Infereni on full or partial parameters based on the standardized signed log likelihood ratio

For parametric models it is shown in general that by adjusting the mean and variance of the signed log likelihood ratio for a single parameter of interest ψ one obtains a statistic which is asymptotically standard normally distributed to order O(n–23) , under repeated sampling. This statistic may also be used as an ancillary in the associated problem of drawing inference on the complementary parameter χ for known value of ψ in which case it entails accuracy to the same order of a simple formula (Barndorff-Nielsen, 1983) for the conditional distribution of the maximum likelihood estimator of χ. By iterated application, these results are extended to the case of multivariate parameters of interest. The asymptotic normality result may be used to set confidence regions for the parameter of interest which are correct to order O(n–) conditionally as well as uncondi tionally. Several examples are discussed. In the course of the argument the concept of the affine ancillary (Barndorff-Nielsen, 1980) is extended from curved exponential families to rather general models.