Paired comparison models, such as Bradley-Terry and Thurstone-Mosteller, are commonly used to estimate relative strengths of pairwise compared items in tournament-style data. We discuss estimation of paired comparison models with a ridge penalty. A new approach is derived which combines empirical Bayes and composite likelihoods without any need to re-fit the model, as a convenient alternative to cross-validation of the ridge tuning parameter. Simulation studies demonstrate much better predictive accuracy of the new approach relative to ordinary maximum likelihood. A widely used alternative, the application of a standard bias-reducing penalty, is also found to improve appreciably the performance of maximum likelihood; but the ridge penalty, with tuning as developed here, yields greater accuracy still. The methodology is illustrated through application to 28 seasons of English Premier League football.

翻译：配对比较模型，如Bradley-Terry模型和Thurstone-Mosteller模型，常用于估计锦标赛式数据中成对比较项目的相对强度。本文讨论了带岭惩罚项的配对比较模型的估计问题。我们推导了一种新方法，该方法结合了经验贝叶斯与复合似然，无需重新拟合模型，为岭调优参数的交叉验证提供了一个便捷的替代方案。模拟研究表明，相较于普通最大似然估计，新方法具有显著更好的预测准确性。一种广泛使用的替代方案——应用标准的偏差减少惩罚——也被发现能显著提升最大似然估计的性能；但采用本文所开发的调优方法的岭惩罚，仍能带来更高的准确性。该方法通过应用于28个赛季的英格兰足球超级联赛数据得到了说明。