Large Sample Covariance Matrices and High-Dimensional Data Analysis: (Series Number 39 Cambridge Series in Statistical and Probabilistic Mathematics)

High-dimensional data appear in many fields, and their analysis has become increasingly important in modern statistics. However, it has long been observed that several well-known methods in multivariate analysis become inefficient, or even misleading, when the data dimension p is larger than, say, several tens. A seminal example is the well-known inefficiency of Hotelling's T2-test in such cases. This example shows that classical large sample limits may no longer hold for high-dimensional data; statisticians must seek new limiting theorems in these instances. Thus, the theory of random matrices (RMT) serves as a much-needed and welcome alternative framework. Based on the authors' own research, this book provides a firsthand introduction to new high-dimensional statistical methods derived from RMT. The book begins with a detailed introduction to useful tools from RMT, and then presents a series of high-dimensional problems with solutions provided by RMT methods.

Table of Contents:
1. Introduction; 2. Limiting spectral distributions; 3. CLT for linear spectral statistics; 4. The generalised variance and multiple correlation coefficient; 5. The T2-statistic; 6. Classification of data; 7. Testing the general linear hypothesis; 8. Testing independence of sets of variates; 9. Testing hypotheses of equality of covariance matrices; 10. Estimation of the population spectral distribution; 11. Large-dimensional spiked population models; 12. Efficient optimisation of a large financial portfolio.

About the Author :
Jianfeng Yao has rich research experience in random matrix theory and its applications to high-dimensional statistics. In recent years, he has published many authoritative papers in these areas and organised several international workshops on related topics. Shurong Zheng is author of several influential results in random matrix theory including a widely used central limit theorem for eigenvalue statistics of a random Fisher matrix. She has also developed important applications of the inference theory presented in the book to real-life high-dimensional statistics. Zhidong Bai is a world-leading expert in random matrix theory and high-dimensional statistics. He has published over 200 research papers and several specialized monographs, including Spectral Analysis of Large Dimensional Random Matrices (with J. W. Silverstein), for which he won the Natural Science Award of China (Second Class).

Review :
'This is the first book which treats systematic corrections to the classical multivariate statistical procedures so that the resultant procedures can be used for high-dimensional data. The corrections have been done by employing asymptotic tools based on the theory of random matrices.' Yasunori Fujikoshi, Hiroshima University, Japan '… this book is the first to cover these topics and can serve both as a good introduction to the topics as well as a comprehensive reference on the state of the art.' Robert Stelzer, MathSciNet 'This book deals with the analysis of covariance matrices under two different assumptions: large-sample theory and high-dimensional-data theory. While the former approach is the classical framework to derive asymptotics, nevertheless the latter has received increasing attention due to its applications in the emerging field of big-data. Due to its novelty and its relevance in the current research, the authors focus mainly on the high-dimensional-data framework. … The theory and the applications are presented under both the large-sample theory and the high-dimensional-data theory, and thus the reader can easily appreciate the differences between the two approaches. The material is presented in a quite simple manner, and the reader only needs some pre-requisites in basic mathematical statistics, linear algebra, and theory of multivariate normal distributions. Some technical prerequisites are collected in two appendices. Therefore, the book can be used by graduate students and researchers in a wide range of disciplines, ranging from mathematics to applied sciences.' Fabio Rapallo, Zentralblatt MATH