The dynamic ranking, due to its increasing importance in many applications, is becoming crucial, especially with the collection of voluminous time-dependent data. One such application is sports statistics, where dynamic ranking aids in forecasting the performance of competitive teams, drawing on historical and current data. Despite its usefulness, predicting and inferring rankings pose challenges in environments necessitating time-dependent modeling. This paper introduces a spectral ranker called Kernel Rank Centrality, designed to rank items based on pairwise comparisons over time. The ranker operates via kernel smoothing in the Bradley-Terry model, utilizing a Markov chain model. Unlike the maximum likelihood approach, the spectral ranker is nonparametric, demands fewer model assumptions and computations, and allows for real-time ranking. We establish the asymptotic distribution of the ranker by applying an innovative group inverse technique, resulting in a uniform and precise entrywise expansion. This result allows us to devise a new inferential method for predictive inference, previously unavailable in existing approaches. Our numerical examples showcase the ranker's utility in predictive accuracy and constructing an uncertainty measure for prediction, leveraging data from the National Basketball Association (NBA). The results underscore our method's potential compared to the gold standard in sports, the Arpad Elo rating system.

翻译：动态排名因其在众多应用中日益增长的重要性而变得至关重要，尤其是在收集大规模时变数据的背景下。体育统计便是此类应用之一，它利用历史和当前数据，通过动态排名帮助预测竞技队伍的表现。尽管动态排名具有实用性，但在需要时变建模的环境中，预测和推断排名仍面临挑战。本文提出一种名为Kernel Rank Centrality的谱排序器，旨在基于时间上的两两比较对项目进行排名。该排序器通过Bradley-Terry模型中的核平滑技术，并利用马尔可夫链模型实现操作。与最大似然方法不同，谱排序器是非参数化的，所需的模型假设和计算量较少，并支持实时排名。我们通过应用创新的群逆技术，建立了该排序器的渐近分布，从而获得统一且精确的逐元素展开。这一结果使我们能够设计出一种新的预测推断方法，而现有方法此前无法实现。数值示例展示了该排序器在预测准确性方面的实用性，并利用美国国家篮球协会（NBA）的数据构建了预测的不确定性度量。结果凸显了我们的方法相较于体育领域的黄金标准——Arpad Elo评分系统的潜力。