Uncertainty quantification in the Bradley-Terry-Luce model

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 l(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.

Automated summary

This paper focuses on the Bradley-Terry-Luce (BTL) model and its uncertainty quantification, specifically in the case of sparse comparison graphs. The maximum likelihood estimator (MLE) and spectral estimator are examined in this context, with the derivation of non-asymptotic expansions and the development of confident intervals and optimal constant of l(2) estimation as main contributions.