An introduction to statistical analysis of random arrays
著者
書誌事項
An introduction to statistical analysis of random arrays
VSP, 1998
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注記
Includes bibliographical references (p. [649]-668) and index
内容説明・目次
内容説明
01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information.
This book contains the results of 30 years of investigation by the author into the creation of a new theory on statistical analysis of observations, based on the principle of random arrays of random vectors and matrices of increasing dimensions. It describes limit phenomena of sequences of random observations, which occupy a central place in the theory of random matrices. This is the first book to explore statistical analysis of random arrays and provides the necessary tools for such analysis. This book is a natural generalization of multidimensional statistical analysis and aims to provide its readers with new, improved estimators of this analysis.
The book consists of 14 chapters and opens with the theory of sample random matrices of fixed dimension, which allows to envelop not only the problems of multidimensional statistical analysis, but also some important problems of mechanics, physics and economics. The second chapter deals with all 50 known canonical equations of the new statistical analysis, which form the basis for finding new and improved statistical estimators. Chapters 3-5 contain detailed proof of the three main laws on the theory of sample random matrices. In chapters 6-10 detailed, strong proofs of the Circular and Elliptic Laws and their generalization are given. In chapters 11-13 the convergence rates of spectral functions are given for the practical application of new estimators and important questions on random matrix physics are considered. The final chapter contains 54 new statistical estimators, which generalize the main estimators of statistical analysis.
目次
List of basic notations and assumptions
Preface and some historical remarks
Chapter 1. Introduction to the theory of sample matrices of fixed dimension
Chapter 2. Canonical equations
Chapter 3. First Law for the eigenvalues and eigenvectors of random symmetric matrices
Chapter 4. The Second Law for the singular values and eigenvectors of random matrices. Inequalities for the spectral radius of large random matrices
Chapter 5. The Third Law for the eigenvalues and eigenvectors of empirical covariance matrices
Chapter 6. The first proof of the Strong Circular Law
Chapter 7. Strong Law for normalized spectral functions of nonselfadjoint random matrices with independent row vectors and simple rigorous proof of the Strong Circular Law
Chapter 8. Rigorous proof of the Strong Elliptic Law
Chapter 9. The Circular and Uniform Laws for eigenvalues of random nonsymmetric complex matrices with independent entries
Chapter 10. Strong V-Law for eigenvalues of nonsymmetric random matrices
Chapter 11. Convergence rate of the expected spectral functions of symmetric random matrices is equal to O(n-1/2)
Chapter 12. Convergence rate of expected spectral functions of the sample covariance matrix is equal to O(n-1/2) under the condition
Chapter 13. The First Spacing Law for random symmetric matrices
Chapter 14. Ten years of General Statistical Analysis (GSA) (The main G-estimators of general statistical analysis)
References
Index
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