, Volume 81, Issue 5, pp 485–492 | Cite as

A complete characterization of bivariate densities using the conditional percentile function

  • Indranil Ghosh


It is well known that joint bivariate densities cannot always be characterized by the corresponding two conditional densities. Hence, additional requirements have to be imposed. In the form of a conjecture, Arnold et al. (J Multivar Anal 99:1383–1392, 2008) suggested using any one of the two conditional densities and replacing the other one by the corresponding conditional percentile function. In this article we establish, in affirmative, this conjecture and provide several illustrative examples.


Bivariate distribution Conditional density Conditional percentile Characterization 

Mathematics Subject Classification

60E05 62E10 62H05 


Compliance with ethical standards

Conflict of interest

The corresponding author states that there is no conflict of interest.


  1. Arnold BC (1983) Pareto distributions. International Cooperative Publishing House, Fairland, MDGoogle Scholar
  2. Arnold BC, Gokhale DV (1994) On uniform marginal representations of contingency tables. Stat Probab Lett 21:311–316MathSciNetCrossRefzbMATHGoogle Scholar
  3. Arnold BC, Gokhale DV (1998) Distributions most nearly compatible with given families of conditional distributions. The finite discrete case. Test 7:377–390MathSciNetCrossRefzbMATHGoogle Scholar
  4. Arnold BC, Castillo E, Sarabia JM (1996) Specification of distributions by combinations of marginal and conditional distributions. Stat Probab Lett 26:153–157MathSciNetCrossRefzbMATHGoogle Scholar
  5. Arnold BC, Castillo E, Sarabia JM (1999) Conditional specification of statistical models. Springer, New YorkzbMATHGoogle Scholar
  6. Arnold BC, Castillo E, Sarabia JM (2001a) Quantification of incompatibility of conditional and marginal information. Commun Stat Theory Methods 30:381–395MathSciNetCrossRefzbMATHGoogle Scholar
  7. Arnold BC, Castillo E, Sarabia JM (2001b) Conditionally specified distributions: an introduction. Stat Sci 16:249–265MathSciNetCrossRefzbMATHGoogle Scholar
  8. Arnold BC, Castillo E, Sarabia JM (2008) Bivariate distributions characterized by one family of conditionals and conditional percentile or mode functions. J Multivar Anal 99:1383–1392MathSciNetCrossRefzbMATHGoogle Scholar
  9. Asimit V, Vernic R, Zitikis R (2013) Evaluating risk measures and capital allocations based on multi-losses driven by a heavy-tailed background risk: the multivariate Pareto-II model. Risk 1(1):14–33Google Scholar
  10. Asimit V, Vernic R, Zitikis R (2016) Background risk models and stepwise portfolio construction. Methodol Comput Appl Probab 18:805–827Google Scholar
  11. Balakrishnan N, Lai CD (2009) Continuous bivariate distributions, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  12. Cacoullos T, Papageorgiou H (1995) Characterizations of discrete distributions by a conditional distribution and a regression function. Ann Inst Stat Math 35:95–103MathSciNetCrossRefzbMATHGoogle Scholar
  13. Castillo E, Galambos J (1987) Lifetime regression models based on a functional equation of physical nature. J Appl Probab 24:160–169MathSciNetCrossRefzbMATHGoogle Scholar
  14. Ghosh I, Balakrishnan N (2014) Partial or complete characterization of a bivariate distribution based on one conditional distribution and partial specification of the mode function of the other conditional distribution. Stat Methodol 16:1–13MathSciNetCrossRefGoogle Scholar
  15. Satterthwaite SP, Hutchinson TP (1978) A generalization of Gumbel’s bivariate logistic distribution. Metrika 25:163–170MathSciNetCrossRefzbMATHGoogle Scholar
  16. Song CC, Li LA, Chen CH (2010) Compatibility of finite discrete conditional distributions. Stat Sin 20:423–440zbMATHGoogle Scholar
  17. Tan M, Tian GL, Ng KW (2010) Bayesian missing data problems: EM, data augmentation and non-iterative computation. Chapman and Hall, Boca RatonzbMATHGoogle Scholar
  18. Tian GL, Tan M, Ng KW, Tang ML (2009) A unified method for checking compatibility and uniqueness for finite discrete conditional distributions. Commun Stat Theory Methods 38:115–129MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of North Carolina at WilmingtonWilmingtonUSA

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