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Almost sure convergence of the Hill estimator

  • Paul Deheuvels (a1), Erich Haeusler (a2) and David M. Mason (a3)

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.

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[1]Beirlant J. and Teugels J.. Asymptotics of Hill estimator. Teoria Verojatnost 26 (1986), 530536.
[2]Chernoff H.. A measure of asymptotic efficiency for tests of hypothesis based on the sums of observations. Ann. Math. Statist. 23 (1952), 493507.
[3]Csáki E.. The law of the iterated logarithm for normalized empirical distribution functions. Z. Wahrsch. Verw. Gebiete 38 (1977), 147167.
[4]Csörgő S., Deheuvels P. and Mason D. M.. Kernel estimates for the tail index of a distribution. Ann. Statist. 13 (1985), 10501077.
[5]Csörgő S. and Mason D. M.. Central limit theorems for sums of extreme values. Math. Proc. Cambridge Philos. Soc. 98 (1985), 547548.
[6]David H.. Order Statistics, 2nd ed. (Wiley, 1980).
[7]Deheuvels P.. Strong laws for the k-th order statistics when kc log2n. Probab. Theory Related Fields 72 (1986), 133154.
[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.)
[9]Deheuvels P. and Mason D. M.. The asymptotic behavior of sums of exponential extreme values. Bull. Sci. Math. (2), 112 (1988), to appear.
[10]Deheuvels P. and Pfeifer D.. A semigroup approach to Poisson approximation. Ann. Probab. 14 (1986), 663676.
[11]Hall P.. On some simple estimates of an exponent of regular variation. J. Roy. Statist. Soc. Ser. B 44 (1982), 3742.
[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), 932953.
[13]Haeusler E. and Teugels J.. On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Statist. 13 (1985), 743756.
[14]Hill B. M.. A simple approach to inference about the tail of a distribution. Ann. Statist. 3 (1975), 11631174.
[15]Kiefer J.. Iterated logarithm analogues for sample quantiles when p n ↓ 0. Proc. Sixth Berkeley Symp. 1 (1972), 227244.
[16]Mason D. M.. Laws of large numbers for sums of extreme values. Ann. Probab. 10 (1982), 756764.
[17]Rényi A.. Théorie des éléments saillants d'une suite d'observations. In Colloquium in Combinatorial Methods in Probability Theory (Aarhus University, 1962), pp. 104115.
[18]Shorack G. R. and Wellner J. A.. Empirical Processes with Applications to Statistics (Wiley, 1986).
[19]Smith R. L.. Estimating tails of probability distributions. Ann. Statist. 15 (1987), 11741207.
[20]Smith R. L. and Weissman I.. Large deviations of tail estimators based on the Pareto approximation. J. Appl. Prob. 24 (1987), 619630.
[21]Teugels J.. Extreme values in insurance mathematics. In Statistical Extremes and Applications (Reidel, 1984), pp. 253259.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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