On the Validity of the Likelihood Ratio and Maximum Likelihood Methods

When the null or alternative hypothesis of a statistical testing problem is a composite of regions of varying dimensionality, the likelihood ratio test is statistically inappropriate. Its inappropriateness is revealed not by its performance under the Neyman-Pearson criterion but by the fact that it yields incorrect inferences in certain regions of the sample space due to its inability to adapt to the di ering dimensions in the composite hypothesis. Maximum likelihood estimators and associated model selection procedures also are inappropriate for such composite models. Tests and estimators based on the p-values associated with the various regions that determine the composite model are more appropriate for this geometry. Similar issues arise when the boundary of the null or alternative hypothesis is a composite of regions of varying dimensionality. Corresponding author: Michael D. Perlman, Department of Statistics, University of Washington, Seattle, WA 98195. email: [email protected].