Asymptotic cumulants of the parameter estimators in item response theory.

The asymptotic cumulants of the parameter estimators for the three-parameter logistic model in item response theory are derived up to the fourth order with the higher-order added asymptotic variances. The asymptotic cumulants of the corresponding Studentized estimators up to the third order are also given. The estimators are obtained by marginal maximum likelihood using the standard normal distribution for the latent variable with and without model misspecification. Numerical examples with fixed guessing parameters show advantages of the asymptotic expansions over the usual normal approximation