MM algorithms for generalized Bradley-Terry models

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The Bradley-Terry model for paired comparisons is a simple and much-studied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs. Among the many studies of the model in the past 75 years, numerous authors have generalized it in several directions, sometimes providing iterative algorithms for obtaining maximum likelihood estimates for the generalizations. Building on a theory of algorithms known by the initials MM, for minorization-maximization, this paper presents a powerful technique for producing iterative maximum likelihood estimation algorithms for a wide class of generalizations of the Bradley-Terry model. While algorithms for problems of this type have tended to be custom-built in the literature, the techniques in this paper enable their mass production. Simple conditions are stated that guarantee that each algorithm described will produce a sequence that converges to the unique maximum likelihood estimator. Several of the algorithms and convergence results herein are new.

Original languageEnglish (US)
Pages (from-to)384-406
Number of pages23
JournalAnnals of Statistics
Issue number1
StatePublished - Feb 2004

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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