Inference in a generalized Bradley-Terry model for paired comparisons with covariates and a growing number of subjects

Motivated by the home-field advantage in sports, we propose a generalized Bradley--Terry model that incorporates covariate information for paired comparisons. It has an n-dimensional merit parameter bsβ and a fixed-dimensional regression coefficient bsγ for covariates. When the number of subjects n approaches infinity and the number of comparisons between any two subjects is fixed, we show the uniform consistency of the maximum likelihood estimator (MLE) (widehatbsβ, widehatbsγ) of (bsβ, bsγ) Furthermore, we derive the asymptotic normal distribution of the MLE by characterizing its asymptotic representation. The asymptotic distribution of widehatbsγ is biased, while that of widehatbsβ is not. This phenomenon can be attributed to the different convergence rates of widehatbsγ and widehatbsβ. To the best of our knowledge, this is the first study to explore the asymptotic theory in paired comparison models with covariates in a high-dimensional setting. The consistency result is further extended to an ErdHos--R'enyi comparison graph with a diverging number of covariates. Numerical studies and a real data analysis demonstrate our theoretical findings.