Multivariate Generalized Marshall–Olkin Distributions and Copulas

  • Jianhua Lin
  • Xiaohu Li


The multivariate generalized Marshall–Olkin distributions, which include the multivariate Marshall–Olkin exponential distribution due to Marshall and Olkin (J Am Stat Assoc 62:30–41, 1967) and multivariate Marshall–Olkin type distribution due to Muliere and Scarsini (Ann Inst Stat Math 39:429–441, 1987) as special cases, are studied in this paper. We derive the survival copula and the upper/lower orthant dependence coefficient, build the order of these survival copulas, and investigate the evolution of dependence of the residual life with respect to age. The main conclusions developed here are both nice extensions of the main results in Li (Commun Stat Theory Methods 37:1721–1733, 2008a, Methodol Comput Appl Probab 10:39–54, 2008b) and high dimensional generalizations of some results on the bivariate generalized Marshall–Olkin distributions in Li and Pellerey (J Multivar Anal 102:1399–1409, 2011).


IFR Marginal distribution NBU Proportional hazard PUOD Residual life Upper/lower orthant tail dependent 

AMS 2000 Subject Classifications

60E15 62H20 


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  1. Balakrishnan N, Lai CD (2009) Continuous bivariate distributions, 2nd edn. Springer, New YorkMATHGoogle Scholar
  2. Barlow RE, Proschan F (1981) Statistical theory of reliability and life testing. To begin with, Silver SpringGoogle Scholar
  3. Boyd SP, Vandenberghe L (2004) Convex optimization. Cambridge University Press, New YorkCrossRefMATHGoogle Scholar
  4. Denuit M, Dhaene J, Goovaerts M, Kaas R (2005) Actuarial theory for dependent risks. Wiley, New YorkCrossRefGoogle Scholar
  5. Frees EW, Carriere J, Valdez E (1996) Annuity valuation with dependent mortality. J Risk Insur 63:229–261CrossRefGoogle Scholar
  6. Galambos J, Kotz S (1978) Characterizations of probability distributions. Springer, BerlinMATHGoogle Scholar
  7. Jaworski P, Durante F, Härdle W, Rychlik T (2010) Copula theory and its applications. Springer, New YorkCrossRefMATHGoogle Scholar
  8. Joe H (1993) Parametric families of multivariate distributions with given maginals. J Multivar Anal 46:262–282CrossRefMATHMathSciNetGoogle Scholar
  9. Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, LondonCrossRefMATHGoogle Scholar
  10. Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions. In: Models and applications, vol 1. Wiley, New YorkGoogle Scholar
  11. Lai CD, Xie M (2006) Stochastic ageing and dependence for reliability. Springer, New YorkMATHGoogle Scholar
  12. Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley, HobokenMATHGoogle Scholar
  13. Li H (2008a) Duality of the multivariate distributions of Marshall–Olkin type and tail dependence. Commun Stat, Theory Methods 37:1721–1733CrossRefMATHGoogle Scholar
  14. Li H (2008b) Tail dependence comparison of survival Marshall–Olkin copulas. Methodol Comput Appl Probab 10:39–54CrossRefMATHMathSciNetGoogle Scholar
  15. Li X, Pellerey F (2011) Generalized Marshall–Olkin distributions, and related bivariate aging properties. J Multivar Anal 102:1399–1409CrossRefMATHMathSciNetGoogle Scholar
  16. Lu JC (1989) Weibull extension of the Freund and Marshall–Olkin bivariate exponential model. IEEE Trans Reliab 38:615–619CrossRefMATHGoogle Scholar
  17. Mai J, Scherer M (2010) The Pickands representation of survival Marshall–Olkin copulas. Stat Probab Lett 80:357–360CrossRefMATHMathSciNetGoogle Scholar
  18. Mai J, Scherer M (2011) Reparameterizing Marshall–Olkin copulas with applications to sampling. J Stat Comput Simul 81:59–78CrossRefMATHMathSciNetGoogle Scholar
  19. Marshall AW, Olkin I (1967) A multivariate exponential distribution. J Am Stat Assoc 62:30–41CrossRefMATHMathSciNetGoogle Scholar
  20. Marshall AW, Olkin I (2007) Life distributions. Springer, New YorkMATHGoogle Scholar
  21. McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management. Princeton University Press, Princeton, New YorkMATHGoogle Scholar
  22. Muliere P, Scarsini M (1987) Characterization of a Marshall–Olkin type class of distributions. Ann Inst Stat Math 39:429–441CrossRefMATHMathSciNetGoogle Scholar
  23. Nelsen RB (2006) An introduction to copulas, 2nd edn, Springer, New YorkMATHGoogle Scholar
  24. Sarhan AM, Hamilton DC, Smitha B, Kundub D (2010) The bivariate generalized linear failure rate distribution and its multivariate extension. Comput Stat Data Anal 55:644–654CrossRefGoogle Scholar
  25. Scarsini M (1984) One measure of concordance. Stochastica 8:201–218MATHMathSciNetGoogle Scholar
  26. Schmidt R (2002) Tail dependence for elliptically contoured distributions. Math Methods Oper Res 55:301–327CrossRefMATHMathSciNetGoogle Scholar
  27. Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New YorkCrossRefMATHGoogle Scholar
  28. Wu C (1997) New characterization of Marshall–Olkin type distributions via bivariate random summation scheme. Stat Probab Lett 34:171–178CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina

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