This book is about the statistical analysis of data, and in particular approximations based on the likelihood function. We emphasize procedures that have been developed using the theory of higher order asymptotic analysis and which provide more precise inferences than are provided by standard theory. Our goal is to illustrate their use in a range of applications that are close to many that arise in practice. We generally restrict attention to parametric models, although extensions of the key ideas to semi-parametric and non-parametric models exist in the literature and are briefly mentioned in contexts where they may be appropriate. Most of our examples consist of a set of independent observations, each of which consists of a univariate response and a number of explanatory variables.

Much application of likelihood inference relies on first order asymptotics, by which we mean the application of the central limit theorem to conclude that the statistics of interest are approximately normally distributed, with mean and variance consistently estimable from the data. There has, however, been great progress over the past twenty-five years or so in the theory of likelihood inference, and two main themes have emerged. The first is that very accurate approximations to the distributions of statistics such as the maximum likelihood estimator are relatively easily derived using techniques adapted from the theory of asymptotic expansions. The second is that even in situations where first order asymptotics is to be used, it is often helpful to use procedures suggested by these more accurate approximations, as they provide modifications to naive approaches that result in more precise inferences.