384 Pages
by
Chapman & Hall
384 Pages
by
Chapman & Hall
Also available as eBook on:
Useful in physics, economics, psychology, and other fields, random matrices play an important role in the study of multivariate statistical methods. Until now, however, most of the material on random matrices could only be found scattered in various statistical journals. Matrix Variate Distributions gathers and systematically presents most of the recent developments in continuous matrix variate distribution theory and includes new results.
After a review of the essential background material, the authors investigate the range of matrix variate distributions, including:
With its inclusion of new results, Matrix Variate Distributions promises to stimulate further research and help advance the field of multivariate statistical analysis.
PRELIMINARIES
Matrix Algebra
Jacobians of Transformations
Integration
Zonal Polynomials
Hypergeometric Functions of Matrix Argument
LaGuerre Polynomials
Generalized Hermite Polynomials
Notion of Random Matrix Problems
MATRIX VARIATE NORMAL DISTRIBUTION
Density Function
Properties
Singular Matrix Variate Normal distribution
Symmetic Matrix Variate Normal Distribution
Restricted Matrix Variate Normal Distribution
Matrix Variate Q-Generalized Normal Distribution
WISHART DISTRIBUTION
Introduction
Density Function
Properties
Inverted Wishart Distribution
Noncentral Wishart Distribution
Matrix Variate Gamma Distribution
Approximations
MATRIX VARIATE t-DISTRIBUTION
Density Function
Properties
Inverted Matrix Variate t-Distribution
Disguised Matrix Variate t-Distribution
Restricted Matrix Variate t-Distribution
Noncentral Matrix Variate t-Distribution
Distribution of Quadratic Forms
MATRIX VARIATE BETA DISTRIBUTIONS
Density Functions
Properties
Related Distributions
Noncentral Matrix Variate Beta Distribution
MATRIX VARIATE DIRICHLET DISTRIBUTIONS
Density Functions
Properties
Related Distributions
Noncentral Matrix Variate Dirichlet Distributions
DISTRIBUTION OF MATRIX QUADRATIC FORMS
Density Function
Properties
Functions of Quadratic Forms
Series Representation of the Density
Noncentral Density Function
Expected Values
Wishartness and Independence of Quadratic Forms of the Type XAX'
Wishartness and Independence of Quadratic Forms of the Type XAX'+1/2(LX'+XL')+C
Wishartness and Independence of Quadratic Forms of the Type XAX'+L1X'+XL'2+C
MISCELLANEOUS DISTRIBUTIONS
Uniform Distribution on Stiefel Manifold
Von Mises-Fisher Distribution
Bingham Matrix Distribution
Generalized Bingham-Von Mises Matrix Distribution
Manifold Normal Distribution
Matrix Angular Central Gaussian Distribution
Bimatix Wishart Distribution
Beta-Wishart Distribution
Confluent Hypergeometric Function Kind 1 Distribution
Confluent Hypergeometric Function Kind 2 Distribution
Hypergeometric Function Distributions
Generalized Hypergeometric Function Distributions
Complex Matrix Variate Distributions
GENERAL FAMILIES OF MATRIX VARIATE DISTRIBUTIONS
Matrix Variate Liouville Distributions
Matrix Variate Spherical Distributions
Matrix Variate Elliptically Contoured Distributions
Orthogonally Invariant and Residual Independent Matrix Distributions
GLOSSARY
REFERENCES
SUBJECT INDEX
Each chapter also includes an Introduction and Problems
@
Matrix Algebra
Jacobians of Transformations
Integration
Zonal Polynomials
Hypergeometric Functions of Matrix Argument
LaGuerre Polynomials
Generalized Hermite Polynomials
Notion of Random Matrix Problems
MATRIX VARIATE NORMAL DISTRIBUTION
Density Function
Properties
Singular Matrix Variate Normal distribution
Symmetic Matrix Variate Normal Distribution
Restricted Matrix Variate Normal Distribution
Matrix Variate Q-Generalized Normal Distribution
WISHART DISTRIBUTION
Introduction
Density Function
Properties
Inverted Wishart Distribution
Noncentral Wishart Distribution
Matrix Variate Gamma Distribution
Approximations
MATRIX VARIATE t-DISTRIBUTION
Density Function
Properties
Inverted Matrix Variate t-Distribution
Disguised Matrix Variate t-Distribution
Restricted Matrix Variate t-Distribution
Noncentral Matrix Variate t-Distribution
Distribution of Quadratic Forms
MATRIX VARIATE BETA DISTRIBUTIONS
Density Functions
Properties
Related Distributions
Noncentral Matrix Variate Beta Distribution
MATRIX VARIATE DIRICHLET DISTRIBUTIONS
Density Functions
Properties
Related Distributions
Noncentral Matrix Variate Dirichlet Distributions
DISTRIBUTION OF MATRIX QUADRATIC FORMS
Density Function
Properties
Functions of Quadratic Forms
Series Representation of the Density
Noncentral Density Function
Expected Values
Wishartness and Independence of Quadratic Forms of the Type XAX'
Wishartness and Independence of Quadratic Forms of the Type XAX'+1/2(LX'+XL')+C
Wishartness and Independence of Quadratic Forms of the Type XAX'+L1X'+XL'2+C
MISCELLANEOUS DISTRIBUTIONS
Uniform Distribution on Stiefel Manifold
Von Mises-Fisher Distribution
Bingham Matrix Distribution
Generalized Bingham-Von Mises Matrix Distribution
Manifold Normal Distribution
Matrix Angular Central Gaussian Distribution
Bimatix Wishart Distribution
Beta-Wishart Distribution
Confluent Hypergeometric Function Kind 1 Distribution
Confluent Hypergeometric Function Kind 2 Distribution
Hypergeometric Function Distributions
Generalized Hypergeometric Function Distributions
Complex Matrix Variate Distributions
GENERAL FAMILIES OF MATRIX VARIATE DISTRIBUTIONS
Matrix Variate Liouville Distributions
Matrix Variate Spherical Distributions
Matrix Variate Elliptically Contoured Distributions
Orthogonally Invariant and Residual Independent Matrix Distributions
GLOSSARY
REFERENCES
SUBJECT INDEX
Each chapter also includes an Introduction and Problems
@
Biography
A K Gupta, D K Nagar
"This book is about probability distributions for random matrices."
Biometrics, Vol. 56, No. 3, September 2000
"I am sure that the book will be welcomed by specialists, because of its systematic and thorough coverage of the different distributions."
Biometrics, Vol. 56, No. 3, September 2000
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