Inference in a generalized Bradley-Terry model for paired comparisons with covariates and a growing number of subjects
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| Format: | Preprint |
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2025
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| _version_ | 1866915417630441472 |
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| author | Yan, Ting |
| author_facet | Yan, Ting |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_22472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inference in a generalized Bradley-Terry model for paired comparisons with covariates and a growing number of subjects Yan, Ting Methodology 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. |
| title | Inference in a generalized Bradley-Terry model for paired comparisons with covariates and a growing number of subjects |
| topic | Methodology |
| url | https://arxiv.org/abs/2507.22472 |