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

Published online by Cambridge University Press:  01 January 2025

Gerhard H. Fischer
Gerhard H. Fischer*
Affiliation:
University of Vienna
*
Requests for reprints should be addressed to Gerhard H. Fischer, Institut für Psychologie, Universität Wien, Liebiggasse 5, A-1010 Wien, Austria.
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Abstract

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.

Keywords

latent trait theoryRasch modelmaximum-likelihood estimation
Type
Original Paper
Information
Psychometrika , Volume 46 , Issue 1 , March 1981 , pp. 59 - 77
Copyright
Copyright © 1981 The Psychometric Society

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Footnotes

Paper read at the European Meeting of the Psychometric Society, Groningen, June 19-21, 1980.

Part of the research reported herein was done while the author was staying at the Pulmologisches Zentrum der Stadt Wien; he is indebted to Professor Dr. F. Muhar and Dr. R. Mutschlechner for providing excellent working conditions.

References

Reference Notes

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