This paper establishes the asymptotic admissibility of the likelihood ratio (LR) test for a general class of testing problems in which a nuisance parameter is present only under the alternative hypothesis. The paper also establishes the finite sample admissibility of the LR test for testing problems of this sort that arise in Gaussian linear regression models with known variance.
"Admissibility of the Likelihood Ratio Test When a Nuisance Parameter is Present Only Under the Alternative." Ann. Statist. 23 (5) 1609 - 1629, October, 1995. https://doi.org/10.1214/aos/1176324316