STABLE AND INVARIANT ADJUSTED DIRECTED LIKELIHOODS

For the purpose of making inferences for a one-dimensional interest pa rameter, or constructing approximate complementary ancillaries or resi duals, the directed likelihood or signed square root of the likelihood ratio statistic can be adjusted so that the resulting modified direct ed likelihood is under ordinary repeated sampling approximately standa rd normal with error of O(n(-3/2)), conditional on a suitable ancillar y statistic and hence unconditionally. In general, suitable specificat ion of the ancillary statistic may be difficult. We introduce two adju sted directed likelihoods which are similar to the modified directed l ikelihood but do not require the specification of the ancillary statis tic. The error of the standard normal approximation to the distributio n of these new adjusted directed likelihoods is O(n(-1)), conditional on any reasonable ancillary statistic, which is still an improvement o ver the unadjusted directed likelihoods.