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Uniform rates of convergence in extreme-value theory

Published online by Cambridge University Press:  01 July 2016

Richard L. Smith
Richard L. Smith*
Affiliation:
Imperial College, London
*
∗Postal address: Department of Mathematics, Imperial College of Science and Technology, Huxley Building, Queen's Gate, London SW7 2BZ, U.K.
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Abstract

Rates of convergence are derived for the convergence in distribution of renormalised sample maxima to the appropriate extreme-value distribution. Related questions which are discussed include the estimation of the principal error term and the optimality of the renormalising constants. Throughout the paper a close parallel is drawn with the theory of slow variation with remainder. This theory is used in proving most of the results. Some applications are discussed, including some models of importance in reliability.

Keywords

EXTREME-VALUE THEORYRATES OF CONVERGENCESLOW VARIATION WITH REMAINDER
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 3 , September 1982 , pp. 600 - 622
Copyright
Copyright © Applied Probability Trust 1982 

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References

Auancic, S., Bojanic, R. and Tomic, M. (1974) Slowly varying functions with remainder term and their applications in analysis. Serbian Acad. Sci. and Arts Monographs, Vol. CDLXII, Sections Nat. math, Sco. No. 41, Belgrade.Google Scholar
Anderson, C. W. (1971) , University of London.Google Scholar
Anderson, C. W. (1976) Extreme value theory and its approximations. Proc. Symp. Reliability Technology. Bradford, UKAEA.Google Scholar
Anderson, C. W. (1978) Super-slowly varying functions in extreme value theory. J. R. Statist. Soc. B 40, 197–202.Google Scholar
Ash, J. M., Erdös, P. and Rubel, L. A. (1974) Very slowly varying functions. Aequationes Math. 10, 1–9.CrossRefGoogle Scholar
Bingham, N. H. and Goldie, C. M. (1982a) Extensions of regular variation, I: Uniformity and quantifiers. Proc. London Math. Soc. Google Scholar
Bingham, N. H. and Goldie, C. M. (1982b) Extensions of regular variation, II: Representations and indices. Proc. London Math. Soc. CrossRefGoogle Scholar
Borges, W. S. (1978) , Cornell University.Google Scholar
Cohen, J. P. (1982) The penultimate form of approximation to normal extremes. Adv. Appl. Prob. 14, 324–339.CrossRefGoogle Scholar
Daniels, H. E. (1945) The statistical theory of the strength of bundles of threads, I. Proc. R. Soc. Lond A 183, 404–435.Google Scholar
Dziubdziela, W. (1978) On convergence rates in the limit laws for extreme order statistics. Trans. 7th Prague Conf. B, Academia, Prague, 119–127.Google Scholar
Fisher, R. A. and Tippett, L. H. C. (1928) Limiting forms of the frequency distributions of largest or smallest member of a sample. Proc. Camb. Phil. Soc. 24, 180–190.CrossRefGoogle Scholar
Galambos, J. (1978) The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.Google Scholar
Gnedenko, B. V. (1943) Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423–453.CrossRefGoogle Scholar
Goldie, C. M. (1980) Slow variation with remainder. Unpublished manuscript, University of Sussex.Google Scholar
Haan, L. De (1970) On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tracts 32, Amsterdam.Google Scholar
Hall, P. (1979) On the rate of convergence of normal extremes. J. Appl. Prob. 16, 433–439.CrossRefGoogle Scholar
Hall, P. (1980) Estimating probabilities for normal extremes. Adv. Appl. Prob. 12, 491–500.CrossRefGoogle Scholar
Hall, W. J. and Wellner, J. A. (1979) The rate of convergence in law of the maximum of an exponential sample. Statist. Neerlandica 33, 151–154.CrossRefGoogle Scholar
Kendall, M. G. and Stuart, A. (1969) The Advanced Theory of Statistics, Vol. 1, 3rd edn, Griffin, London.Google Scholar
Seneta, E. (1976) Regularly Varying Functions. Lectures Notes in Mathematics 508, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Smith, R. L. (1982) The asymptotic distribution of the strength of a series-parallel system with equal load-sharing. Ann. Prob. 10, 137–171.CrossRefGoogle Scholar
Suh, M. W., Bhattacharyya, B. B. and Grand Age, A. (1970) On the distribution and moments of the strength of a bundle of fibres. J. Appl. Prob. 7, 712–720.CrossRefGoogle Scholar