  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 on assessments or questionnaires. In this
paper, we introduce a new likelihood-based estimator -- random pairing maximum
likelihood estimator ($\mathsf{RP\text{-}MLE}$) and its bootstrapped variant
multiple random pairing MLE ($\mathsf{MRP\text{-}MLE}$) that 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
$\ell_{\infty}$ estimation error. Lastly, $\mathsf{RP\text{-}MLE}$ admits
precise distributional characterization that allows uncertainty quantification
on the item parameters, e.g., construction of confidence intervals of the item
parameters. The main idea underlying $\mathsf{RP\text{-}MLE}$ and
$\mathsf{MRP\text{-}MLE}$ 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.
