Random Pairing MLE for Estimation of Item Parameters in Rasch Model
提出随机配对极大似然估计（RP-MLE）及其自助法变体（MRP-MLE），用于Rasch模型的项目参数估计，适用于稀疏观测数据，具有极小极大最优性和不确定性量化能力。
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.