On the Existence and Uniqueness of Maximum-Likelihood Estimates in the Rasch Model

Necessary and sufficient conditions for the existence and uniqueness of a solution of the so-called “unconditional” (UML) and the “conditional” (CML) maximum-likelihood estimation equations in the dichotomous Rasch model are given. The basic critical condition is essentially the same for UML and CML estimation. For complete data matrices A , it is formulated both as a structural property of A and in terms of the sufficient marginal sums. In case of incomplete data, the condition is equivalent to complete connectedness of a certain directed graph. It is shown how to apply the results in practical uses of the Rasch model.