## The Annals of Statistics

### MM algorithms for generalized Bradley-Terry models

David R. Hunter

#### Abstract

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

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

Dates
First available in Project Euclid: 12 March 2004

https://projecteuclid.org/euclid.aos/1079120141

Digital Object Identifier
doi:10.1214/aos/1079120141

Mathematical Reviews number (MathSciNet)
MR2051012

Zentralblatt MATH identifier
1105.62359

#### Citation

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

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