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 matricesA, it is formulated both as a structural property ofA 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.

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