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February 2004 MM algorithms for generalized Bradley-Terry models
David R. Hunter
Ann. Statist. 32(1): 384-406 (February 2004). DOI: 10.1214/aos/1079120141


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.


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David R. Hunter. "MM algorithms for generalized Bradley-Terry models." Ann. Statist. 32 (1) 384 - 406, February 2004.


Published: February 2004
First available in Project Euclid: 12 March 2004

zbMATH: 1105.62359
MathSciNet: MR2051012
Digital Object Identifier: 10.1214/aos/1079120141

Primary: 62F07 , 65D15

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

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.32 • No. 1 • February 2004
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