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On the Sampling Interpretation of Confidence Intervals and Hypothesis Tests in the Context of Conditional Maximum Likelihood Estimation

Published online by Cambridge University Press:  01 January 2025

E. Maris
E. Maris*
Affiliation:
Department of Mathematical Psychology, Nijmegen Institute for Cognition and Information (NICI) National Institute for Educational Measurement (CITO)
*
Requests for reprints should be sent to Eric Maris, Department of Mathematical Psychology (NICI), Catholic University of Nijmegen, P.O. Box 9104, 6500 HE Nijmegen, THE NETHERLANDS.
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Abstract

In the context of conditional maximum likelihood (CML) estimation, confidence intervals can be interpreted in three different ways, depending on the sampling distribution under which these confidence intervals contain the true parameter value with a certain probability. These sampling distributions are (a) the distribution of the data given the incidental parameters, (b) the marginal distribution of the data (i.e., with the incidental parameters integrated out), and (c) the conditional distribution of the data given the sufficient statistics for the incidental parameters. Results on the asymptotic distribution of CML estimates under sampling scheme (c) can be used to construct asymptotic confidence intervals using only the CML estimates. This is not possible for the results on the asymptotic distribution under sampling schemes (a) and (b). However, it is shown that the conditional asymptotic confidence intervals are also valid under the other two sampling schemes.

Keywords

CML estimationconfidence intervalsconditional inference
Type
Original Paper
Information
Psychometrika , Volume 63 , Issue 1 , March 1998 , pp. 65 - 71
Copyright
Copyright © 1998 The Psychometric Society

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Footnotes

I am indebted to Theo Eggen, Norman Verhelst and one of Psychometrika's reviewers for their helpful comments.

References

Andersen, E. B. (1970). Asymptotic properties of conditional maximum likelihood estimators. Journal of the Royal Statistical Society, Series B, 32, 283–301.CrossRefGoogle Scholar
Andersen, E. B.. (1973). Conditional inference and models for measuring, Copenhagen: Mentalhygiejnisk Forlag.Google Scholar
Andersen, E. B., & Madsen, M. (1977). Estimating the parameters of the latent population distribution. Psychometrika, 42, 357–374.CrossRefGoogle Scholar
Cramèr, H.. (1946). Mathematical methods of statistics, Princeton: Princeton University Press.Google Scholar
de Leeuw, J., & Verhelst, N. (1986). Maximum likelihood estimation in generalized Rasch models. Journal of Educational Statistics, 11, 183–196.CrossRefGoogle Scholar
Fischer, G. H. (1981). On the existence and uniqueness of maximum likelihood estimates in the Rasch model. Psychometrika, 53, 525–546.Google Scholar
Follman, D. A. (1988). Consistent estimation in the Rasch model based on nonparametric margins. Psychometrika, 53, 553–562.CrossRefGoogle Scholar
Graybill, F. A.. (1976). Theory and application of the linear model, North Scituate, MA: Duxbury Press.Google Scholar
Kiefer, J., & Wolfowitz, J. (1956). Consistency of the maximum likihood estimator in the presence of infinitely many nuisance parameters. Annals of Mathematical Statistics, 27, 887–906.CrossRefGoogle Scholar
Lehmann, E. L.. (1983). Theory of point estimation, New York: Wiley.CrossRefGoogle Scholar
Lindsay, B., Clogg, C. C., & Grego, J. (1991). Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis. Journal of the American Statistical Association, 86, 96–107.CrossRefGoogle Scholar
Pfanzagl, J. (1993). A case of asymptotic equivalence between conditional and marginal maximum likelihood estimators. Journal of Statistical Planning and Inference, 35, 301–307.CrossRefGoogle Scholar
Pfanzagl, J. (1994). On item parameter estimation in certain latent trait models. In Fischer, G. H., & Laming, D. (Eds.), Contributions to mathematical psychology, psychometrics and methodology (pp. 249–263), New York: Springer.CrossRefGoogle Scholar