This paper studies the performance of the spectral method in the estimation and uncertainty quantification of the unobserved preference scores of compared entities in a very general and more realistic setup in which the comparison graph consists of hyper-edges of possible heterogeneous sizes and the number of comparisons can be as low as one for a given hyper-edge. Such a setting is pervasive in real applications, circumventing the need to specify the graph randomness and the restrictive homogeneous sampling assumption imposed in the commonly-used Bradley-Terry-Luce (BTL) or Plackett-Luce (PL) models. Furthermore, in the scenarios when the BTL or PL models are appropriate, we unravel the relationship between the spectral estimator and the Maximum Likelihood Estimator (MLE). We discover that a two-step spectral method, where we apply the optimal weighting estimated from the equal weighting vanilla spectral method, can achieve the same asymptotic efficiency as the MLE. Given the asymptotic distributions of the estimated preference scores, we also introduce a comprehensive framework to carry out both one-sample and two-sample ranking inferences, applicable to both fixed and random graph settings. It is noteworthy that it is the first time effective two-sample rank testing methods are proposed. Finally, we substantiate our findings via comprehensive numerical simulations and subsequently apply our developed methodologies to perform statistical inferences on statistics journals and movie rankings.

翻译：本文研究谱方法在未观测偏好评分估计与不确定性量化中的性能，研究场景设定为非常通用且更符合实际的比较图由可能异构大小的超边构成，且给定超边的比较次数可低至一次。此设定在现实应用中广泛存在，无需指定图随机性以及常用Bradley-Terry-Luce（BTL）或Plackett-Luce（PL）模型所施加的限制性同质采样假设。此外，在BTL或PL模型适用的场景中，我们揭示了谱估计量与最大似然估计量（MLE）之间的关系。我们发现，通过等权重朴素谱方法估计最优权重后应用的两步谱方法，能够达到与MLE相同的渐近效率。基于估计偏好评分的渐近分布，我们进一步引入了一套综合框架，用于执行单样本和双样本的排序推断，该框架适用于固定图与随机图设定。值得注意的是，这是首次提出有效的双样本排序检验方法。最后，我们通过全面的数值模拟验证了研究发现，并将所开发的方法应用于统计学期刊与电影排名的统计推断。