Computation of distribution functions from likelihood information near observed data

Likelihood is widely used in statistical applications, both for the full parameter by obvious direct calculation and for component interest parameters by recent asymptotic theory. Often, however, we want more detailed information concerning an inference procedure, information such as say the distribution function of a measure of departure which would then permit power calculations or a detailed display of p-values for a range of parameter values. We investigate how such distribution function approximations can be obtained from minimal information concerning the likelihood function, a minimum that is often available in many applications. The resulting expressions clearly indicate the source of the various ingredients from likelihood, and they also provide a basis for understanding how nonnormality of the likelihood function affects related p-values. Moreover they provide the basis for removing a computational singularity that arises near the maximum likelihood value when recently developed significance function formulas are used.