Moment estimation in paired comparison models with a growing number of subjects
Qiuping Wang, Lu Pan, Ting Yan
TL;DR
This work develops a moment-estimation framework for paired comparison models on sparse Erdős--Rényi graphs, establishing uniform consistency and asymptotic normality of the estimator as the number of subjects grows and the graph becomes increasingly sparse. The authors leverage a Newton-Kantorovich analysis to bound the convergence rate of the estimator and derive an explicit asymptotic distribution, with explicit results for the Thurstone model. They extend the theory to fixed sparse designs and provide rich numerical validation, including a real-world NFL data example. The results offer practical inference tools for ranking and merit-parameter estimation in sparse paired comparison settings, with clear conditions on sparsity and model regularity. Overall, the paper provides a unified, theoretically grounded approach to moment-based inference in high-dimensional, sparsely connected comparison graphs.
Abstract
When the number of subjects, $n$, is large, paired comparisons are often sparse. Here, we study statistical inference in a class of paired comparison models parameterized by a set of merit parameters, under an Erdös--Rényi comparison graph, where the sparsity is measured by a probability $p_n$ tending to zero. We use the moment estimation base on the scores of subjects to infer the merit parameters. We establish a unified theoretical framework in which the uniform consistency and asymptotic normality of the moment estimator hold as the number of subjects goes to infinity. A key idea for the proof of the consistency is that we obtain the convergence rate of the Newton iterative sequence for solving the estimator. We use the Thurstone model to illustrate the unified theoretical results. Further extensions to a fixed sparse comparison graph are also provided. Numerical studies and a real data analysis illustrate our theoretical findings.