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 $\text{bs}\beta$ and a fixed-dimensional regression coefficient $\text{bs}\gamma$ 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) $(\widehat{\text{bs}\beta },\widehat{\text{bs}\gamma })$ of $(\text{bs}\beta ,\text{bs}\gamma )$ Furthermore, we derive the asymptotic normal distribution of the MLE by characterizing its asymptotic representation. The asymptotic distribution of $\widehat{\text{bs}\gamma }$ is biased, while that of $\widehat{\text{bs}\beta }$ is not. This phenomenon can be attributed to the different convergence rates of $\widehat{\text{bs}\gamma }$ and $\widehat{\text{bs}\beta }$. 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.