Asymptotic cumulants of ability estimators using fallible item parameters

The asymptotic cumulants of ability estimators using fallible or estimated item parameters in an ability test based on item response theory are given up to the fourth order with higher-order asymptotic variance. The ability estimators cover those obtained by maximum likelihood, Bayes, and pseudo Bayes modal estimation. For estimation of item parameters, the marginal maximum likelihood and Bayes methods are used. Asymptotic cumulants with higher-order asymptotic variance are given with and without model misspecification, and before and after studentization. Three conditions for the relative size of the number of items for ability estimation to that of examinees for item parameter calibration are presented; two of them give some justification for neglecting sampling variation of estimated item parameters. Numerical illustration with simulations is shown using the two-parameter logistic model.