Dependence Comparison of Multivariate Extremes via Stochastic Tail Orders

Part of the Lecture Notes in Statistics book series (LNS, volume 208)


A stochastic tail order is introduced to compare right tails of distributions and related closure properties are established. The stochastic tail order is then used to compare the dependence structure of multivariate extreme value distributions in terms of upper tail behaviors of their underlying samples.


Random Vector Tail Dependence Tail Index Univariate Margin Random Variable Versus 
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.



The author would like to thank Alfred Müller for his comments on this paper during IWAP 2012 in Jerusalem, Israel, and especially for his comment on several references that are closely related to this work. Haijun Li is supported by NSF grants CMMI 0825960 and DMS 1007556.


  1. [12]
    Anderson, T. W.: The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proceedings of the American Mathematical Society, 6, 170–176 (1955)MathSciNetMATHCrossRefGoogle Scholar
  2. [34]
    Barbe, P., Fougères, A. L. and Genest, C.: On the tail behavior of sums of dependent risks. ASTIN Bulletin, 36, 361–374 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. [43]
    Bartoszewicz, J.: Tail orderings and the total time on test transforms. Applicationes Mathematicae 24, 77–86 (1996)MathSciNetMATHGoogle Scholar
  4. [61]
    Bingham, N. H., Goldie, C. M. and Teugels, J. L.: Regular Variation. Cambridge University Press, Cambridge (1987)MATHGoogle Scholar
  5. [84]
    Charpentier, A. and Segers, J.: Tails of multivariate Archimedean copulas. Journal of Multivariate Analysis, 100, 1521–1537 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. [177]
    Genest, C. and Rivest, L.-P.: A characterization of Gumbel’s family of extreme value distributions. Statistics and Probability Letters, 8, 207–211 (1989)MathSciNetMATHCrossRefGoogle Scholar
  7. [184]
    Gnedenko, B. V.: Sur la distribution limite du terme maximum d’une série aléatoire. Annals of Mathematics, 44, 423–453 (1943)MathSciNetMATHCrossRefGoogle Scholar
  8. [207]
    Jaworski, P.: On uniform tail expansions of multivariate copulas and wide convergence of measures. Applicationes Mathematicae, 33(2), 159–184 (2006)MathSciNetMATHCrossRefGoogle Scholar
  9. [211]
    Joe, H.: Multivariate Models and Dependence Concepts. Chapman and Hall, London (1997)MATHCrossRefGoogle Scholar
  10. [213]
    Joe, H., Li, H. and Nikoloulopoulos, A. K.: Tail dependence functions and vine copulas. Journal of Multivariate Analysis, 101, 252–270 (2010)MathSciNetMATHCrossRefGoogle Scholar
  11. [242]
    Klüppelberg, C.: Asymptotic ordering of distribution functions and convolution semigroups. Semigroup Forum, 40(1), 77–92 (1990)MathSciNetMATHCrossRefGoogle Scholar
  12. [243]
    Klüppelberg, C.: Asymptotic ordering of risks and ruin probabilities. Insurance: Mathematics and Economics, 12, 259–264 (1993)MathSciNetMATHCrossRefGoogle Scholar
  13. [244]
    Klüppelberg, C., Kuhn, G. and Peng, L.: Semi-parametric models for the multivariate tail dependence function – the asymptotically dependent. Scandinavian Journal of Statistics, 35(4), 701–718 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. [245]
    Klüppelberg, C. and Resnick, S. I.: The Pareto copula, aggregation of risks, and the emperor’s socks. Journal of Applied Probability, 45(1), 67–84 (2008)MathSciNetMATHCrossRefGoogle Scholar
  15. [280]
    Li, H.: Orthant tail dependence of multivariate extreme value distributions. Journal of Multivariate Analysis 100, 243–256 (2009)MathSciNetMATHCrossRefGoogle Scholar
  16. [281]
    Li, H. and Sun, Y.: Tail dependence for heavy-tailed scale mixtures of multivariate distributions. Journal of Applied Probability 46(4), 925–937 (2009)MathSciNetMATHCrossRefGoogle Scholar
  17. [284]
    Li, W. V. and Shao, Q.-M.: Gaussian processes: Inequalities, small ball probabilities and applications. Stochastic Processes: Theory and Methods, edited by C.R. Rao and D. Shanbhag. Handbook of Statistics, Elsevier, New York, 19, 533–598 (2001)MathSciNetGoogle Scholar
  18. [300]
    Mainik, G. and Rüschendorf, L.: Ordering of multivariate risk models with respect to extreme portfolio losses. Statistics & Risk Modeling, 29(1), 73–106 (2012)MathSciNetMATHCrossRefGoogle Scholar
  19. [307]
    Marshall, A. W. and Olkin, I.: A multivariate exponential distribution. Journal of the American Statistical Association, 2, 84–98 (1967)MathSciNetGoogle Scholar
  20. [308]
    Marshall, A. W. and Olkin, I.: Inequalities: theory of majorization and its applications. Academic Press, Inc., New York (1979)MATHGoogle Scholar
  21. [309]
    Marshall, A. W. and Olkin, I.: Domains of attraction of multivariate extreme value distributions. Annuals of Probability, 11, 168–177 (1983)MathSciNetMATHCrossRefGoogle Scholar
  22. [335]
    Müller, A. and Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, Chichester (2002)MATHGoogle Scholar
  23. [360]
    Nikoloulopoulos, A. K., Joe, H. and Li, H.: Extreme value properties of multivariate t copulas. Extremes, 12, 129–148 (2009)MathSciNetMATHCrossRefGoogle Scholar
  24. [374]
    Pickands, J.: Multivariate negative exponential and extreme value distributions. Unpublished manuscript, University of Pennsylvania (1980)Google Scholar
  25. [387]
    Resnick, S. I.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York (2007)MATHGoogle Scholar
  26. [389]
    Rojo, J.: A pure tail ordering based on the ratio of the quantile functions. Annals of Statistics, 20(1), 570–579 (1992)MathSciNetMATHCrossRefGoogle Scholar
  27. [391]
    Rojo, J.: Relationships between pure tail orderings of lifetime distributions and some concepts of residual life. Annals of the Institute of Statistical Mathematics, 48(2), 247–255 (1996)MathSciNetMATHCrossRefGoogle Scholar
  28. [426]
    Shaked, M. and Shanthikumar, J. G.: Stochastic Orders. Springer, New York (2007)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsWashington State UniversityPullmanUSA

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