Random Pairing MLE for Estimation of Item Parameters in Rasch Model

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 article, we introduce a new likelihood-based estimator—random pairing maximum likelihood estimator (RP‐MLE) and its bootstrapped variant multiple random pairing MLE (MRP‐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, for example, construction of confidence intervals for the item parameters. The main idea underlying RP‐MLE and MRP‐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. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.