Semiparametric analysis for paired comparisons with covariates

TL;DR

This paper introduces a flexible semiparametric framework for paired comparison analysis with covariates, employing kernel-based estimation to handle high-dimensional data and unknown distributions.

Contribution

It develops the first semiparametric approach for high-dimensional paired comparison models, addressing model misspecification and proving estimator consistency and asymptotic normality.

Findings

Estimators are consistent as the number of items grows.

The method achieves asymptotic normality under fixed comparisons per pair.

Simulation results demonstrate good finite-sample performance.

Abstract

Statistical inference in parametric models (e.g., the Bradley--Terry model and its variants) for paired-comparison data has been explored in the high-dimensional regime, in which the number of items involving in paired comparisons diverges. However, parametric models are highly susceptible to model misspecification. To relax the assumption of known distributions and provide flexibility, we propose a semiparametric framework for modeling the merits of items and covariate effects (e.g., home-field advantage) by introducing latent random variables with unspecified distributions. As the number of parameters increases with the number of items, semiparametric inference is highly nontrivial. To address this issue, we employ a kernel-based least squares approach to estimate all unknown parameters. When each pair of items has a fixed number of comparisons and the number of items tends to infinity, we prove the consistency of all resulting estimators and derive their asymptotic normal distributions. To the best of our knowledge, this is the first study to conduct a semiparametric analysis of paired comparisons with an increasing dimension. We conduct simulations to evaluate the finite-sample performance of the proposed method and illustrate its practical utility by analyzing an NBA dataset.