# Statistical inference for pairwise comparison models

arXiv: https://arxiv.org/abs/2401.08463 · HTML: https://mlsanity.com/paper/2401.08463/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 55 references that are in the map.
- First posted on arXiv 16 Jan 2024.

## Builds on (most cited first, 17 of 17)

- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence](https://mlsanity.com/paper/1505.01462/) (2015)
- [Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking](https://mlsanity.com/paper/1707.09971/) (2017)
- [Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons](https://mlsanity.com/paper/1504.07218/) (2015)
- [Efficient computation of rankings from pairwise comparisons](https://mlsanity.com/paper/2207.00076/) (2022)
- [Uncertainty quantification in the Bradley-Terry-Luce model](https://mlsanity.com/paper/2110.03874/) (2021)
- [Ranking Inferences Based on the Top Choice of Multiway Comparisons](https://mlsanity.com/paper/2211.11957/) (2022)
- [Lagrangian Inference for Ranking Problems](https://mlsanity.com/paper/2110.00151/) (2021)
- [Optimal Full Ranking from Pairwise Comparisons](https://mlsanity.com/paper/2101.08421/) (2021)
- [An application of incomplete pairwise comparison matrices for ranking top tennis players](https://mlsanity.com/paper/1611.00538/) (2016)
- [A unified analysis of likelihood-based estimators in the Plackett--Luce model](https://mlsanity.com/paper/2306.02821/) (2023)
- [A General Pairwise Comparison Model for Extremely Sparse Networks](https://mlsanity.com/paper/2002.08853/) (2020)
- [On Sinkhorn's Algorithm and Choice Modeling](https://mlsanity.com/paper/2310.00260/) (2023)
- [Statistical Modelling of Citation Exchange Between Statistics Journals](https://mlsanity.com/paper/1312.1794/) (2013)
- [Modeling outcomes of soccer matches](https://mlsanity.com/paper/1807.01623/) (2018)
- [Analysis of Networks via the Sparse β-Model](https://mlsanity.com/paper/1908.03152/) (2019)

## Built on by (most cited first, 3 of 3)

- [Statistical ranking with dynamic covariates](https://mlsanity.com/paper/2406.16507/) (2024)
- [Hypothesis Testing for Generalized Thurstone Models](https://mlsanity.com/paper/2512.02912/) (2025)
- [Bradley-Terry model under general comparison graphs](https://mlsanity.com/paper/2610.06237/) (2026)

## Abstract (arXiv)

Pairwise comparison models have been widely used for utility evaluation and rank aggregation across various fields. The increasing scale of modern problems underscores the need to understand statistical inference in these models when the number of subjects diverges, a topic that is currently underexplored in the literature. To 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. 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 theoretical foundations for performing statistical inference in a wide range of pairwise comparison models beyond the Bradley--Terry model. Simulations utilizing synthetic data are conducted to validate the asymptotic normality result, followed by a hypothesis test using a tennis competition dataset.