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Penultimate Approximations in Extreme Value Theory

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Abstract

Under a general von Mises condition and a second order regular variation condition, we derive exact penultimate approximation rates for maxima w.r.t. the variational distance. The approach is not restricted to the Gumbel domain of attraction. Moreover, it is shown that the improvement by penultimate approximation is of order n -1, if the penultimate rate is an algebraic function of the ultimate rate.

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References

  • Bingham, N.H., Goldie, C.M., and Teugels, J.L., Regular variation, Cambridge, Cambridge University Press, 1987.

    Google Scholar 

  • Cohen, J.P., “Convergence rates for ultimate and penultimate approximation in extreme value theory,” Adv. Appl. Probab. 14, 833-854, (1982).

    Google Scholar 

  • Fisher, R.A. and Tippett, L.H.C., “Limiting forms of the frequency distributions of the largest or smallest member of a Sample,” Proc. Camb. Phil. Soc. 24, 180-190, (1928).

    Google Scholar 

  • Geluk, J.L. and Haan, L.de, Regular Variation, Extensions and Tauberian Theorems, CWI Tracts 40. Center for Mathematics and Computer Science, Amsterdam, 1987.

    Google Scholar 

  • Gomes, M.I., “Penultimate limiting forms in extreme value theory,” Ann. Inst. Statist. Math., Ser. A 36, 71-85, (1984).

    Google Scholar 

  • Gomes, M.I. and Haan, L.de, “Approximation by penultimate extreme value distributions,” Extremes 2, 71-85, (1999).

    Google Scholar 

  • Haan, L.de and Resnick, S., “Second order regular variation and rates of convergence in extreme value theory,” Ann. Probab. 24, 97-124, (1996).

    Google Scholar 

  • Radtke, M., Konvergenzraten unter von Mises Bedingungen der Extremwerttheorie, Ph.D. Theses, University of Siegen, 1988.

  • Reiss, R.-D., Approximate Distributions of Order Statistics, Springer, New York, 1989.

    Google Scholar 

  • Reiss, R.-D., Extreme life spans. In: R.-D. Reiss and M. Thomas, (1997). Statistical Analysis of Extreme Values, Study 2, 241-244, Birkhäuser, Basel, (1997).

    Google Scholar 

  • Seneta, E., Regularly Varying functions. Lecture Notes in Mathematics 508, Springer, Berlin, 1976.

    Google Scholar 

Download references

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität-GH Siegen, 57068, Siegen, Germany

    Edgar Kaufmann

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  1. Edgar Kaufmann
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Kaufmann, E. Penultimate Approximations in Extreme Value Theory. Extremes 3, 39–55 (2000). https://doi.org/10.1023/A:1009971120468

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  • DOI: https://doi.org/10.1023/A:1009971120468

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