Asymptotic cumulants of the parameters estimators in item response theory (Q549616)

A logistic model of item response theory (IRT) with one latent variable is considered in which the probability of success for the \(i\)-th examinee in the \(j\)-th item is \[ c_j+{1-c_j\over 1+\exp(-Da_j(\vartheta_i-b_j))}, \] where \(c_j\) are known, \(D=1.7\) is a normalizing constant, \(a_j\) and \(b_j\) are unknown parameters to be estimated and \(\vartheta_j\sim N(0,1)\) is the latent variable representing the ability of the \(i\)-th examinee. Maximum likelihood estimation of \(a_j\) and \(b_j\) is considered. The author derives a two-term Edgeworth expansion for the densities of normalized estimates and describes the computation of the involved asymptotic cumulants. Numerical examples with simulated data and real life data are considered.