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) $(\widehat{\bsβ}, \widehat{\bsγ})$ of $(\bsβ, \bsγ)$ Furthermore, we derive the asymptotic normal distribution of the MLE by characterizing its asymptotic representation. The asymptotic distribution of $\widehat{\bsγ}$ is biased, while that of $\widehat{\bsβ}$ is not. This phenomenon can be attributed to the different convergence rates of $\widehat{\bsγ}$ and $\widehat{\bsβ}$. 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 Erdős--Rényi comparison graph with a diverging number of covariates. Numerical studies and a real data analysis demonstrate our theoretical findings.