Maximum likelihood estimation in the sparse Rasch model

TL;DR

The paper addresses scalable inference for the Rasch model under sparse observations by adopting an Erdős–Rényi sampling scheme. It develops a leave-one-out technique to prove uniform consistency of the maximum likelihood estimator when both the number of individuals $r$ and items $t$ grow, with a mild lower bound on the sampling probability $p$ ensuring graph connectivity. It also establishes an asymptotic normality result by approximating the inverse Fisher information with a simple matrix, enabling practical variance estimation. Extensive simulations and a real data analysis on a large item response dataset demonstrate good finite-sample performance, accurate coverage for confidence intervals, and meaningful parameter estimates for individuals and items. This work provides sharp theoretical guarantees for MLE in sparse Rasch models and supports their use in large-scale psychometric applications where data are inherently sparse.

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 Erdős\textendash Rényi 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 Erdős\textendash Rényi 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.

Figures (3)

• Figure 1: The plots are the errors of the MLEs under different $p$, $r$, and $t$. The horizontal axis shows the number of individuals $r$ in the first row or the number of items $t$ in the second row. The vertical axis in each plot shows the average values of $\|\hat{\theta} - \theta \|_\infty$ in the first column, $\|\hat{\alpha} - \alpha \|_\infty$ in the second column, and $\|\hat{\beta} - \beta \|_\infty$ in the third column.
• Figure 2: QQ-plots of $\xi_{i,j}$ under $(r,t)=(1000,1000)$.
• Figure 3: The histograms for the MLEs $\hat{\alpha}_i$ in the left and $\hat{\beta}_j$ in the right. The red line is the density curve.

Theorems & Definitions (19)

• Theorem 1: Existence and entrywise error of the MLE
• Remark 1
• Remark 2
• Theorem 2: Central limit theorem
• Remark 3
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5