Multivariate Extremes: Supplementary Concepts and Results



In this chapter we will deal with exceedances and upper order statistics (besides maxima), with the point process approach being central for these investigations. Extremes will be asymptotically represented by means of Poisson processes with intensity measures given by max-Lévy measures as introduced in Section 4.3.


Poisson Process Point Process Empirical Process Generalize Pareto Distribution Central Sequence 
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  1. [13]
    Andersen, P.K., Borgan, Ø., Gill, R.D., and Keiding, N. (1993). Statistical Models based on Counting Processes. Springer Series in Statistics, Springer, New York. 484Google Scholar
  2. [23]
    Balkema, A.A., Haan, L. de, and Karandikar, R.L. (1993). Asymptotic distribution of the maximum of n independent stochastic processes. J. Appl. Prob. 30, 66-81.zbMATHCrossRefGoogle Scholar
  3. [30]
    Barndorff-Nielsen, O., and Sobel, M. (1966). On the distribution of the number of admissible points in a random vector sample. Theor. Probab. Appl. 11, 283-305.CrossRefMathSciNetGoogle Scholar
  4. [31]
    Barnett, V. (1976). The ordering of multivariate data. J. R. Statist. Soc. Ser. A 139, 318-344.CrossRefGoogle Scholar
  5. [33]
    Berezovskiy, B.A., and Gnedin, A. (1984). The Best Choice Problem. Akad. Nauk, Moscow (in Russian).Google Scholar
  6. [55]
    Brown, B.M., and Resnick, S.I. (1977). Extreme values of independent stochastic processes. J. Appl. Prob. 14, 732-739.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [56]
    Brozius, H. and Haan, L. de (1987). On limiting laws for the convex hull of a sample. J. Appl. Prob. 24, 852-862.zbMATHCrossRefGoogle Scholar
  8. [57]
    Bruss, F.T., and Rogers, L.C.G. (1991). Embedding optimal selection problems in a Poisson process. Stoch. Proc. Appl. 38, 267-278.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [70]
    Coles, S.G. (1993). Regional modelling of extreme storms via max-stable processes. J. R. Statist. Soc. B 55, 797-816.zbMATHMathSciNetGoogle Scholar
  10. [90]
    Daley, D.J., and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer Series in Statistics, Springer, New York.Google Scholar
  11. [119]
    Eddy, W.F., and Gale, J.D. (1981). The convex hull of a spherically symmetric sample. Adv. Appl. Prob. 13, 751-763.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [120]
    Einmahl, J.H.J., Haan, L. de, and Huang, X. (1993). Estimating a multidimensional extreme-value distribution. J. Mult. Analysis 47, 35-47.zbMATHCrossRefGoogle Scholar
  13. [135]
    Falk, M. (1995 c). On testing the extreme valued index via the POT-method. Ann. Statist. 23, 2013-2035.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [136]
    Falk, M. (1998). Local asymptotic normality of truncated empirical processes. Ann. Statist. 26, 692-718.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [140]
    Falk, M., and Liese, F. (1998). LAN of thinned empirical processes with an application to fuzzy set density estimation. Extremes 1, 323-349.CrossRefMathSciNetGoogle Scholar
  16. [142]
    Falk, M., and Marohn, F. (1993 a). Asymptotic optimal tests for conditional distributions. Ann. Statist. 21, 45-60.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [144]
    Falk, M., and Marohn, F. (2000). On the loss of information due to nonrandom truncation. J. Mult. Analysis 72, 1-21.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [180]
    Groeneboom, P. (1988). Limit laws for convex hulls. Probab. Th. Rel. Fields 79, 327-368.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [187]
    Haan, L. de (1984). A spectral representation of max-stable processes. Ann. Probab. 12, 1194-1204.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [188]
    Haan, L. de (1985). Extremes in higher dimensions: The model and some statistics. In Proc. 45th Session ISI (Amsterdam), 26.3.Google Scholar
  21. [196]
    Hájek, J. (1970). A characterization of the limiting distributions of regular estimates. Z. Wahrsch. Verw. Geb. 12, 21-55.Google Scholar
  22. [219]
