Estimation of Ability with Reduced Asymptotic Mean Square Error in Item Response Theory

Abstract

A method of the weighted score or penalized likelihood for estimation of ability reducing the asymptotic mean square error is derived. In this method, associated item parameters are assumed to be given or estimated by using a separate calibration sample with the size of an appropriate order. The method can be seen as an extension of the weighted likelihood method that removes the asymptotic bias of the maximum likelihood estimator. In the proposed method, some bias is retained while variance is reduced by using a multiplicative constant for the weight in the weighted score. A lower bound of the constant minimizing the asymptotic mean square error is found under the logistic model having identical items. The lower bound is numerically also shown to be reasonable in the case of the 3-parameter logistic model, with and without model misspecification.