The Joint Distribution of the Discrete Random Set Vector and Bivariate Coarsening at Random Models

  • Zheng Wei
  • Baokun Li
  • Tonghui WangEmail author
Part of the Studies in Computational Intelligence book series (SCI, volume 892)


In this paper, the characterization of the joint distribution of random set vector by the belief function is investigated. A method for constructing the joint distribution of discrete bivariate random sets through copula is given, and a routine of calculating the corresponding bivariate coarsening at random model of finite random sets is obtained. Several examples are given to illustrate our results.


Random set Copula Coarsening at random model Joint belief function Jointly monotone of infinite order 



The authors would like to thank Professor Hung T. Nguyen for introducing this interesting topic to us.


  1. 1.
    D.A. Alvarez, A Monte Carlo-based method for the estimation of lower and upper probabilities of events using infinite random sets of indexable type. Fuzzy Sets Syst. 160, 384–401 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 28, 325–339 (1967)MathSciNetCrossRefGoogle Scholar
  3. 3.
    R.D. Gill, M.J. Van der Laan, J.M. Robins, Coarsening at random: characterizations, conjectures, counter-examples. Springer Lect. Notes Stat. 123, 149–170 (1997)zbMATHGoogle Scholar
  4. 4.
    R.D. Gill, P.D. Grunwald, An algorithmic and a geometric characterization of coarsening at random. Ann. Stat. 36(5), 2049–2422 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Grunwald, J. Halpern, Updating probabilities. J. Artif. Intell. Res. 19, 243–278 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    J. Harry, Multivariate Models and Dependence Concepts (Chapman & Hall, London, 1997)Google Scholar
  7. 7.
    D.F. Heitjan, D.B. Rubin, Ignorability and coarse data. Ann. Stat. 19(4), 2244–2253 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Jaeger, Ignorability for categorical data. Ann. Stat. 33, 1964–1981 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    B. Li, T. Wang, Computational aspects of the coarsening at random model and the Shapley value. Inf. Sci. 177, 3260–3270 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    I.S. Molchanov, Theory of Random Sets (Springer, Berlin, 2005)Google Scholar
  11. 11.
    R.B. Nelsen, An introduction to Copulas, 2nd edn. (Springer, New York, 2006)Google Scholar
  12. 12.
    H.T. Nguyen, An Introduction to Random Sets (CRC Press, Boca Raton, FL, 2006)Google Scholar
  13. 13.
    H.T. Nguyen, T. Wang, Belief functions and random sets, in The IMA Volumes in Mathematics and Its Applications, vol. 97 (Springer, New York, 1997), pp. 243–255Google Scholar
  14. 14.
    H.T. Nguyen, Lecture Notes in: Statistics with copulas for applied research, Department of Economics, Chiang Mai University, Chiang Mai, Thailand (2013)Google Scholar
  15. 15.
    B. Schmelzer, Characterizing joint distributions of random sets by multivariate capacities. Intern. J. Approx. Reason. 53, 1228–1247 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    B. Schmelzer, Joint distributions of random sets and their relation to copulas, Modeling Dependence in Econometrics. Advances in Intelligent Systems and Computing, vol. 251 (Springer International Publishing, Switzerland, 2014).
  17. 17.
    B. Schmelzer, Joint distributions of random sets and their relation to copulas. Intern. J. Approx. Reason. (2015). In PressGoogle Scholar
  18. 18.
    G. Shafer, A Mathematical Theory of Evidence (Princeton University Press, New Jersey, Princeton, 1976)Google Scholar
  19. 19.
    A. Sklar, Fonctions de répartition án dimensions et leurs marges. Publ. Inst. Statist. Univ. 8, 229–231 (1959). ParisGoogle Scholar
  20. 20.
    A.A. Tsiatis, Semiparametric Theory and Missing Data (Springer, New York, 2006)Google Scholar
  21. 21.
    A.A. Tsiatis, M. Davidian, W. Cao, Improved doubly robust estimation when data are monotonely carsened, with application to longitudinal studies with dropout. Biometrics 67(2), 536–545 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Z. Wei, T. Wang, W. Panichkitkosolkul, Dependence and association concepts through copulas, in Modeling Dependence in Econometrics. Advances in Intelligent Systems and Computing, vol. 251 (Springer International Publishing, Switzerland, 2014), pp. 113–126.
  23. 23.
    Z. Wei, T. Wang, B. Li, P.A. Nguyen, The joint belief function and Shapley value for the joint cooperative game, in Econometrics of Risk, Studies in Computational Intelligence, vol. 583 (Springer International Publishing, Switzerland, 2015), pp. 115–133.
  24. 24.
    Z. Wei, T. Wang, P.A. Nguyen, Multivariate dependence concepts through copulas. Intern. J. Approx. Reason. (2015). In PressGoogle Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA
  2. 2.School of StatisticsSouthwestern University of Finance and EconomyChengduChina
  3. 3.Department of Mathematics and StatisticsUniversity of MaineOronoUSA

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