Maximum likelihood estimation in the sparse Rasch model

TL;DR

This paper establishes the uniform consistency and asymptotic normality of the maximum likelihood estimator in the sparse Rasch model under Erdős–Rényi sampling, providing theoretical guarantees for large-scale psychometric data analysis.

Contribution

It introduces a leave-one-out method for the Rasch model and proves the MLE's consistency and normality in sparse sampling regimes, extending theoretical understanding.

Findings

MLE is uniformly consistent under sparse sampling

Sampling probability can be as low as rac{rac{rac{ ext{max}\{ ext{log} ext{r}/ ext{r}, ext{log} ext{t}/ ext{t} ight)

Simulation studies support theoretical results

Abstract

The Rasch model has been widely used to analyse item response data in psychometrics and educational assessments. When the number of individuals and items are large, it may be impractical to provide all possible responses. It is desirable to study sparse item response experiments. Here, we propose to use the Erd\H{o}s\textendash R\'enyi random sampling design, where an individual responds to an item with low probability $p$. We prove the uniform consistency of the maximum likelihood estimator %by developing a leave-one-out method for the Rasch model when both the number of individuals, $r$, and the number of items, $t$, approach infinity. Sampling probability $p$ can be as small as $\max\{\log r/r, \log t/t\}$ up to a constant factor, which is a fundamental requirement to guarantee the connection of the sampling graph by the theory of the Erd\H{o}s\textendash R\'enyi graph. The key technique behind this significant advancement is a powerful leave-one-out method for the Rasch model. We further establish the asymptotical normality of the MLE by using a simple matrix to approximate the inverse of the Fisher information matrix. The theoretical results are corroborated by simulation studies and an analysis of a large item-response dataset.