标题： 具有协变量和不断增长的受试者数量的配对比较广义 Bradley-Terry 模型中的推断 显示英文标题

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

摘要： 受体育比赛中主场优势的启发，我们提出了一种广义的Bradley--Terry模型，该模型结合了配对比较中的协变量信息。它有一个$n$维的优劣参数$\bs{\beta}$和一个固定维数的协变量回归系数$\bs{\gamma}$。当主体数量$n$趋于无穷大且任意两个主体之间的比较次数固定时，我们证明了最大似然估计量（MLE）$(\widehat{\bs{\beta}}, \widehat{\bs{\gamma}})$关于$(\bs{\beta}, \bs{\gamma})$的均匀一致性。此外，我们通过描述其渐近表示推导出了MLE的渐近正态分布。 $\widehat{\bs{\gamma}}$的渐近分布是偏的，而$\widehat{\bs{\beta}}$的渐近分布不是。 这种现象可以归因于$\widehat{\bs{\gamma}}$和$\widehat{\bs{\beta}}$不同的收敛速度。 据我们所知，这是首次在高维设置下研究具有协变量的配对比较模型的渐近理论。 一致性结果进一步扩展到一个发散协变量数量的 Erdős--Rényi 比较图。 数值研究和实际数据分析证明了我们的理论结果。 显示英文摘要

摘要： 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.