Cambridge Series in Statistical and Probabilistic Mathematics 23. Cambridge: Cambridge University Press (ISBN 978-0-521-84703-2/hbk). viii, 236 p. £ 35.00; $ 65.00 (2007).

In fields such as biology, medical sciences, sociology, and economics, researchers often face the situation where the number of available observations, or the amount of available information, is sufficiently small so that approximations based on the normal distribution may be unreliable. Theoretical works over the last quarter-century have yielded new likelihood-based methods that lead to very accurate approximations in finite samples, but these works have had limited impact on statistical practice. This book illustrates by means of realistic examples and case studies how to use the new theory, and investigates how and when it makes a difference to the resulting inference. The treatment is orientated towards practice and is accompanied by code in the R language that enables the methods to be applied in a range of situations of interest to practitioners. The analysis includes some comparisons of higher order likelihood inference with bootstrap and Bayesian methods.
The Contents are the following: 1. Introduction; 2. Uncertainty and approximation; 3. Simple illustrations; 4. Discrete data; 5. Regression with continuous responses; 6. Some case studies; 7. Further topics; 8. Likelihood approximations; 9. Numerical implementation; 10. Problems and further results. In the Appendix A we find: Convergence of sequences; The sample mean; Laplace approximation and chi-square approximations.
This monograph is very useful for researchers in medicine and biology, as well as in economics and sociology.