Score tests when a nuisance parameter is unidentified under the null hypothesis

Rao's score test normally replaces nuisance parameters by their maximum likelihood estimates under the null hypothesis about the parameter of interest. In some models, however, a nuisance parameter is not identified under the null, so that this approach cannot be followed. This paper suggests replacing the nuisance parameter by its maximum likelihood estimate from the unrestricted model and making the appropriate adjustment to the variance of the estimated score. This leads to a rather natural modification of Rao's test, which is examined in detail for a regression-type model. It is compared with the approach, which has featured most frequently in the literature on this problem, where a test statistic appropriate to a known value of the nuisance parameter is treated as a function of that parameter and maximised over its range. It is argued that the modified score test has considerable advantages, including robustness to a crucial assumption required by the rival approach.