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On the Existence and Uniqueness of JML Estimates for the Partial Credit Model

Published online by Cambridge University Press:  01 January 2025

Lucio Bertoli-Barsotti
Lucio Bertoli-Barsotti*
Affiliation:
University di Bergamo, Italy
*
Requests for reprints should be sent to Prof. Lucio Bertoli-Barsotti, Dipartimento di Matematica, Statistica, Informatica e Applicazioni, Università di Bergamo, Via Dei Caniana, 2, I-24127 Bergamo, Italy. E-mail: [email protected]
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Abstract

A necessary and sufficient condition is given in this paper for the existence and uniqueness of the maximum likelihood (the so-called joint maximum likelihood) estimate of the parameters of the Partial Credit Model. This condition is stated in terms of a structural property of the pattern of the data matrix that can be easily verified on the basis of a simple iterative procedure. The result is proved by using an argument of Haberman (1977).

Keywords

rasch modelpartial credit modelmaximum likelihood estimateexponential familyconvexity.
Type
Original Paper
Information
Psychometrika , Volume 70 , Issue 3 , September 2005 , pp. 517 - 531
Copyright
Copyright © 2005 The Psychometric Society

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Footnotes

The author wishes to thank the Editor and the anonymous reviewers for their comments that helped to substantially improve the final version of this paper.

This research was supported in part by a MURST grant (ex 60%).

References

Adams, R.A., Khoo, S.T. (1993). QUEST [computer program]. Camberwell, Victoria: Australian Council for Educational Research.Google Scholar
Andersen, E.B. (1973). Conditional inference for multiple-choice questionnaires. British Journal of Mathematical and Statistical Psychology, 26, 31–44.CrossRefGoogle Scholar
Andersen, E.B. (1980). Discrete Statistical Models with Social Science Applications. Amsterdam: North-Holland.Google Scholar
Andersen, E.B. (1995). Polytomous Rasch models and their estimation. In Fischer, G.H., Molenaar, I. (Eds.), Rasch models. Foundations, Recent Developements, and Applications (pp. 271–291). Berlin Heidelberg New York: Springer.Google Scholar
Andersen, E.B. (1995). Residual analysis in the polytomous Rasch model. Psychometrika, 60, 375–393.CrossRefGoogle Scholar
Andersen, E.B. (1997). The rating scale model. In van der Linden, W.J., Hambleton, R.K. (Eds.), Handbook of Modern Item Response Theory (pp. 67–84). Berlin Heidelberg New York: Springer.CrossRefGoogle Scholar
Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. New York: Wiley.Google Scholar
Barndorff-Nielsen, O. (1988). Parametric Statistical Models and Likelihood. Lecture Notes in Statistics, 50. Berlin Heildelberg New York: Springer.CrossRefGoogle Scholar
Fischer, G.H. (1981). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 46, 59–77.CrossRefGoogle Scholar
Fischer, G.H., Parzer, P. (1991). An extension of the rating scale model with an application to the measurement of change. Psychometrika, 56, 637–651.CrossRefGoogle Scholar
Fischer, G.H., Ponocny, I. (1994). An extension of the partial credit model with an application to the measurement of change. Psychometrika, 59, 177–192.CrossRefGoogle Scholar
Fischer, G.H., Ponocny-Seliger, E. (1998). Structural Rasch modeling. Groningen, NL: Handbook of the usage of LPCM-WIN 1.0. ProGAMMA and SciencePlus.Google Scholar
Ghosh, M. (1995). Inconsistent maximum likelihood estimators for the Rasch model. Statistics & Probability Letters, 23, 165–170.CrossRefGoogle Scholar
Haberman, S.J. (1974). The Analysis of Frequency Data. Chicago: University of Chicago Press.Google Scholar
Haberman, S.J. (1977). Maximum likelihood estimates in exponential response models. Annals of Statistics, 5, 815–841.CrossRefGoogle Scholar
Jacobsen, M. (1989). Existence and unicity of MLEs in discrete exponential family distributions. Scandinavian Journal of Statistics, 16, 335–349.Google Scholar
Linacre, J.M., Wright, B.D. (2000). WINSTEPS: Multiple-choice, Rating Scale and Partial Credit Rasch Analysis [computer program]. Chicago: MESA Press.Google Scholar
Marshall, A.W., Olkin, I. (1979). Inequalities: Theory of Majorization and its Applications. New York: Academic Press.Google Scholar
Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.CrossRefGoogle Scholar
Rockafellar, R.T. (1970). Convex Analysis. Princeton: Princeton University Press.CrossRefGoogle Scholar
van der Linden, W.J., Hambleton, R.K. (1997). Item response theory: Brief history, common models, and extensions. In van der Linden, W.J., Hambleton, R.K. (Eds.), Handbook of Modern Item Response Theory (pp. 1–28). Berlin Heidelberg New York: Springer.CrossRefGoogle Scholar
Wilson, M., Masters, G.N. (1993). The Partial Credit Model and null categories. Psychometrika, 58, 87–99.CrossRefGoogle Scholar
Wright, B.D., Masters, G.N. (1982). Rating Scale Analysis. Chicago: MESA Press.Google Scholar
Zehna, P.W. (1966). Invariance of maximum likelihood. The Annals of Mathematical Statistics, 37, 744.CrossRefGoogle Scholar