    Höpfner, R. (1997). On tail parameter estimation in certain point process models. J. Statist. Plann. Inference 60, 169-187.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [220]
    Höpfner, R., and Jacod, J. (1994). Some remarks on the joint estimation of the index and the scale parameter for stable processes. In Asymptotic Statistics (P. Mandl and M. Huskova eds.). Proceedings of the Fifth Prague Symposium 1993, Physica, Heidelberg, 273-284.Google Scholar
  24. [268]
    Ibragimov, I.A., and Has’minskii, R.Z. (1981). Statistical Estimation. Springer, New York.zbMATHGoogle Scholar
  25. [285]
    Kaufmann, E., and Reiss, R.-D. (1993). Strong convergence of multivariate point processes of exceedances. Ann. Inst. Statist. Math. 45, 433-444.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [286]
    Kaufmann, E., and Reiss, R.-D. (1995). Approximation rates for multivariate exceedances. J. Statist. Plann. Inference 45, 235-245.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [308]
    LeCam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer Series in Statistics. Springer, New York.Google Scholar
  28. [309]
    LeCam, L., and Yang, G.L. (1990). Asymptotics in Statistics (Some Basic Concepts). Springer Series in Statistics. Springer, New York.Google Scholar
  29. [320]
    Marohn, F. (1995). Contributions to a Local Approach in Extreme Value Statistics. Habilitation thesis, Katholische Universität Eichstätt.Google Scholar
  30. [321]
    Marohn, F. (1999). Local asymptotic normality of truncation models. Statist. Decisions 17, 237-253.zbMATHMathSciNetGoogle Scholar
  31. [367]
    Pfanzagl, J. (1994). Parametric Statistical Theory. De Gruyter, Berlin.zbMATHGoogle Scholar
  32. [370]
    Pickands III, J. (1971). The two-dimensional Poisson process and extremal processes. J. Appl. Probab. 8, 745-756.zbMATHCrossRefMathSciNetGoogle Scholar
  33. [385]
    Reiss, R.-D. (1989). Approximate Distributions of Order Statistics. (With Applications to Nonparametric Statistics). Springer Series in Statistics, Springer, New York.Google Scholar
  34. [386]
    Reiss, R.-D. (1990). Asymptotic independence of marginal point processes of exceedances. Statist. Decisions 8, 153-165.zbMATHMathSciNetGoogle Scholar
  35. [387]
    Reiss, R.-D. (1993). A Course on Point Processes. Springer Series in Statistics, Springer, New York.Google Scholar
  36. [389]
    Reiss, R.-D. and Thomas, M. (2001). Statistical Analysis of Extreme Values 2nd ed., Birkhäuser, Basel.zbMATHGoogle Scholar
  37. [391]
    Rényi, A., and Sulanke, R. (1963). Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrsch. Verw. Geb. 2, 75-84.zbMATHCrossRefGoogle Scholar
  38. [393]
    Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Prob. Vol. 4, Springer, New York.Google Scholar
  39. [430]
    Strasser, H. (1985). Mathematical Theory of Statistics. De Gruyter Studies in Math. 7. De Gruyter, Berlin.Google Scholar
  40. [436]
    Tajvidi, N. (1996). Characterization and some statistical aspects of univariate and multivariate generalised Pareto distributions. PhD Thesis, Dept. Math., University of Güoteborg.Google Scholar
  41. [449]
    Van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge University Press.Google Scholar
  42. [452]
    Vatan, P. (1985). Max-infinite divisibility and max-stability in infinite dimensions. In Probability in Banach Spaces V, Lect. Notes Mathematics 1153, 400-425, Springer, New York.Google Scholar
  43. [462]
    Witte, H.-J. (1993). Extremal Points, Extremal Processes, Greatest Convex Minorants, Martingales and Point Processes. Habilitationsschrift, University of Oldenburg.Google Scholar
  44. [468]
    Zadeh, L.A. (1965). Fuzzy Sets. Information and Control, Vol. 8. Academic Press, New York, 338-353.Google Scholar
  45. [474]
    Zimmermann, H.-J. (1996). Fuzzy Set Theory - And its Applications. 3rd ed., Kluwer Academic Publishers, Dordrecht.zbMATHGoogle Scholar

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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany
  2. 2.Department of Mathematical Statistics and Actuarial ScienceUniversity of BerneBernSwitzerland
  3. 3.Department of MathematicsUniversity of SiegenSiegenGermany

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