Moment Estimation in Paired Comparison Models with a Growing Number of Subjects

TL;DR

This paper introduces a statistical framework for estimating subject merits in sparse paired comparison models as the number of subjects grows.

Contribution

A unified theoretical framework for moment estimation in sparse paired comparison models with growing subjects.

Findings

The moment estimator is uniformly consistent and asymptotically normal as n increases.

The Newton iterative sequence's convergence rate is key to proving consistency.

Numerical and real data analyses validate the theoretical results.

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 pn 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 real data analysis illustrate our theoretical findings.