Oxford Statistical Science Series. 22. Oxford: Oxford University Press. xii, 380 p. £ 57.50 (2000).

This is book gives an introduction to likelihood inference. Included is a discussion of modern aspects as saddlepoint approximations, conditional and marginal inference. The emphasis of the book is on the development of statistical methods and a description of the underlying theory. It is suitable for students and statisticians who are not specialists in higher-order asymptotics.
Large sample approximations are presented in chapter 2. This includes Edgeworth series approximation, saddlepoint approximations, stochastic asymptotic expansions, and Laplace approximations. The following chapter gives an introduction to the likelihood itself. First, some properties of the likelihood function are discussed. Then the likelihood principle, regular models, information, and methods of inference are presented. The classical theory of likelihood inference is based on first-order asymptotic theory. This is the topic of chapter 4. Here, maximum likelihood estimates and the likelihood ratio statistic play the central role.
In chapter 5, higher-order asymptotic theory is considered. This theory uses approximations based on either an Edgeworth expansion or a saddlepoint approximation. In chapter 6, the approximation of conditional distributions given an ancillary statistic is discussed. It also deals with approximate ancillarity and the approximation of sample space derivatives. The modified signed likelihood ratio statistic is derived in chapter 7, along with several approximations to that statistic. Conditional and marginal likelihood functions are presented in chapter 8, and chapter 9 considers the modified profile likelihood function along with several approximations.
Each chapter has a section with discussion and references and one with exercises.