Abstract
It is a well-known result in extreme value theory that the von Mises conditions imply the strong convergence of extreme order statistics. We extend this result to extreme generalized order statistics. A characterization of strong domains of attraction of joint distributions of a fixed number of extreme generalized order statistics by means of the corresponding result for generalized maxima is given. In particular, we determine the asymptotic joint distribution of (upper and lower) extreme generalized order statistics. Finally, we show that the Hill estimator based on extreme generalized order statistics is asymptotic normal.
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References
De Haan, L., “Extreme value statistics” in Extreme Value Theory and Applications, (J. Galambos, J. Lechner and E. Simiu, eds) Kluwer Academic Press, Dordrecht, 93–122, 1994.
Falk, M., “Uniform Convergence of Extreme Order Statistics,” Habilitationsschrift, University of Siegen, (1985).
Falk, M., “Rates of uniform convergence of extreme order statistics,” Ann. Inst. Statist. Math. 38, 245–262, (1986).
Falk, M., Hüsler, J., and Reiss, R.-D., Laws of Small Numbers: Extremes and Rare Events, DMV-Seminar, Birkhäuser, Basel, 1994.
Falk, M. and Marohn, F., “Von Mises conditions revisited,” Ann. Probab. 21, 1310–1328, (1993).
Hill, B.M., “A simple general approach to inference about the tail of a distribution,” Ann. Statist. 3, 1163–1174, (1975).
Kamps, U., A Concept of Generalized Order Statistics, Teubner, Stuttgart, 1995.
Nasri-Roudsari, D., “Extreme value theory of generalized order statistics,” J. Statist. Plann. Inference 55, 281–297, (1996).
Nasri-Roudsari, D., “Limit distributions of generalized order statistics under power normalization,” Comm. Statist. Theory Methods 28, 1379–1389, (1999).
Nasri-Roudsari, D. and Cramer, E., “On the convergence rates of extreme generalized order statistics,” Extremes 2, 421–447, (1999).
Pickands, J., “The continuous and differentiable domains of attractions of the extreme value distributions,” Ann. Probab. 14, 996–1004, (1986).
Reiss, R.-D., Approximate Distributions of Order Statistics, Springer Series in Statistics, New York, 1989.
Reiss, R.-D. and Thomas, M., Statistical Analysis of Extreme Values, Second Edition, Birkhäuser, Basel, 2001.
Resnick, S.I., Extreme Values, Regular Variation, and Point Processes, Springer, New York, 1987.
Sweeting, T.J., “On domains of uniform local attraction in extreme value theory,” Ann. Probab. 13, 196–205, (1985).
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Marohn, F. Strong Domain of Attraction of Extreme Generalized Order Statistics. Extremes 5, 369–386 (2002). https://doi.org/10.1023/A:1025176209750
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DOI: https://doi.org/10.1023/A:1025176209750