Third-order local power properties of tests for a composite hypothesis, II

The Bartlett-type adjustment is a higher-order asymptotic method for improving the chi-squared approximation to the null distributions of various test statistics, which ensures that the resulting test has size α+o(N−1), where 0<α<1 is the significance level and N is the sample size. We continue our recent works on the third-order average local power properties of several Bartlett-type adjusted tests. Strengthening the results in the 1990s, the third-order optimality of the adjusted Rao test in a sense has been established even if both the interest parameter and the nuisance parameter are multi-dimensional. We briefly discuss adjusted profile likelihood inference for handling the nuisance parameter.