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On the validity of the likelihood ratio and maximum likelihood methods. (English) Zbl 1022.62048
Summary: When the null or alternative hypothesis of a statistical testing problem is a union of finitely many 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 differing 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 each of the regions that constitute the composite model are more appropriate for this geometry. Similar issues arise when the boundary of the null hypothesis is a union of finitely many regions of varying dimensionality.

##### MSC:
 62H15 Multivariate hypothesis testing
Full Text:
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