Moment estimation in paired comparison models with a growing number of subjects

TL;DR

This paper develops a theoretical framework for moment estimation in large, sparse paired comparison models, establishing consistency and asymptotic normality of estimators as the number of subjects grows, with applications to the Thurstone model.

Contribution

It introduces a unified approach for analyzing moment estimators in sparse paired comparison models with growing subjects, including convergence rates and extensions to fixed graphs.

Findings

Proves uniform consistency and asymptotic normality of the estimator.

Establishes convergence rates for the Newton iterative solution.

Validates theoretical results through numerical studies and real data.

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 Erd\"{o}s--R\'{e}nyi 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.