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

Abstract

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 $\bs{\beta}$ and a fixed-dimensional regression coefficient $\bs{\gamma}$ 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) $(\widehat{\bs{\beta}}, \widehat{\bs{\gamma}})$ of $(\bs{\beta}, \bs{\gamma})$ Furthermore, we derive the asymptotic normal distribution of the MLE by characterizing its asymptotic representation. The asymptotic distribution of $\widehat{\bs{\gamma}}$ is biased, while that of $\widehat{\bs{\beta}}$ is not. This phenomenon can be attributed to the different convergence rates of $\widehat{\bs{\gamma}}$ and $\widehat{\bs{\beta}}$. 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 Erd\H{o}s--R\'{e}nyi comparison graph with a diverging number of covariates. Numerical studies and a real data analysis demonstrate our theoretical findings.