Methodology and Computing in Applied Probability

, Volume 20, Issue 3, pp 1013–1027 | Cite as

Operator Tail Dependence of Copulas

  • Haijun LiEmail author


A notion of tail dependence based on operator regular variation is introduced for copulas, and the standard tail dependence used in the copula literature is included as a special case. The non-standard tail dependence with marginal power scaling functions having possibly distinct tail indexes is investigated in detail. We show that the copulas with operator tail dependence, incorporated with regularly varying univariate margins, give rise to a rich class of the non-standard multivariate regularly varying distributions. We also show that under some mild conditions, the copula of a non-standard multivariate regularly varying distribution has the standard tail dependence of order 1. Some illustrative examples are given.


Operator regular variation Tail dependence Extreme value analysis Tail risk 

Mathematics Subject Classification (2010)

62H20 62E20 


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The author would like to sincerely thank referees and an associate editor for their insightful comments, which led to an improvement of the presentation and motivation of this paper. The author would also like to thank Mark Meerschaert for a useful discussion.


  1. Arnold VI, Gusein-Zade SM, Varchenko AN (2012) Quasihomogeneous and semiquasihomogeneous singularities. In: Singularities of differentiable maps, vol 1. Modern Birkhäuser Classics. Birkhäuser, BostonGoogle Scholar
  2. Balkema G, Embrechts P (2007) High risk scenarios and extremes: a geometric approach. European Mathematical Society, ZürichCrossRefzbMATHGoogle Scholar
  3. Hua L, Joe H (2011) Tail order and intermediate tail dependence of multivariate copulas. J Multivar Anal 102:1454–1471MathSciNetCrossRefzbMATHGoogle Scholar
  4. Hua L, Joe H, Li H (2014) Relations between hidden regular variation and tail order of copulas. J Appl Probab 51(1):37–57MathSciNetCrossRefzbMATHGoogle Scholar
  5. Jaworski P (2006) On uniform tail expansions of multivariate copulas and wide convergence of measures. Applicationes Mathematicae 33(2):159–184MathSciNetCrossRefzbMATHGoogle Scholar
  6. Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, LondonCrossRefzbMATHGoogle Scholar
  7. Joe H, Li H (2011) Tail risk of multivariate regular variation. Methodol Comput Appl Probab 13:671–693MathSciNetCrossRefzbMATHGoogle Scholar
  8. Joe H, Li H, Nikoloulopoulos AK (2010) Tail dependence functions and vine copulas. J Multivar Anal 101:252–270MathSciNetCrossRefzbMATHGoogle Scholar
  9. Klüppelberg C, Kuhn G, Peng L (2008) Semi-parametric models for the multivariate tail dependence function – the asymptotically dependent. Scand J Stat 35(4):701–718MathSciNetCrossRefzbMATHGoogle Scholar
  10. Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions, vol 1. WileyGoogle Scholar
  11. Ledford AW, Tawn JA (1996) Statistics for near independence in multivariate extreme values. Biometrika 83(1):169–187MathSciNetCrossRefzbMATHGoogle Scholar
  12. Ledford AW, Tawn JA (1998) Concomitant tail behaviour for extremes. Adv Appl Probab 30(1):197–215MathSciNetCrossRefzbMATHGoogle Scholar
  13. Li H (2008) Tail dependence comparison of survival Marshall-Olkin copulas. Methodol Comput Appl Probab 10:39–54MathSciNetCrossRefzbMATHGoogle Scholar
  14. Li H (2009) Orthant tail dependence of multivariate extreme value distributions. J Multivar Anal 100:243–256MathSciNetCrossRefzbMATHGoogle Scholar
  15. Li H (2013) Toward a copula theory for multivariate regular variation. In: Durante F, Härdle W, Jaworski P (eds) Lecture notes in statistics, vol 213. Springer, pp 177–199Google Scholar
  16. Li H, Hua L (2015) Higher order tail densities of copulas and hidden regular variation. J Multivar Anal 138:143–155MathSciNetCrossRefzbMATHGoogle Scholar
  17. Li H, Sun Y (2009) Tail dependence for heavy-tailed scale mixtures of multivariate distributions. J Appl Prob 46(4):925–937MathSciNetCrossRefzbMATHGoogle Scholar
  18. Li H, Wu P (2013) Extremal dependence of copulas: a tail density approach. J Multivar Anal 114:99–111MathSciNetCrossRefzbMATHGoogle Scholar
  19. Marshall AW, Olkin I (1967) A multivariate exponential distribution. J Amer Statist Assoc 2:84–98MathSciNetzbMATHGoogle Scholar
  20. McNeil AJ, Frey R, Embrechts P (2015) Quantitative risk management: concepts, techniques, and tools. Princeton University Press, PrincetonzbMATHGoogle Scholar
  21. Meerschaert MM, Scheffler H-P (1999) Moment estimator for random vectors with heavy tails. J Multivar Anal 71:145–159MathSciNetCrossRefzbMATHGoogle Scholar
  22. Meerschaert MM, Scheffler H-P (2001) Limit distributions for sums of independent random vectors. WileyGoogle Scholar
  23. Meerschaert MM, Scheffler H-P, Stoev S (2013) Extreme value theory with operator norming. Extremes 16:407–428MathSciNetCrossRefzbMATHGoogle Scholar
  24. Nikoloulopoulos AK, Joe H, Li H (2009) Extreme value properties of multivariate t copulas. Extremes 12:129–148MathSciNetCrossRefzbMATHGoogle Scholar
  25. Noland J, Panorska A, McCulloch JH (2001) Estimation of stable spectral measures. Math Comput Model 34:1113–1122MathSciNetCrossRefzbMATHGoogle Scholar
  26. Resnick S (1987) Extreme values, regularly variation, and point processes. Springer, New YorkCrossRefzbMATHGoogle Scholar
  27. Resnick S (2002) Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5(4):303–336MathSciNetCrossRefzbMATHGoogle Scholar
  28. Resnick S (2007) Heavy-tail phenomena: probabilistic and statistical modeling. Springer, New YorkzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsWashington State UniversityPullmanUSA

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