OPTIMAL TESTS WHEN A NUISANCE PARAMETER IS PRESENT ONLY UNDER THE ALTERNATIVE

This paper derives asymptotically optimal tests for testing problems i n which a nuisance parameter exists under the alternative hypothesis b ut not under the null. For example, the results apply to tests of one- time structural change with unknown change-point. Several other exampl es are discussed in the paper. The results of the paper are of interes t, because the testing problem considered is nonstandard and the class ical asymptotic optimality results for the Lagrange multiplier (LM), W ald, and likelihood ratio (LR) tests do not apply. A weighted average power criterion is used here to generate optimal tests. This criterion is similar to that used by Wald (1943) to obtain the classical asympt otic optimality properties of Wald tests in ''regular'' testing proble ms. In fact, the optimal tests introduced here reduce to the standard LM, Wald, and LR tests when standard regularity conditions hold. Never theless, in the nonstandard cases of main interest, new optimal tests are obtained and the LR test is not found to be an optimal test.