Characterizations of discrete distributions by a conditional distribution and a regression function

  • T. Cacoullos
  • H. Papageorgiou


The bivariate distribution of (X, Y), whereX andY are non-negative integer-valued random variables, is characterized by the conditional distribution ofY givenX=x and a consistent regression function ofX onY. This is achieved when the conditional distribution is one of the distributions: a) binomial, Poisson, Pascal or b) a right translation of these. In a) the conditional distribution ofY is anx-fold convolution of another random variable independent ofX so thatY is a generalized distribution. A main feature of these characterizations is that their proof does not depent on the specific form of the regression function. It is also indicated how these results can be used for good-ness-of-fit purposes.


Conditional Distribution Regression Function Negative Binomial Distribution Discrete Distribution Negative Binomial 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Cacoullos, T. and Papageorgiou, H. (1980). On some bivariate probability models applicable to traffic accidents and fatalities,Int. Statist. Rev.,48, 345–356.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Cacoullos, T. and Papageorgiou, H. (1981). On bivariate discrete distribution generated by compounding,Statistical Distributions in Scientific Work, Vol. 4, D. Reidel Publishing Company, Holland, 197–212.MATHGoogle Scholar
  3. [3]
    Charalambides, Ch. H. (1977). On the generalized discrete distributions and the Bell polynomials,Sankhyã, B,39 36–44.MathSciNetMATHGoogle Scholar
  4. [4]
    Dahiya, R. C. and Korwar, R. M. (1977). On characterizing some bivariate discrete distributions by linear regression,Sankhyã, A39, 124–129.MathSciNetMATHGoogle Scholar
  5. [5]
    Khatri, C. G. (1978a). Characterization of some discrete distributions by linear regression,J. Indian Statist. Ass.,16, 49–58.MathSciNetGoogle Scholar
  6. [6]
    Khatri, C. G. (1978b). Characterization of some multivariate distributions by conditional distributions and linear regression,J. Indian Statist. Ass.,16, 59–70.MathSciNetGoogle Scholar
  7. [7]
    Korwar, R. M. (1975). On characterizing some discrete distributions by linear regression,Commun. Statist.,4, 1133–1147.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Papageorgiou, H. (1983). On characterizing some discrete distributions,Austral. J. Statist.,25, in press.Google Scholar
  9. [9]
    Seshadri, V. and Patil, G. P. (1964). A characterization of a bivariate distribution by the marginal and the conditional distributions of the same component.Ann. Inst. Statist. Math.,15, 215–221.CrossRefGoogle Scholar
  10. [10]
    Teicher, H. (1961). Identifiability of mixtures,Ann. Math. Statist.,32, 244–248.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Xekalaki, E. (1980). On characterizing the bivariate Poisson, binomial and negative binomial distributions,Colloquia Mathematica Societatis János Bolyai,21, North-Holland, 369–379.MathSciNetGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1983

Authors and Affiliations

  • T. Cacoullos
    • 1
  • H. Papageorgiou
    • 1
  1. 1.University of AthensAthensGreece

Personalised recommendations