, Volume 60, Issue 3, pp 261–277 | Cite as

The association between two random elements: A complete characterization and odds ratio models

  • Gerhard OsiusEmail author


For random elements X and Y (e.g. vectors) a complete characterization of their association is given in terms of an odds ratio function. The main result establishes for any odds ratio function and any pre-specified marginals the unique existence of a corresponding joint distribution (the joint density is obtained as a limit of an iterative procedure of marginal fittings). Restricting only the odds ratio function but not the marginals leads to semi-parmetric association models for which statistical inference is available for samples drawn conditionally on either X or Y. Log-bilinear association models for random vectors X and Y are introduced which generalize standard (regression) models by removing restrictions on the marginals. In particular, the logistic regression model is recognized as a log-bilinear association model. And the joint distribution of X and Y is shown to be multivariate normal if and only if both marginals are normal and the association is log-bilinear.


Association Marginal fitting Multivariate normal distribution Odds ratio Semi-parametric models 


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Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Institut für Statistik, Fachbereich 3Universität BremenBremenGermany

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