Advertisement

Extremes

, Volume 11, Issue 1, pp 55–80 | Cite as

It was 30 years ago today when Laurens de Haan went the multivariate way

  • Michael FalkEmail author
Article

Abstract

Since the publication of his masterpiece on regular variation and its application to the weak convergence of (univariate) sample extremes in 1970, Laurens de Haan (Thesis, Mathematical Centre Tract vol. 32, University of Amsterdam, 1970) is among the leading mathematicians in the world, with a particular focus on extreme value theory (EVT). On the occasion of his 70th birthday it is a great pleasure and a privilege to follow his route through multivariate EVT, which started only seven years later in 1977, when Laurens de Haan published his first paper on multivariate EVT, jointly with Sid Resnick.

Keywords

Max infinitely divisibility Multivariate extreme value distributions Exponent measure Stable tail dependence function Angular measure Extreme value statistics Max-stable processes 

AMS 2000 Subject Classification

Primary—60G70 Secondary—62G32 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Balkema, A.A., de Haan, L.: Almost sure continuity of stable moving average processes with index less than one. Ann. Probab. 16, 333–343 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  2. Balkema, A.A., Resnick, S.I.: Max-infinite divisibility. J. Appl. Prob. 14, 309–319 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  3. Barão, I., de Haan, L., Li, D.: Comparison of estimators in multivariate EVT. International J. Statistics and Systems (2003) (to appear)Google Scholar
  4. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes. Wiley, Chichester (2004)zbMATHGoogle Scholar
  5. Billingsley, P.: Weak Convergence of Measures: Applications in Probability. SIAM Monograph No. 5, Philadelphia (1971)Google Scholar
  6. Cheng, S., de Haan, L., Yang, J.: Asymptotic distributions of multivariate intermediate order statistics. Theory Probab. Appl. 41, 646–656 (1997)CrossRefGoogle Scholar
  7. Coles, S.: An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics, Springer, New York (2001)zbMATHGoogle Scholar
  8. de Haan, L.: On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Thesis, Mathematical Centre Tract vol. 32, University of Amsterdam (1970)Google Scholar
  9. de Haan, L.: A characterization of multidimensional extreme-value distributions. Sankhya Ser. A 40, 85–88 (1978)zbMATHMathSciNetGoogle Scholar
  10. de Haan, L.: A spectral representation of max-stable processes. Ann. Probab. 12, 1194-1204 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  11. de Haan, L.: Multivariate regular variation and applications in probability theory. In: Krishnaiah, P.R. (ed.) Multivariate Analysis – VI, pp. 281–288. Elsevier Science Publishers B.V., Amsterdam (1985)Google Scholar
  12. de Haan, L., de Ronde, J.: Sea and wind: multivariate extremes at work. Extremes 1, 7–45 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  13. de Haan, L., Ferreira, A.: Extreme Value Theory. An Introduction. Springer, New York (2006)zbMATHGoogle Scholar
  14. de Haan, L., Huang, X.: Large quantile estimation in a multivariate setting. J. Multivariate Anal. 53, 247–263 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  15. de Haan, L., Lin T.: On convergence towards an extreme value distribution in C[0,1]. Ann. Probab. 29, 467–483 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  16. de Haan, L., Peng, L.: Rates of convergence for bivariate extremes. J. Multivariate Anal. 61, 195–230 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  17. de Haan, L., Pereira, T.T.: Spatial extremes: models for the stationary case. Ann. Statist. 34, 146-168 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. de Haan, L., Pickands, J. III: Stationary min-stable processes. Probab. Th. Rel. Fields 72, 477–492 (1986)zbMATHCrossRefGoogle Scholar
  19. de Haan, L., Resnick, S.I.: Limit theory for multivariate sample extremes. Z. Wahrsch. Verw. Gebiete 40, 317–337 (1977)zbMATHCrossRefGoogle Scholar
  20. de Haan, L. Resnick, S.I.: Derivatives of regularly varying functions in \( \mathbb{R}^{2} \) and domain of attraction of stable distributions. Stochastic Process. Appl. 8, 349–355 (1979)zbMATHCrossRefGoogle Scholar
  21. de Haan, L., Resnick, S.I.: Estimating the limit distribution of multivariate extremes. Comm. Statist. – Stochastic Models 9, 275–309 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  22. de Haan, L., Sinha, A.K.: Estimating the probability of a rare event. Ann. Statist. 27, 732–759 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  23. Deheuvels, P.: Caractèrisation complète des lois extrème multivariées et de la convergene des types extrèmes. Publ. Inst. Statist. Univ. Paris 23, 1–36 (1978)zbMATHMathSciNetGoogle Scholar
  24. Deheuvels, P.: Point processes and multivariate extreme values. J. Multivariate Anal. 13, 257–272 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  25. Deheuvels, P.: Probabilistic aspects of multiariate extremes. In: Tiago de Oliveira, J. (ed.) Statistical Extremes and Applications, pp. 117–130. D. Reidel Publishing Company (1984)Google Scholar
