This paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null. The results of the paper are of interest, because the testing problem considered in non-standard and the classical asymptotic optimality results for the Wald, Lagrange multiplier (LM), and likelihood ratio (LR) tests do not apply. In the non-standard cases of main interest, new optimal tests are obtained and the LR test is not found to be an optimal test.
Andrews, Donald W.K. and Ploberger, Werner, "Optimal Tests When a Nuisance Parameter is Present Only under the Alternative" (1992). Cowles Foundation Discussion Papers. 1258.