pp 1–16 | Cite as

Estimation and testing of multiplicative models for frequency data

  • Antonoio ForcinaEmail author


This paper is about models for a vector of probabilities whose elements must have a multiplicative structure and sum to 1 at the same time; in certain applications, like basket analysis, these models may be seen as a constrained version of quasi-independence. After reviewing the basic properties of the models, their geometric features as a curved exponential family are investigated. An improved algorithm for computing maximum likelihood estimates is introduced and new insights are provided on the underlying geometry. The asymptotic distribution of three statistics for hypothesis testing are derived and a small simulation study is presented to investigate the accuracy of asymptotic approximations.


Curved exponential families Mixed parametrization Log linear models Quasi-independence 



The author would like to thank A. Klimova and T. Rudas for sharing ideas concerning Relational models and for several very enlightening discussions, A. Salvan for comment on the nature of the curved exponential family and P. Giudici for providing the basked data.

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Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di EconomiaUniversity of PerugiaPerugiaItaly

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