# Random pairing MLE for estimation of item parameters in Rasch model

arXiv: https://arxiv.org/abs/2406.13989 · HTML: https://mlsanity.com/paper/2406.13989/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 52 references that are in the map.
- First posted on arXiv 20 Jun 2024.

## Builds on (most cited first, 19 of 19)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [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)
- [Uncertainty quantification in the Bradley-Terry-Luce model](https://mlsanity.com/paper/2110.03874/) (2021)
- [Minimax-optimal Inference from Partial Rankings](https://mlsanity.com/paper/1406.5638/) (2014)
- [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)
- [Item Response Theory -- A Statistical Framework for Educational and Psychological Measurement](https://mlsanity.com/paper/2108.08604/) (2021)
- [Uncertainty Quantification of MLE for Entity Ranking with Covariates](https://mlsanity.com/paper/2212.09961/) (2022)
- [Spectral Ranking Inferences based on General Multiway Comparisons](https://mlsanity.com/paper/2308.02918/) (2023)
- [Top-K Ranking from Pairwise Comparisons: When Spectral Ranking is Optimal](https://mlsanity.com/paper/1603.04153/) (2016)
- [ℓ∞-Bounds of the MLE in the BTL Model under General Comparison Graphs](https://mlsanity.com/paper/2110.10825/) (2021)
- [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)
- [Ranking from Pairwise Comparisons in General Graphs and Graphs with Locality](https://mlsanity.com/paper/2304.06821/) (2023)
- [A Spectral Approach to Item Response Theory](https://mlsanity.com/paper/2210.04317/) (2022)
- [Top-K ranking with a monotone adversary](https://mlsanity.com/paper/2402.07445/) (2024)
- [Optimal and Private Learning from Human Response Data](https://mlsanity.com/paper/2303.06234/) (2023)

## Abstract (arXiv)

The Rasch model, a classical model in the item response theory, is widely used in psychometrics to model the relationship between individuals' latent traits and their binary responses to assessments or questionnaires. In this paper, we introduce a new likelihood-based estimator -- random pairing maximum likelihood estimator (mathrmRP-MLE) and its bootstrapped variant multiple random pairing MLE (mathrmMRP-MLE) which faithfully estimate the item parameters in the Rasch model. The new estimators have several appealing features compared to existing ones. First, both work for sparse observations, an increasingly important scenario in the big data era. Second, both estimators are provably minimax optimal in terms of finite sample ℓ∞ estimation error. Lastly, both admit precise distributional characterization that allows uncertainty quantification on the item parameters, e.g., construction of confidence intervals for the item parameters. The main idea underlying mathrmRP-MLE and mathrmMRP-MLE is to randomly pair user-item responses to form item-item comparisons. This is carefully designed to reduce the problem size while retaining statistical independence. We also provide empirical evidence of the efficacy of the two new estimators using both simulated and real data.