtmüller, U., “Generalized regular variation of second order,” J. Austral. Math. Soc. (Series A) 61, 381–395, (1996).
de Haan, L. and de Ronde, J., “Sea and wind: Multivariate extremes at work,” Extremes 1(1), 7–45, (1998).
Dekkers, A.L.M. and de Haan, L., “Optimal choice of sample fraction in extreme value estimation,” Journal of Multivariate Analysis 47(2), 173–195, (1993).
Dekkers, A.L.M., Einmahl, J.H.J., and de Haan, L., “A moment estimator for the index of an extreme value distribution,” Ann. Statis. 17(4), 1833–1855, (1989).
Drees, H., “On smooth statistical tail functionals,” Scandinavian Journal of Statistics 25(1), 187–210, (1998).
Drees, H. and Kaufmann, E., “Selecting the optimal sample fraction in univariate extreme value estimation,” Stock: Proc. Appl. 75, 149–172, (1998).
Feller, W., An Introduction to Probability Theory and its Applications, Volume II Wiley, New York, 1966.
Hall, P., “On simple estimates of an exponent of regular variation,” J. Royal Statistical Society B 44, 37–42, (1982).
Hall, P., “Using the bootstrap to estimate means squared error and select smoothing parameter in nonparametric problems,” Journal of Multivariate Analysis 32, 177–203, (1990).
Geluk J. and de Haan, L., Regular variation, extensions and tauberian theorems, Technical Report CWI Tract 40, CWI, Amsterdam, (1987).
Pickands III, J., “Statistical inference using extreme order statistics,” Ann. Statis. 3, 119–131, (1975).
Pickands III, J., “The continuous and differentiable domains of attraction of the extreme value distributions,” Ann. Probab., 14, 9996–1004, (1986).
Omey, E. and Willekens, E., “p-Variation with remainder,” J. London Math. Soc. 37, 105–118, (1988).
Pereira, T.T., “Second order behavior of domains of attraction and the bias of generalized pickands' estimator,” In J. Lechner, J. Galambos and E. Simin, eds, Extreme Value Theory and Applications III, Proc. Gaithersburg Conference (NIST special publ), (1993).
Petrov, V., Sums of Independent Random Variables, Springer, Berlin, 1975.
Shorack, G. and Wellner, J., Empirical processes with Applications to Statistics, John Wiley & Sons, 1986.
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Draisma, G., de Haan, L., Peng, L. et al. A Bootstrap-based Method to Achieve Optimality in Estimating the Extreme-value Index. Extremes 2, 367–404 (1999). https://doi.org/10.1023/A:1009900215680
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DOI: https://doi.org/10.1023/A:1009900215680