Advertisement

Penultimate Behaviour of the Extremes

  • M. Ivette Gomes

Abstract

Let {X n } n ≥ 1 be a sequence of random variables (r. v.’ s), and let M n = {max1≤i≤n X i }n≥1 be the associated sequence of Maximum Values. Let F n (x) = P[M n x] denote the distribution function (d. f.) of M n , n ≥ 1.

Keywords

Asymptotic Theory Generalize Extreme Value Parametric Classis Generalize Extreme Extreme Order Statistic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, C.W. (1971), Contributions to the Asymptotic Theory of Extreme Values, Ph.D. Thesis, University of London.Google Scholar
  2. Anderson, C.W. (1976), Extreme value theory and its approximations, Proc. Symp. Reliability Technology, Bradford, U.K.Google Scholar
  3. Anderson, C.W. (1984), Large deviations of extremes, In J. Tiago de Oliveira (ed.), Statistical Extremes and Applications, D. Reidel, 325–340.Google Scholar
  4. Balkema, A. A. and De Haan, L. (1990), A convergence rate in extreme value theory, J. Appl. Probab. 27, 577–585.CrossRefMATHMathSciNetGoogle Scholar
  5. Canto e Castro, L. (1990), Rate of Convergence to the GEV model: an unifying approach, Notas e Comunicações 11/90, C.E.A.U.L.Google Scholar
  6. Canto e Castro, L. (1992), Sobre a Teoria Assintótica de Extremos, Ph.D. Thesis, DEIO, University of Lisbon.Google Scholar
  7. Cohen, J.P. (1982a), The penultimate form of approximation to normal extremes, Adv. Appl Probab. 14, 324–339.CrossRefMATHGoogle Scholar
  8. Cohen, J.P. (1982b), Convergence rates for the ultimate and penultimate approximations in extreme value theory, Adv. Appl Probab. 14, 833–854.CrossRefMATHGoogle Scholar
  9. De Haan, L. (1970), On Regular Variation and Its Applications to the Weak Convergence of Sample Extremes, Math. Centrum, Amsterdam.Google Scholar
  10. De Haan, L. (1984), Slow variation and characterization of domains of attraction, In J. Tiago de Oliveira (ed.), Statistical Extremes and Applications, D. Reidel, 31–48.Google Scholar
  11. De Haan, L. and Rachev, S. T. (1989), Estimates of the rate of convergence for max-stable processes, Ann. Probab. 17, 651–677.CrossRefMATHMathSciNetGoogle Scholar
  12. Falk, M. (1986), Rates of uniform convergence of extreme order statistics, Ann. Inst. Stat Math. 38, 245–262.CrossRefMATHMathSciNetGoogle Scholar
  13. Fisher, R.A. and Tippett, L.H.C. (1928), Limiting forms of the frequency distributions of the largest or smallest member of a sample, Proc. Camb. Phil Soc. 24, 180–190.CrossRefMATHGoogle Scholar
  14. Galambos, J. (1978), The Asymptotic Theory of Extreme Order Statistics, Wiley, New York.MATHGoogle Scholar
  15. Galambos, J. (1984). Rates of convergence in extreme value theory, In J. Tiago de Oliveira (ed.), Statistical Extremes and Applications, D. Reidel, 347–352.Google Scholar
  16. Galambos, J. (1987), The Asymptotic Theory of Extreme Order Statistics ( 2nd edition ), Krieger, Malabar, FL.MATHGoogle Scholar
  17. Gnedenko, B.V. (1943), Sur la distribution limite du terme maximum d’une série aléatoire, Ann. Math. 44, 423–453.CrossRefMATHMathSciNetGoogle Scholar
  18. Gomes, M.I. (1978), Some Probabilistic and Statistical Problems in Extreme Value Theory, Ph.D. Thesis, Univ. Sheffield.Google Scholar
  19. Gomes, M. I. (1981). Closeness of penultimate approximations in extreme value theory, Abs. l4th European Meeting Statisticians, Wroclaw, 149–151.Google Scholar
  20. Gomes, M.I. (1984), Penultimate limiting forms in extreme value theory, Ann. Inst Statist Math. 36, 71–85.CrossRefMATHMathSciNetGoogle Scholar
  21. Gomes, M. I. (1986), Comparison of ultimate and penultimate approximations through simulation techniques. Computational Statistics and Data Analysis 4, 257–267.CrossRefMATHGoogle Scholar
  22. Gomes, M.I. and Pestana, D.D. (1987), Nonstandard domains of attraction and rates of convergence, In L. Puri et al (eds.). New Perspectives in Theoretical and Applied Statistics, Wiley, 467–477.Google Scholar
  23. Hall, P. (1979), On the rate of convergence of normal extremes, J. Appl. Probab. 16, 433–439.CrossRefMATHMathSciNetGoogle Scholar
  24. 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.CrossRefMATHMathSciNetGoogle Scholar
  25. Kohne, W. and Reiss, R.-D. (1983), A note on uniform approximation to extreme order statistics, Ann. Inst. Statist. Math. 35, 343–345.CrossRefMATHMathSciNetGoogle Scholar
  26. Mejzler, D. (1956), On the problem of the limit distribution for the maximal term of a variational series, L’vov Politechn. Inst. Naucn Zp. (Fiz.-Mat) 38, 90–109 (in Russian).MathSciNetGoogle Scholar
  27. Omey, E. and Rachev, S.T. (1988), Rates of convergence in multivariate extreme value theory, J. Multivariate Analysis 38, 36–50.CrossRefMathSciNetGoogle Scholar
  28. Omey, E. and Rachev, S.T. (1989), On the rate of convergence in extreme value theory, Theory Probab. Appl. 33, 560–566.CrossRefMathSciNetGoogle Scholar
  29. Pickands, J. (1986), The continuous and differentiable domains of attraction of the extreme-value distributions, Ann. Probab. 14, 996–1004.CrossRefMATHMathSciNetGoogle Scholar
  30. Reiss, R.-D. (1981), Uniform approximation to the distribution of extreme order statistics, Adv. Appl. Probab. 13, 533–547.CrossRefMATHMathSciNetGoogle Scholar
  31. Resnick, S. (1988), Uniform rates of convergence to extreme value distributions, In J. Srivastava (ed.), Probability and Statistics: Essays in Honor of Franklin Graybill, North-Holland, Amsterdam.Google Scholar
  32. Smith, R.L. (1982), Uniform rates of convergence in extreme value theory, Adv. Appl. Probab. 14, 600–622.CrossRefMATHGoogle Scholar
  33. Smith, R.L. (1987), Approximations in extreme value theory, Preprint, Univ. North-Carolina.Google Scholar
  34. Smith, R.L. (1988), Extreme value theory for dependent sequences via the Stein-Chen method of Poisson approximation, Stoch. Proc. Appl. 30, 317–327.CrossRefMATHGoogle Scholar
  35. Zolotarev, V.M. and Rachev, S.T. (1985), Rate of convergence in limit theorems for the max-scheme, In Lecture Notes in Mathematics 155, Springer-Verlag, 415–442.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • M. Ivette Gomes
    • 1
  1. 1.DEIO, Faculty of SciencesUniversity of LisbonLisboaPortugal

Personalised recommendations