Vaart, Aad van der; Wellner, Jon A. Weak convergence and empirical processes. With applications to statistics. (English) [B] Springer Series in Statistics. New York, NY: Springer, Berlin, 1996. 508 pp., 34.50 English Pounds. [ISBN 0-387-94640-3]
This book is part of the Springer series in statistics. It aims to develop a rigorous mathematical approach to the modern theory of stochastic convergence, in particular convergence in distribution. The main tool used throughout is the concept of weak convergence, but treated in such a way that results will hold in settings that are sufficiently broad to include very general classes of empirical processes. This approach allows the classical types of stochastic convergence, convergence in distribution, convergence in probability, and convergence almost surely, to be extended to include the large class of empirical processes which have become increasingly important in statistics.
The book is divided into three parts. The first concentrates on the theory of stochastic convergence in a setting which prepares the ground for the rest of the book. The second part is concerned with the concepts of empirical measures and empirical processes indexed by quite general classes of functions. With both these parts the approach is predominantly mathematical. They are well written an have commendable clarity which allows a non-expert reader to work through the necessary abstraction. The authors take great pains to prepare the reader for definitions and they manage to keep the overall context of the mathematics clear. The final section, which consists of about half of the book, applies the previous ideas to a wide range of well- and less-well-known statistical tools. In particular general $M-$ and $Z-$estimators are treated. Well-known classical results concerning rates of convergence and the $\delta-$method are extended and explored. The bootstrap in various forms is shown to fit into the previously developed structure, as are Kac processes and random sample size problems.
Although the book takes a strict mathematical line of development, it is clear and well written. I particularly like the amount of thought that has gone into the structure of the book; there are plenty of signposts telling the reader where the flow is going. Good notes at the end of each chapter give a historical perspective and well-prepared indexes and glossaries make it much easier for the non-expert to be able to derive much from this book.
Paul Marriott
University of Surrey
Guildford