  26. Dekkers, A.L.M., Einmahl, J., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Statist. 17, 1833–1855 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  27. Draisma, G., Drees, H., Ferreira, A., de Haan, L.: Bivariate tail estimation: dependence in asymptotic independence. Bernoulli 10, 251–280 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  28. Drees, H., Huang, X.: Best attainable rates of convergence for estimators of the stable tail dependence function. J. Multivariate Anal. 64, 25–47 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  29. Einmahl, J., de Haan, L., Huang, X.: Estimating a multidimensional extreme-value distribution. J. Multivariate Anal. 47, 35–47 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  30. Einmahl, J., de Haan, L., Sinha, A.: Estimation of the spectral measure of an extreme-valued distribution. Stoch. Proc. Appl. 70, 143–171 (1997)zbMATHCrossRefGoogle Scholar
  31. Einmahl, J., de Haan, L., Piterbarg, V.: Nonparametric estimation of the spectral measure of an extreme value distribution. Ann. Statist. 29, 1401–1423 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  32. Einmahl, J., de Haan, L., Li, D.: Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition. Ann. Statist. 34, 1987–2014 (2006)CrossRefMathSciNetGoogle Scholar
  33. Embrechts, P., de Haan, L., Huang, X.: Modelling multivariate Extremes. In: Embrechts, P. (ed.), Extremes and Integrated Risk Management. Risk Books, London (2000)Google Scholar
  34. Falk, M., Hüsler, J., Reiss, R.-D.: Laws of Small Numbers: Extremes and Rare Events, 2nd edn. Birkhäuser, Basel (2004)zbMATHGoogle Scholar
  35. Finkelshteyn, B.V.: Limiting distribution of extremes of a variational series of a two-dimensional random variable. Dokl. Ak. Nauk. S.S.S.R. 91, 209–211 (1953) (in Russian)Google Scholar
  36. Fougères, A.-L.: Multivariate extremes. In: Finkenstädt, B., Rootzén, B.H. (eds.) Extreme Values in Finance, Telecommunications, and the Environment, pp. 373–388. Chapman and Hall, Boca Raton (2004)Google Scholar
  37. Galambos, J.: The Asymptotic Theory of Extreme Order Statistics, 2nd edn. Krieger, Malabar (1987)zbMATHGoogle Scholar
  38. Geffroy, J.: Contribution à la théorie des valeurs extrêmes. Publ. l’Inst. Statist. l’Univ. Paris 7, 37–121 (1958)MathSciNetGoogle Scholar
  39. Geffroy, J.: Contribution à la théorie des valeurs extrêmes, II. Publ. l’Inst. Statist. l’Univ. Paris 8, 3–65 (1959)MathSciNetGoogle Scholar
  40. Giné, E., Hahn, M.G., Vatan, P.: Max-infinitely divisible and max-stable sample continuous processes. Probab. Th. Rel. Fields 87, 139–165 (1990)zbMATHCrossRefGoogle Scholar
  41. Gumbel, E.J., Goldstein, N.: Analysis of empirical bivariate extremal distributions. J. Amer. Statist. Assoc. 59, 794–816 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  42. Huang, X.: Statistics of Bivariate Extremes. Ph.D. Thesis, Erasmus University, Rotterdam, Tinbergen Institute Research series No. 22 (1992)Google Scholar
  43. Kotz, S., Nadarajah, S.: Extreme Value Distributions. Imperial College Press, London (2000)zbMATHGoogle Scholar
  44. Ledford, A., Tawn, J.A.: Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  45. Ledford, A., Twen, J.A.: Modelling dependence within joint tail regions. J. Royal Statist. Soc. Ser. B 59, 475–499 (1997)zbMATHCrossRefGoogle Scholar
  46. Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  47. Pickands III, J.: Multivariate extreme value distributions. Bull. Internat. Statist. Inst. Proc. 43th Session ISI, pp. 859–878 (1981) (Buenos Aires)Google Scholar
  48. Reiss, R.-D., Thomas, M.: Statistical Analysis of Extreme Values, 3rd edn. Birkhauser, Basel (2007)zbMATHGoogle Scholar
  49. Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Applied Prob., vol. 4. Springer, New York (1987)zbMATHGoogle Scholar
  50. Resnick, S.I.: Heavy Tail Phenomena. Probabilistic and Statistical Modeling. Springer, New York (2007)zbMATHGoogle Scholar
  51. Resnick, S.I., Roy, R.: Random usc functions, max-stable processes and continuous choice. Ann. App. Probab. 2, 267–292 (1991)CrossRefMathSciNetGoogle Scholar
  52. Rvaçera, E.L.: On the domains of attraction of multidimensional distributions. Select. Translat. Math. Statist. Prob. 2, 183–207 (1962)Google Scholar
  53. Sibuya, M.: Bivariate extreme statistics. Ann. Inst. Stat. Math. 11, 195–210 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  54. Tiago de Oliveira, J.: Extremal distributions. Rev. Fac. Ciências Lisboa 2 ser., A, Math. VIII, 299–310 (1958)Google Scholar
  55. Tiago de Oliveira, J.: Structure theory of bivariate extremes: extensions. Estudos de Math. Estat. Econom. 7, 165–195 (1962/1963)Google Scholar
  56. Tiago de Oliveira, J.L: Statistical decisions for bivariate extremes. In: Hüsler, J., Reiss R.-D., (eds.) Extreme Value Theory. Lecture Notes Statistics, vol. 51, pp. 246–259, Springer, New York (1989)Google Scholar
  57. Vatan, P.: Max-infinite divisibility and max-stability in infinite dimensions. In: Beck, A. et al. (eds.) Probability in Banach Spaces V, Lect. Notes Mathematics, vol. 1153, pp. 400–425, Springer, New York (1985)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany

Personalised recommendations