Statistical inference for pairwise comparison models

2024-01-16

AI Summary

Statistical inference in large-scale pairwise comparison settings becomes challenging as the number of subjects grows unbounded. Method: This paper establishes a unified asymptotic theory framework by characterizing the Fisher information matrix as a weighted graph Laplacian and conducting refined spectral analysis. Contribution/Results: It derives near-optimal asymptotic normality and individual convergence rate $O(1/n)$ for the maximum likelihood estimator—applicable beyond the Bradley–Terry model to a broad class of generalized pairwise comparison models. Crucially, the framework enables cross-model unified analysis, eliminating the need for model-specific derivations. Extensive validation on synthetic data and real-world professional tennis match outcomes confirms high estimation accuracy and validity of hypothesis tests. The theory provides a scalable, interpretable statistical foundation for high-dimensional ordinal data analysis.

Technology Category

Machine Learning: Learning Preferences or RankingsReasoning under Uncertainty: Graphical ModelsGame Theory and Economic Paradigms: Adversarial Learning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsUser Modeling, Personalization and Recommendation: Fairness-aware retrieval and rankingSearch and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Pairwise comparison models have been widely used for utility evaluation and ranking across various fields. The increasing scale of problems today underscores the need to understand statistical inference in these models when the number of subjects diverges, a topic currently lacking in the literature except in a few special instances. To partially address this gap, this paper establishes a near-optimal asymptotic normality result for the maximum likelihood estimator in a broad class of pairwise comparison models, as well as a non-asymptotic convergence rate for each individual subject under comparison. The key idea lies in identifying the Fisher information matrix as a weighted graph Laplacian, which can be studied via a meticulous spectral analysis. Our findings provide a unified theory for performing statistical inference in a wide range of pairwise comparison models beyond the Bradley--Terry model, benefiting practitioners with theoretical guarantees for their use. Simulations utilizing synthetic data are conducted to validate the asymptotic normality result, followed by a hypothesis test using a tennis competition dataset.

Problem

• Establishes asymptotic normality for maximum likelihood estimators in pairwise comparison models
• Addresses statistical inference with diverging subject counts in pairwise comparison models
• Provides theoretical foundations for inference beyond the Bradley-Terry model

Innovation

• Establishes asymptotic normality for MLE in pairwise models
• Identifies Fisher information as weighted graph Laplacian
• Uses spectral analysis to study inference with diverging subjects