It is well known that anomalies are sometimes observed when using the likelihood ratio test (LRT) for testing restricted hypotheses in a normal model. This paper considers a general framework for these anomalies to occur. We provide a condition, that relates the null and the alternative hypotheses, under which the dominance of the LRT is obtained. Conditions are also given which guarantee the equivalence between the LRT and a simpler test. The situations of known and unknown variances are considered and examples are given to illustrate the results.
"Dominance of Likelihood Ratio Tests under Cone Constraints." Ann. Statist. 20 (4) 2087 - 2099, December, 1992. https://doi.org/10.1214/aos/1176348904