The Bradley-Terry-Luce (BTL) model is a benchmark model for pairwise comparisons between individuals. Despite recent progress on the first-order asymptotics of several popular procedures, the understanding of uncertainty quantification in the BTL model remains largely incomplete, especially when the underlying comparison graph is sparse. In this paper, we fill this gap by focusing on two estimators that have received much recent attention: the maximum likelihood estimator (MLE) and the spectral estimator. Using a unified proof strategy, we derive sharp and uniform non-asymptotic expansions for both estimators in the sparsest possible regime (up to some poly-logarithmic factors) of the underlying comparison graph. These expansions allow us to obtain: (i) finite-dimensional central limit theorems for both estimators; (ii) construction of confidence intervals for individual ranks; (iii) optimal constant of $\ell_2$ estimation, which is achieved by the MLE but not by the spectral estimator. Our proof is based on a self-consistent equation of the second-order remainder vector and a novel leave-two-out analysis.
翻译:Bradley-Terri-Luce(BTL)模型是个人之间进行对称比较的基准模式。尽管最近在若干流行程序的第一阶低位方面有所进展,但对于BTL模型中不确定性量化的理解仍基本不完全,特别是当基本比较图缺乏时。在本文件中,我们填补这一差距的方法是侧重于两个最近才得到极大关注的估测器:最大可能性估测器(MLE)和光谱估测器。我们使用统一的证据战略,为基本比较图中最稀疏的系统(除某些多logricies之外)的估测器获得敏锐和统一的非抽测扩张。这些扩张使我们得以获得:(一) 两个估测器的有限维中央定律; (二) 建立个人级别的信任间隔;(三) 最高常数为$ell_2美元,这是由MLE实现的,而不是光谱估测仪实现的。我们的证据是基于第二个矢量矢量的自我一致方程式和第二次矢量分析。