Annals of Statistics

MM algorithms for generalized Bradley-Terry models

David R. Hunter

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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 or 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.

Article information

Ann. Statist., Volume 32, Number 1 (2004), 384-406.

First available in Project Euclid: 12 March 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F07: Ranking and selection 65D15: Algorithms for functional approximation

Bradley-Terry model Luce's choice axiom maximum likelihood estimation MM algorithm Newton-Raphson Plackett-Luce model


Hunter, David R. MM algorithms for generalized Bradley-Terry models. Ann. Statist. 32 (2004), no. 1, 384--406. doi:10.1214/aos/1079120141.

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