ABSTRACT
The traditional approach to the comparison of alternative test procedures is via the study of power functions. There are two broad stages to the choice of an inference procedure for 0, the selection of a test statistic which in complex problems largely determines the 'shape' of the resulting solution and the resolution of a distributional problem that calibrates the confidence regions according to the confidence coefficient involved. It is an advantage, if this distributional problem has, if not an 'exact' solution, then a simple reasonably accurate approximation. The higher-order asymptotic theory of procedures based on likelihood can be approached from a number of viewpoints. The higher-order asymptotic distribution theory associated with Edgeworth expansions and saddlepoint expansions is summarized by Barndorff-Nielsen and Cox.
