The Asymptotic Distribution of Ability Estimates

The maximum likelihood estimate (MLE) of the ability parameter of an item response theory model with known item parameters was proved to be asymptotically normally distributed under a set of regularity conditions for tests involving dichotomous items and a unidimensional ability parameter (Klauer, 1990; Lord, 1983). This article first considers the more general case of tests that include a mix of dichotomous and polytomous items. A proof is given of the asymptotic normality of the MLE of the ability parameter for such tests under a set of regularity conditions. Then, it is proved that a similar result holds for the weighted likelihood estimate and the posterior mode of the ability parameter. Multidimensional ability parameters are considered next. Numerical illustrations are provided to demonstrate the asymptotic results.