Some formulas for the standard error of the weighted likelihood estimator of ability with small psychometric tests

The weighted likelihood estimator of ability (WLE, [3]) was introduced as an asymptotically unbiased estimator of ability in item response theory (IRT) models. Moreover, its standard error was shown to be asymptotically equal to that of the maximum likelihood (ML) estimator [2]. Although this asymptotic framework is most often encountered in psychometric and educational studies, there are several practical applications for which an "exact" formula for the standard error would be useful. For instance, such a formula would certainly be convenient at the early steps of a computerized adaptive test (CAT), whenever only a few items are administered. The purpose of this paper is to derive two possible formulas for the standard error of the WLE, by starting from the objective function to be optimized and deriving the standard error in a similar approach of the ML framework (see e.g., [1]). The two potential formulas are then compared through both, a small simulation study and a practical analysis with realistic, yet arti cial data. It is concluded that one of the formulas must be preferred to the other, both for mathematical consistency and on the basis of the simulated results.
References
[1] Baker, F. B., & Kim, S.-H. (2004). Item Response Theory: Parameter Estimation
Techniques (2nd edition). New York: Marcel Dekker.
[2] Lord, F. M. (1980) Applications of Item Response Theory to Practical Testing
Problems. Hillsdale, NJ: Lawrence Erlbaum.
[3] Warm, T.A. (1989). Weighted likelihood estimation of ability in item response
models. Psychometrika, 54, 427-450.