The Plackett--Luce model is a popular approach for ranking data analysis, where a utility vector is employed to determine the probability of each outcome based on Luce's choice axiom. In this paper, we investigate the asymptotic theory of utility vector estimation by maximizing different types of likelihood, such as the full-, marginal-, and quasi-likelihood. We provide a rank-matching interpretation for the estimating equations of these estimators and analyze their asymptotic behavior as the number of items being compared tends to infinity. In particular, we establish the uniform consistency of these estimators under conditions characterized by the topology of the underlying comparison graph sequence and demonstrate that the proposed conditions are sharp for common sampling scenarios such as the nonuniform random hypergraph model and the hypergraph stochastic block model; we also obtain the asymptotic normality of these estimators and discuss the trade-off between statistical efficiency and computational complexity for practical uncertainty quantification. Both results allow for nonuniform and inhomogeneous comparison graphs with varying edge sizes and different asymptotic orders of edge probabilities. We verify our theoretical findings by conducting detailed numerical experiments.

翻译：Plackett--Luce模型是一种流行的排序数据分析方法，其中通过效用向量基于Luce选择公理确定每个结果的概率。本文研究了通过最大化不同类型似然（如完全似然、边际似然和拟似然）进行效用向量估计的渐近理论。我们为这些估计量的估计方程提供了秩匹配解释，并分析了当被比较项目数量趋于无穷大时它们的渐近行为。特别地，我们在底层比较图序列拓扑所刻画的条件框架下建立了这些估计量的一致相合性，并证明了所提条件对于常见采样场景（如非均匀随机超图模型和超图随机块模型）是精确的；同时，我们获得了这些估计量的渐近正态性，并讨论了实际不确定性量化中统计效率与计算复杂度之间的权衡。这两个结果均允许比较图具有非均匀和非同质结构，包含不同边大小及边概率的不同渐近阶数。通过详细的数值实验验证了我们的理论发现。