(Almost) Equivalence Between Conditional and Mixture Maximum Likelihood Estimates for Some Models of the Rasch Type

Abstract

It has been known for several years that conditional and mixture maximum likelihood estimates do agree for the dichotomous Rasch model (RM). This equivalence may be attained by a sufficient number of mixing components in a specifically restricted latent-class model. Because the principle of such a semiparametric approach is a general one (Kiefer & Wolfowitz, 1956), it also applies to some other models of the Rasch type having simple sufficient statistics. Exact equivalence regarding the parameter estimates can be shown, for example, for the linear logistic test model, a RM with linearly constrained item parameters, and for the polytomous RM; only almost equivalence is found for the mixed RM. As formal proofs of these results are difficult, the presentation focuses on numerical examples demonstrating the said (almost) equivalence.