Testing a model parameter when another is unidentified under the null

Some standard test procedures, such as Score and Likelihood Ratio, replace nuisance parameters by their maximum likelihood estimates under the null hy pothesis about the parameter of interest. In some models, however, a nuisan ce parameter is not identified under the null, so that these approaches nee d modification. By taking a mathematically tractable case, this paper illus trates the issues that arise and the solutions that have been proposed in t he literature. The rival tests are compared in terms of power and robustnes s to misspecification.