A test-length correction to the estimation of extreme proficiency levels

In this paper, the estimation of extremely large or extremely small proficiency levels, given
the item parameters of a logistic item response model, is investigated. On the one hand, the
estimation of proficiency levels by maximum likelihood (ML), despite being asymptotically
unbiased, may yield infinite estimates. On the other hand, with an appropriate prior
distribution, the Bayesian approach of maximum a posteriori (MAP) yields finite estimates,
but it suffers from severe estimation bias at the extremes of the proficiency scale. In a first
step, we propose a simple correction to the MAP estimator in order to reduce this estimation
bias. The correction factor is determined through a simulation study and depends only on the
length of the test. In a second step, some additional simulations highlight that the corrected
estimator behaves like the ML estimator and outperforms the standard MAP method for
extremely small or extremely large abilities. Although based on the Rasch model, the method
could be adapted to other logistic item response models.