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，尤其是在收集大量时变数据的背景下。体育统计是一个典型应用，其中动态排名借助历史与当前数据，有助于预测竞技队伍的表现。尽管其具有实用价值，但在需要时变建模的环境中，预测与推断排名仍面临挑战。本文提出一种名为核秩中心性的谱排序器，旨在基于随时间变化的成对比较对项目进行排序。该排序器通过在布莱德利-特里模型中运用核平滑技术，并借助马尔可夫链模型实现。与最大似然方法不同，谱排序器是非参数的，所需模型假设和计算量更少，且支持实时排名。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，从而得到一致且精确的元素级展开。这一结果使我们能够设计一种新的预测推断方法，这在现有方法中尚不可用。我们的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比于体育领域的黄金标准——阿帕德·埃洛评分系统的潜力。