Higher-order asymptotics and the likelihood principle: One-parameter models

SUMMARY: Inferences based on higher-order asymptotics, in contrast to those based on standard first-order methods, approximate exact inferences closely enough to depend on the reference set. We use this feature to quantify the effect of the reference set on exact inferences. In particular, the difference in P-values for two reference sets can be at most O(n^-2-Jan), and it is seen that one can often identify the coefficient of n^-2-Jan. Although this is useful from a foundational viewpoint, the question arises of whether higher-order adjustments to first-order P-values reflect only the choice of reference set. It is indicated that for one-parameter problems this may largely be the case, but that when there are nuisance paramenters the situation is likely to be quite different. The nuisance parameter setting is not investigated thoroughly in this paper, but we believe that in that case the most important part of the higher-order corrections largely conforms to the likelihood principle.