SUMMARY For the purpose of making inferences for a one-dimensional interest parameter, or constructing approximate complementary ancillaries or residuals, the directed likelihood or signed square root of the likelihood ratio statistic can be adjusted so that the resulting modified directed likelihood is under ordinary repeated sampling approximately standard normal with error of O(n-312), conditional on a suitable ancillary statistic and hence unconditionally. In general, suitable specification of the ancillary statistic may be difficult. We introduce two adjusted directed likelihoods which are similar to the modified directed likelihood but do not require the specification of the ancillary statistic. The error of the standard normal approximation to the distribution of these new adjusted directed likelihoods is O(n -1), conditional on any reasonable ancillary statistic, which is still an improvement over the unadjusted directed likelihoods.
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Barndorff–Nielsen et al. (1994) studied this question.