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Hierarchical Aitchison–Silvey models for incomplete binary sample spaces

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  • Klimova, Anna
  • Rudas, Tamás

Abstract

Multivariate sample spaces may be incomplete Cartesian products, when certain combinations of the categories of the variables are not possible. Traditional log-linear models, which generalize independence and conditional independence, do not apply in such cases, as they may associate positive probabilities with the non-existing cells. To describe the association structure in incomplete sample spaces, this paper develops a class of hierarchical multiplicative models which are defined by setting certain non-homogeneous generalized odds ratios equal to one and are named after Aitchison and Silvey who were among the first to consider such ratios. These models are curved exponential families that do not contain an overall effect and, from an algebraic perspective, are non-homogeneous toric ideals. The relationship of this model class with log-linear models and quasi log-linear models is studied in detail in terms of both statistics and algebraic geometry. The existence of maximum likelihood estimates and their properties, as well as the relevant algorithms are also discussed.

Suggested Citation

  • Klimova, Anna & Rudas, Tamás, 2022. "Hierarchical Aitchison–Silvey models for incomplete binary sample spaces," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    • Handle: RePEc:eee:jmvana:v:187:y:2022:i:c:s0047259x21000865
    DOI: 10.1016/j.jmva.2021.104808
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    References listed on IDEAS

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    1. Agresti, Alan, 1983. "A simple diagonals-parameter symmetry and quasi-symmetry model," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 313-316, October.
    2. Henrik Nyman & Johan Pensar & Timo Koski & Jukka Corander, 2016. "Context-specific independence in graphical log-linear models," Computational Statistics, Springer, vol. 31(4), pages 1493-1512, December.
    3. Klimova, Anna & Rudas, Tamás, 2016. "On the closure of relational models," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 440-452.
    4. Antonoio Forcina, 2019. "Estimation and testing of multiplicative models for frequency data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(7), pages 807-822, October.
    5. Klimova, Anna & Rudas, Tamás & Dobra, Adrian, 2012. "Relational models for contingency tables," Journal of Multivariate Analysis, Elsevier, vol. 104(1), pages 159-173, February.
    6. Anna Klimova & Tamás Rudas, 2015. "Iterative Scaling in Curved Exponential Families," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(3), pages 832-847, September.
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