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E-raamat: Matrix Variate Distributions

(Bowling Green State University, Bowling Green, Ohio, USA),
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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.

Arvustused

"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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Preliminaries
1(55)
Introduction
1(1)
Matrix Algebra
2(10)
Jacobians of Transformations
12(6)
Integration
18(11)
Zonal Polynomials
29(5)
Hypergeometric Functions of Matrix Argument
34(7)
Laguerre Polynomials
41(1)
Generalized Hermite Polynomials
42(2)
Notion of Random Matrix
44(11)
Problems
47(8)
Matrix Variate Normal Distribution
55(32)
Introduction
55(1)
Density Function
55(1)
Properties
56(12)
Singular Matrix Variate Normal Distribution
68(2)
Symmetric Matrix Variate Normal Distribution
70(4)
Restricted Matrix Variate Normal Distribution
74(3)
Matrix Variate θ-Generalized Normal Distribution
77(10)
Problems
82(5)
Wishart Distribution
87(46)
Introduction
87(1)
Density Function
87(3)
Properties
90(21)
Inverted Wishart Distribution
111(2)
Noncentral Wishart Distribution
113(9)
Matrix Variate Gamma Distribution
122(2)
Approximations
124(9)
Problems
127(6)
Matrix Variate t-Distribution
133(32)
Introduction
133(1)
Density Function
134(1)
Properties
135(7)
Inverted Matrix Variate t-Distribution
142(1)
Disguised Matrix Variate t-distribution
143(8)
Restricted Matrix Variate t-Distribution
151(1)
Noncentral Matrix Variate t-distribution
152(4)
Distribution of Quadratic Forms
156(9)
Problems
159(6)
Matrix Variate Beta Distributions
165(34)
Introduction
165(1)
Density Functions
165(6)
Properties
171(11)
Related Distributions
182(6)
Noncentral Matrix Variate Beta Distribution
188(11)
Problems
193(6)
Matrix Variate Dirichlet Distributions
199(26)
Introduction
199(1)
Density Functions
199(5)
Properties
204(10)
Related Distributions
214(4)
Noncentral Matrix Variate Dirichlet Distributions
218(7)
Problems
222(3)
Distribution of Quadratic Forms
225(54)
Introduction
225(1)
Density Function
225(3)
Properties
228(5)
Functions of Quadratic Forms
233(5)
Series Representation of the Density
238(8)
Noncentral Density Function
246(5)
Expected Values
251(2)
Wishartness and Independence of Quadratic Forms of the Type XAX'
253(9)
Wishartness and Independence of Quadratic Forms of the Type XAX' + 1/2(LX' + XL') + C
262(8)
Wishartness and Independence of Quadratic Forms of the type XAX' + L1X' + XL2' + C
270(9)
Problems
273(6)
Miscellaneous Distributions
279(32)
Introduction
279(1)
Uniforms Distribution on Stiefel Manifold
279(2)
Von Mises-Fisher Distribution
281(3)
Bingham Matrix Distribution
284(1)
Generalized Bingham-Von Mises Matrix Distribution
285(2)
Manifold Normal Distribution
287(1)
Matrix Angular Central Gaussian Distribution
288(1)
Bimatrix Wishart Distribution
289(1)
Beta-Wishart Distribution
290(1)
Confluent Hypergeometric Function Kind 1 Distribution
291(4)
confluent Hypergeometric Function Kind 2 Distribution
295(3)
Hypergeometric Function Distributions
298(3)
Generalized Hypergeometric Function Distributions
301(2)
Complex Matrix Variate Distributions
303(8)
Problems
304(7)
General Families of Matrix Variate Distributions
311(20)
Introduction
311(1)
Matrix Variate Liouville Distributions
311(4)
Matrix Variate Spherical Distributions
315(7)
Matrix Variate Elliptically Contoured Distributions
322(1)
Orthogonally Invariant and Residual Independent Matrix Distributions
323(8)
Problems
328(3)
Glossary of Notations and Abbreviations 331(12)
References 343(21)
Subject Index 364
A K Gupta, D K Nagar
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