Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-17T12:09:09.913Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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