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Almost sure convergence of the Hill estimator

Published online by Cambridge University Press:  24 October 2008

Paul Deheuvels ,
Erich Haeusler  and
David M. Mason
Paul Deheuvels
Affiliation:
Université Paris VI, Paris, France
Erich Haeusler
Affiliation:
University of Delaware, Newark, Delaware, U.S.A.
David M. Mason
Affiliation:
Universität M¨nchen, Munich, Germany
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Abstract

In this note we characterize those sequences kn such that the Hill estimator of the tail index based on the kn upper order statistics of a sample of size n from a Pareto-type distribution is strongly consistent.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 2 , September 1988 , pp. 371 - 381
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Beirlant, J. and Teugels, J.. Asymptotics of Hill estimator. Teoria Verojatnost 26 (1986), 530–536.Google Scholar
[2]Chernoff, H.. A measure of asymptotic efficiency for tests of hypothesis based on the sums of observations. Ann. Math. Statist. 23 (1952), 493–507.CrossRefGoogle Scholar
[3]Csáki, E.. The law of the iterated logarithm for normalized empirical distribution functions. Z. Wahrsch. Verw. Gebiete 38 (1977), 147–167.CrossRefGoogle Scholar
[4]Csörgő, S., Deheuvels, P. and Mason, D. M.. Kernel estimates for the tail index of a distribution. Ann. Statist. 13 (1985), 1050–1077.CrossRefGoogle Scholar
[5]Csörgő, S. and Mason, D. M.. Central limit theorems for sums of extreme values. Math. Proc. Cambridge Philos. Soc. 98 (1985), 547–548.CrossRefGoogle Scholar
[6]David, H.. Order Statistics, 2nd ed. (Wiley, 1980).Google Scholar
[7]Deheuvels, P.. Strong laws for the k-th order statistics when k ≤ c log2n. Probab. Theory Related Fields 72 (1986), 133–154.CrossRefGoogle Scholar
[8]Deheuvels, P., Haeusler, E. and Mason, D. M.. On the almost sure behavior of sums of extreme values from a distribution in the domain of attraction of a Gumbel law. (Preprint, 1986.)Google Scholar
[9]Deheuvels, P. and Mason, D. M.. The asymptotic behavior of sums of exponential extreme values. Bull. Sci. Math. (2), 112 (1988), to appear.Google Scholar
[10]Deheuvels, P. and Pfeifer, D.. A semigroup approach to Poisson approximation. Ann. Probab. 14 (1986), 663–676.CrossRefGoogle 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]Haeusler, E. and Mason, D. M.. A law of the iterated logarithm for sums of extreme values from a distribution with a regularly varying upper tail. Ann. Probab. 15 (1987), 932–953.CrossRefGoogle Scholar
[13]Haeusler, E. and Teugels, J.. On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Statist. 13 (1985), 743–756.CrossRefGoogle Scholar
[14]Hill, B. M.. A simple approach to inference about the tail of a distribution. Ann. Statist. 3 (1975), 1163–1174.CrossRefGoogle Scholar
[15]Kiefer, J.. Iterated logarithm analogues for sample quantiles when p n ↓ 0. Proc. Sixth Berkeley Symp. 1 (1972), 227–244.Google Scholar
[16]Mason, D. M.. Laws of large numbers for sums of extreme values. Ann. Probab. 10 (1982), 756–764.CrossRefGoogle Scholar
[17]Rényi, A.. Théorie des éléments saillants d'une suite d'observations. In Colloquium in Combinatorial Methods in Probability Theory (Aarhus University, 1962), pp. 104–115.Google Scholar
[18]Shorack, G. R. and Wellner, J. A.. Empirical Processes with Applications to Statistics (Wiley, 1986).Google Scholar
[19]Smith, R. L.. Estimating tails of probability distributions. Ann. Statist. 15 (1987), 1174–1207.CrossRefGoogle Scholar
[20]Smith, R. L. and Weissman, I.. Large deviations of tail estimators based on the Pareto approximation. J. Appl. Prob. 24 (1987), 619–630.CrossRefGoogle Scholar
[21]Teugels, J.. Extreme values in insurance mathematics. In Statistical Extremes and Applications (Reidel, 1984), pp. 253–259.CrossRefGoogle Scholar