ABSTRACT
This chapter discusses higher-order refinements of the classical theory of the limiting behaviour in distribution of the most important likelihood quantities. It considers a remarkably simple formula for the conditional distribution of a maximum likelihood estimator giving either a very close approximation to that distribution or, indeed, in some cases the exact distribution. The chapter also considers the general case that concerns the testing of composite hypotheses. All of the adjusted versions as well as the directed likelihood itself may be used as test statistics, to calculate approximate interval probabilities, as components of ancillary statistics and in residual analysis. The formula determines a probability density, with respect to Lebesgue measure, if the range space of 0 is an open subset of and with respect to counting measure when the range space is discrete.
