Expansions and asymptotics for statistics, by Christopher G. Small, Boca Raton, Chapman & Hall/CRC Press, 2010, xiv + 343 pp., £59.99 or US$94.95 (hardback), ISBN 978-1-58488-590-0
This book is a welcome addition to anyone learning asymptotics. The material is presented in a very intuitive manner while at the same time not sacrificing rigour. The book begins by introducing the basic mathematical foundations necessary for asymptotics. By perusing the different types of series, the reader gets a good grasp of their usage in the field of asymptotics. Methods such as the Delta method, optimality and likelihood asymptotics are used in a very methodical manner.
The author introduces Pade approximants and continued fractions which are hardly ever investigated in traditional books on asymptotics. The book introduces the Laplace approximation and shows the application to statistical inference. Saddle point approximations such as the method of Darboux and Hayman's approximation and application of these methods cause the reader to be an active participant rather than a passive learner. The final chapter is devoted to the summation of series and addresses methods for accelerating the speed of convergence for these methods.
Probably the major goal of this book is to introduce the hows and whys of asymptotic theory which are seldom taught in the traditional asymptotic courses at the doctoral level. This book gives the reader a thorough appreciation of asymptotics through the use of lots of practical examples and down-to-earth explanations. While it may not be able to serve as an essential text, students may find it very useful as a reference book.