Third-order average local powers of Bartlett-type adjusted tests: Ordinary versus adjusted profile likelihood

Statistical inference in the presence of a nuisance parameter is often based on profile likelihood. Because it is not a genuine likelihood function, several adjustments to the profile likelihood function for eliminating score/information bias were proposed in the 1980s and 1990s, under the so-called global parameter orthogonality. On the basis of Stern's (1997) adjusted profile likelihood, which is applicable even without the global parameter orthogonality, we discuss higher-order average local power properties after several Bartlett-type adjustments. It turns out that Rao's statistic arising from Stern's adjusted profile likelihood continues to enjoy desirable average local power properties, as in the ordinary likelihood inference. We also investigate, using a simulation, the performance of Rao's test, compared with the likelihood ratio test and Wald's test.