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Central limit theorems for sums of extreme values

Published online by Cambridge University Press:  24 October 2008

Sándor Csörgoő  and
David M. Mason
Sándor Csörgoő
Affiliation:
Szeged University, Szeged, Hungary
David M. Mason
Affiliation:
University of Delaware, Newark, Delaware, U.S.A.
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Abstract

Given a sequence of non-negative independent and identically distributed random variables, we determine conditions on the common distribution such that the sum of appropriately normalized and centred upper kn extreme values based on the first n random variables converges in distribution to a normal random variable, where kn → ∞ and kn/ n → 0 as n → ∞. The probabilistic problem is motivated by recent statistical work on the estimation of the exponent of a regularly varying distribution function. Our main tool is a new Brownian bridge approximation to the uniform empirical and quantile processes in weighted supremum norms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 3 , November 1985 , pp. 547 - 558
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Balkema, A. and de Haan, L.. Limit laws for order statistics. In Limit Theorems of Probability (ed. Révész, P.), pp. 17–22. Colloquia Math. Soc. J. Bolyai, no. 11 (North-Holland, 1975).Google Scholar
[2]Csörgoő, S.. Adaptive estimation of the parameters of stable laws. In Limit Theorems in Probability and Statistics (ed. Révész, P.), pp. 305–368. Colloquia Math. Soc. J. Bolyai, no. 36. (North-Holland, 1984).Google Scholar
[3]Csörgoő, M., Csörgoő, S., Horváth, L. and Mason, D. M.. Weighted empirical and quantile processes. Ann. Probab. To appear.Google Scholar
[4]Csörgoő, S., and Mason, D. M.. The asymptotic distribution of sums of extreme values from a regularly varying distribution. Ann. Probab. To appear.Google Scholar
[5]Csörgoő, S., Deheuvels, P. and Mason, D. M.. Kernel estimates of the tail index of a distribution. Ann. Statist. To appear.Google Scholar
[6]Csörgoő, S., Horváth, L. and Mason, D. M.. What portion of the sample makes a partial sum asymptotically stable or normal? Z. Warsch. verw. Gebiete. To appear.Google Scholar
[7]Davis, R. and Resnick, S.. Tail estimates motivated by extreme value theory Ann. Statist. 12 (1984), 1467–1487.CrossRefGoogle Scholar
[8]de Haan, L.. On regular variation and its application to the weak convergence of sample extremes. Mathematical Centre Tracts no. 32 (Amsterdam, 1970).Google Scholar
[9]Haeusler, E. and Teugels, J. L.. On the asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Statist. To appear.Google Scholar
[10]Hall, P.. On the relative stability of large order statistics. Math. Proc. Cambridge Philos. Soc. 86 (1979), 467–475.Google Scholar
[11]Hall, P.. On some simple estimates of an exponent of regular variation. J. Roy. Statist. Soc. Ser. B 44 (1982), 37–42.Google Scholar
[12]Hill, B. M.. A simple general approach to inference about the tail of a distribution. Ann. Statist. 3 (1975), 1163–1174.Google Scholar
[13]Mason, D. M.. Laws of large numbers for sums of extreme values. Ann. Probab. 10 (1982), 754–764.CrossRefGoogle Scholar
[14]de Meyer, A. and Teugels, J. L.. Limit theorems for Paretotype distributions (Katholieke Universiteit te Leuven, preprint, 1981).Google Scholar