Large Sample Covariance Matrices and High-Dimensional Data Analysis

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.

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.