Introduction

In this chapter we give a brief overview of the main theoretical results and approximations used in this book. These approximations are derived from the theory of higher order likelihood asymptotics. We present these fairly concisely, with few details on the derivations. There is a very large literature on theoretical aspects of higher order asymptotics, and the bibliographic notes give guidelines to the references we have found most helpful.

The building blocks for the likelihood approximations are some basic approximation techniques: Edgeworth and saddlepoint approximations to the density and distribution of the sample mean, Laplace approximation to integrals, and some approximations related to the chi-squared distribution. These techniques are summarized in Appendix A, and the reader wishing to have a feeling for the mathematics of the approximations in this chapter may find it helpful to read that first.

We provide background and notation for likelihood, exponential family models and transformation models in Section 8.2 and describe the limiting distributions of the main likelihood statistics in Section 8.3. Approximations to densities, including the very important p* approximation, are described in Section 8.4. Tail area approximations for inference about a scalar parameter are developed in Sections 8.5 and 8.6. These tail area approximations are illustrated in most of the examples in the earlier chapters. Approximations for Bayesian posterior distribution and density functions are described in Section 8.7. Inference for vector parameters, using adjustments to the likelihood ratio statistic, is described in Section 8.8.