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

TL;DR

This paper develops a generalized Bradley-Terry model incorporating covariates for paired comparisons, establishing the asymptotic properties of the MLE in high-dimensional settings with growing subjects and covariates.

Contribution

It introduces the first asymptotic theory for paired comparison models with covariates in high-dimensional regimes, including consistency and distribution results.

Findings

MLE is uniformly consistent as subjects grow large.

Asymptotic normality of the MLE is derived, with bias in covariate coefficient estimates.

Results extend to Erdős–Rényi comparison graphs with many covariates.

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.