The Exact Standard Errors Of The MLE And WLE Of Proficiency Levels In GPC Model

Under given the pre-calibrated item parameters, we have obtained the maximum like-lihood estimation and weighted likelihood estimation of proficiency level. These two estimates have the same asymptotic standard deviation, and easily obtained, that the sqrt of expectations of the Fisher information. However, when the test length is short, standard deviation may not converges to the true value. The purpose of this article is to obtain the exact standard deviation and asymptotic standard of the potential ability parameter estimates with short test. Exact standard deviation is obtained by the exact distribution of the estimates, and therefore can only be calculated precisely the standard deviation of the shorter test (except Rasch model). We compared by stochastic simulation method compare the difference between the exact standard deviation and the true stan-dard deviation.In a short test, exact standard deviation have smaller deviation than the asymptotic standard deviation However, in the long test, both the estimated asymptotic standard deviation is the same.