发表机构
The Hong Kong Polytechnic University; University of Kentucky（香港理工大学; 肯塔基大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在一般比较图下建立了Bradley-Terry模型MLE的一致性和渐近正态性，误差界由有效电阻和谱间隙刻画，适用于随机图设计。
AI 中文摘要
Bradley-Terry模型是一种用于从成对比较中进行排序的参数模型。在对象数量增长的场景中，现有的关于最大似然估计（MLE）的渐近理论通常需要对比较图施加同质性假设或相容性条件，这限制了其在实际许多场景中的适用性。在本工作中，我们建立了在一般确定性比较设计下MLE的一致一致性。我们的成对误差界由两部分组成：一个是由对应对象之间的有效电阻决定的特定对项，另一个是由未归一化图拉普拉斯算子的谱间隙决定的全局项。对于某些图序列，该界在至多对数因子的意义下是紧的。对于任何满足额外平衡条件的预定对序列，我们进一步建立了相应估计效用差的渐近正态性。作为一个应用，当最小边概率超过Erdős-Rényi连通性阈值一个对数因子时，我们获得了独立边随机图设计的一致一致性，并在额外平衡条件下获得了渐近正态性。
英文摘要
The Bradley-Terry model is a parametric model for ranking from pairwise comparisons. Existing asymptotic theory for the maximum likelihood estimator (MLE) in regimes where the number of objects grows often requires homogeneity assumptions or compatibility conditions on comparison graphs, which limits its applicability to many practical settings. In this work, we establish uniform consistency of the MLE under general deterministic comparison designs. Our pairwise error bound consists of a pair-specific term governed by the effective resistance between the corresponding objects and a global term governed by the spectral gap of the unnormalized graph Laplacian. The bound is tight up to logarithmic factors for certain graph sequences. For any prespecified sequence of pairs satisfying an additional balancing condition, we further establish asymptotic normality of the corresponding estimated utility differences. As an application, we obtain uniform consistency for independent-edge random graph designs whenever the minimum edge probability exceeds the Erdős-Rényi connectivity threshold by a logarithmic factor, as well as asymptotic normality under additional balancing conditions.
Comments36 pages
