ABSTRACT
This chapter discusses Jacobians of transformations to such structures occur as important elements in some of the developments. The conjunction of an exact or approximate ancillary statistic and the maximum likelihood estimator constitutes what is termed a conditionality structure. The chapter considers several other aspects of higher-order asymptotics that relate in a direct manner to material. It addresses the questions by examining a number of special cases connected with Brownian motion and with Poisson processes and survival analysis: these lend themselves to fairly simple direct calculations. From the viewpoint of likelihood-based inference, to evaluate the performance of an approximate ancillary it is essential to compare the conditional likelihood functions given the ancillary to the corresponding unconditional likelihoods.
