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On the asymptotic normality of Hill's estimator

Published online by Cambridge University Press:  24 October 2008

Sándor Csörgő  and
László Viharos
Sándor Csörgő
Affiliation:
Department of Statistics, University of Michigan, Ann Arbor, MI 48109-1027, U.S.A.
László Viharos
Affiliation:
Bolyai Institute, University of Szeged, 6720 Szeged, Hungary
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Extract

Let X, X1, X2, …, be independent random variables with a common distribution function F(x) = P {X ≤ x}, x∈ℝ, and for each n∈ℕ, let X1, n ≤ … ≤ Xn, n denote the order statistics pertaining to the sample X1, …, Xn. We assume that 1–F(x) = x−1/cl(x), 0 < x < ∞, where l is some function slowly varying at infinity and c > 0 is any fixed number. The class of all such distribution functions will be denoted by .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 2 , September 1995 , pp. 375 - 382
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

REFERENCES

[1]Bingham, N. H., Goldie, C. M. and Teugels, J. L.. Regular Variation. Encyclopedia of Math. and its Appl. 27 (Cambridge University Press, 1987).CrossRefGoogle Scholar
[2]Csörgő, S., Deheuvels, P. and Mason, D. M.. Kernel estimates of the tail index of a distribution. Ann. Statist. 13 (1985), 1050–1077.CrossRefGoogle Scholar
[3]Csörgő, S. and Mason, D. M.. Central limit theorems for sums of extreme values. Math. Proc. Cambridge Philos. Soc. 98 (1985), 547–558.CrossRefGoogle Scholar
[4]Davis, R. and Resnick, S.. Tail estimates motivated by extreme value theory. Ann. Statist. 12 (1984), 1467–1487.CrossRefGoogle Scholar
[5]Deheuvels, P., Haeusler, E. and Mason, D. M.. Almost sure convergence of the Hill estimator. Math. Proc. Cambridge Philos. Soc. 104 (1988), 371–381.CrossRefGoogle Scholar
[6]Haeusler, E. and Teugels, J. L.. On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Statist. 13 (1985), 743–756.CrossRefGoogle Scholar
[7]Hall, P.. On some simple estimates of an exponent of regular variation. J. Roy. Statist. Soc. Ser. B 44 (1982), 37–42.Google Scholar
[8]Hill, B. M.. A simple approach to inference about a tail of a distribution. Ann. Statist. 3 (1975), 1163–1174.CrossRefGoogle Scholar
[9]Mason, D. M.. Laws of large numbers for sums of extreme values. Ann. Probab. 10 (1982), 754–764.CrossRefGoogle Scholar