1st Edition

Matrix Variate Distributions




ISBN 9781584880462
Published October 22, 1999 by Chapman and Hall/CRC
384 Pages

USD $200.00

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Book Description

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:

  • matrix variate normal distribution
  • Wishart distribution
  • Matrix variate t-distribution
  • Matrix variate beta distribution
  • F-distribution
  • Matrix variate Dirichlet distribution
  • Matrix quadratic forms
    With its inclusion of new results, Matrix Variate Distributions promises to stimulate further research and help advance the field of multivariate statistical analysis.
  • Table of Contents

    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

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    Reviews

    "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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