Large Sample Covariance Matrices and High-Dimensional Data Analysis (Cambridge Series in Statistical and Probabilistic Mathematics 39)
By: Shurong Zheng (author), Jianfeng Yao (author), Zhidong Bai (author)Hardback
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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.
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).
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.
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- ID: 9781107065178
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