Statistics for high-dimensional data, by Peter Bühlmann and Sara van de Geer, Berlin, Springer-Verlag, 2011, xvii + 556 pp., £81.00 or US$99.00 (hardback), ISBN 978-3-642-20191-2
During the last few years, we have witnessed a revolution of computational and methodological advances which allow statistical inference for high-dimensional data. Such data are now common in areas such as bioinformatics and information technology. The terminology ‘high-dimensional data’ refers to the situation where the number of observations (samples in the data) is of (usually much) smaller order of magnitude that the number of unknown parameters (variables of the involved model) to be estimated from the data. Although classical statistical inference is not appropriate for dealing with high-dimensional settings, statistical inference for such settings is still possible based on assuming certain notions of sparsity.
Motivated by real data problems, this book presents theoretical advances for high-dimensional statistical inference. Chapters 1–5 provide the motivation for high-dimensional statistical inference as well as a survey of theoretical results and computational algorithms for the least absolute shrinkage and selection operator (known as LASSO) method and some of its variants and generalizations, like the elastic net, the adaptive LASSO and the group LASSO. The mathematical tools for these methods are fully developed in Chapters 6–8, while Chapters 9–13 contains results and methodologies in areas such as generalized linear models, boosting and graphical models. Finally, Chapter 14 contains a detailed exposition of some important probability and moment inequalities for the maximal difference between averages and expectations that are needed to develop the mathematical theory in the previous chapters. At the end of Chapters 2–14, a number of problems are provided which will be extremely useful for the serious reader of the subject. The book also contains 10 pages or so of relevant references. Author and subject indices are also provided.
This book, written by two specialists, provides an up-to-date treatment of statistics for high-dimensional data by bringing together methodological and computational algorithms for high-dimensional statistical inference in a self-contained fashion. In my view, it will prove to be a valuable source for those researchers and graduate students who are interested in the mathematical theory of high-dimensional statistics as well as in techniques of modern asymptotics such as oracle inequalities for sparse data. It can also serve as an excellent textbook for a one-semester graduate course on the study of l 1-penalty-based statistical methods and their variants and generalizations. Overall, this book is a timely and welcome exposition on statistical inference for high-dimensional data and it is highly recommended.