This paper develops some basic asymptotictheory for such a simple statistical model. For this special case we determine canonical versions of thebest approximation at a data point, by an exponential type or location type model. We also examinethe standard parameterization-invariant test quantities for these models and determine the connectionsamong them. The results lead to some simple proofs for key inference formulas and provide the basisfor the multi-parameter, many variable contexts....

For composite testing of several canonical components in an exponential model, a conical or directional test provides a one-dimensional measure of departure. We approximate the distribution of this test statistic to third order by numerical inversion of a profile likelihood. The method extends the Lugannani & Rice type formula to kernels other than the normal.

The Edgeworth expansion is well known as a means for obtaining approximate tail probabilities from information concerning the moments of the distribution. Recent saddlepoint and asymptotic methods lead to several alternative approximations. These alternatives are developed and compared by means of average relative error.

Saddlepoint methods, extended to distribution functions, can provide highly accurate tail probabilities for testing real parameters in exponential models. For extensions, asymptotic connections among various test quantities are needed. For five quantities, the maximum likelihood departure standardized by observed and expected information, the score function standardized by observed and expected information, and the signed square root of the likelihood ratio statistic, the needed connections to third order are recorded. Their use is illustrated by a simple integration proof of the Lugannani and Rice formula.