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Multivariate Statistical Analysis in the Real and Complex Domains.

Bibliographic Details
Main Author: Mathai, Arak M.
Other Authors: Provost, Serge B., Haubold, Hans J.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2022.
Edition:1st ed.
Subjects:
Electronic books.
Online Access:Click to View
Table of Contents:
  • Intro
  • Preface/Special features
  • Contents
  • List of Symbols
  • 1 Mathematical Preliminaries
  • 1.1 Introduction
  • 1.2 Determinants
  • 1.2.1 Inverses by row operations or elementary operations
  • 1.3 Determinants of Partitioned Matrices
  • 1.4 Eigenvalues and Eigenvectors
  • 1.5 Definiteness of Matrices, Quadratic and Hermitian Forms
  • 1.5.1 Singular value decomposition
  • 1.6 Wedge Product of Differentials and Jacobians
  • 1.7 Differential Operators
  • 1.7.1 Some basic applications of the vector differential operator
  • References
  • 2 The Univariate Gaussian and Related Distributions
  • 2.1 Introduction
  • 2.1a The Complex Scalar Gaussian Variable
  • 2.1.1 Linear functions of Gaussian variables in the real domain
  • 2.1a.1 Linear functions in the complex domain
  • 2.1.2 The chisquare distribution in the real domain
  • 2.1a.2 The chisquare distribution in the complex domain
  • 2.1.3 The type-2 beta and F distributions in the real domain
  • 2.1a.3 The type-2 beta and F distributions in the complex domain
  • 2.1.4 Power transformation of type-1 and type-2 beta random variables
  • 2.1.5 Exponentiation of real scalar type-1 and type-2 beta variables
  • 2.1.6 The Student-t distribution in the real domain
  • 2.1a.4 The Student-t distribution in the complex domain
  • 2.1.7 The Cauchy distribution in the real domain
  • 2.2 Quadratic Forms, Chisquaredness and Independence in the Real Domain
  • 2.2a Hermitian Forms, Chisquaredness and Independence in the Complex Domain
  • 2.2.1 Extensions of the results in the real domain
  • 2.2a.1 Extensions of the results in the complex domain
  • 2.3 Simple Random Samples from Real Populations and Sampling Distributions
  • 2.3a Simple Random Samples from a Complex Gaussian Population
  • 2.3.1 Noncentral chisquare having n degrees of freedom in the real domain.
  • 2.3.1.1 Mean value and variance, real central and non-central chisquare
  • 2.3a.1 Noncentral chisquare having n degrees of freedom in the complex domain
  • 2.4 Distributions of Products and Ratios and Connection to Fractional Calculus
  • 2.5 General Structures
  • 2.5.1 Product of real scalar gamma variables
  • 2.5.2 Product of real scalar type-1 beta variables
  • 2.5.3 Product of real scalar type-2 beta variables
  • 2.5.4 General products and ratios
  • 2.5.5 The H-function
  • 2.6 A Collection of Random Variables
  • 2.6.1 Chebyshev's inequality
  • 2.7 Parameter Estimation: Point Estimation
  • 2.7.1 The method of moments and the method of maximum likelihood
  • 2.7.2 Bayes' estimates
  • 2.7.3 Interval estimation
  • References
  • 3 The Multivariate Gaussian and Related Distributions
  • 3.1 Introduction
  • 3.1a The Multivariate Gaussian Density in the Complex Domain
  • 3.2 The Multivariate Normal or Gaussian Distribution, Real Case
  • 3.2.1 The moment generating function in the real case
  • 3.2a The Moment Generating Function in the Complex Case
  • 3.2a.1 Moments from the moment generating function
  • 3.2a.2 Linear functions
  • 3.3 Marginal and Conditional Densities, Real Case
  • 3.3a Conditional and Marginal Densities in the Complex Case
  • 3.4 Chisquaredness and Independence of Quadratic Forms in the Real Case
  • 3.4.1 Independence of quadratic forms
  • 3.4a Chisquaredness and Independence in the ComplexGaussian Case
  • 3.4a.1 Independence of Hermitian forms
  • 3.5 Samples from a p-variate Real Gaussian Population
  • 3.5a Simple Random Sample from a p-variate Complex Gaussian Population
  • 3.5.1 Some simplifications of the sample matrix in the real Gaussian case
  • 3.5.2 Linear functions of the sample vectors
  • 3.5a.1 Some simplifications of the sample matrix in the complex Gaussian case.
  • 3.5a.2 Linear functions of the sample vectors in the complex domain
  • 3.5.3 Maximum likelihood estimators of the p-variate real Gaussian distribution
  • 3.5a.3 MLE's in the complex p-variate Gaussian case
  • 3.5a.4 Matrix derivatives in the complex domain
  • 3.5.4 Properties of maximum likelihood estimators
  • 3.5.5 Some limiting properties in the p-variate case
  • 3.6 Elliptically Contoured Distribution, Real Case
  • 3.6.1 Some properties of elliptically contoureddistributions
  • 3.6.2 The density of u=r2
  • 3.6.3 Mean value vector and covariance matrix
  • 3.6.4 Marginal and conditional distributions
  • 3.6.5 The characteristic function of an elliptically contoured distribution
  • References
  • 4 The Matrix-Variate Gaussian Distribution
  • 4.1 Introduction
  • 4.2 Real Matrix-variate and Multivariate Gaussian Distributions
  • 4.2a The Matrix-variate Gaussian Density, Complex Case
  • 4.2.1 Some properties of a real matrix-variate Gaussian density
  • 4.2a.1 Some properties of a complex matrix-variate Gaussian density
  • 4.2.2 Additional properties in the real and complex cases
  • 4.2.3 Some special cases
  • 4.3 Moment Generating Function and Characteristic Function, Real Case
  • 4.3a Moment Generating and Characteristic Functions, Complex Case
  • 4.3.1 Distribution of the exponent, real case
  • 4.3a.1 Distribution of the exponent, complex case
  • 4.3.2 Linear functions in the real case
  • 4.3a.2 Linear functions in the complex case
  • 4.3.3 Partitioning of the parameter matrix
  • 4.3.4 Distributions of quadratic and bilinear forms
  • 4.4 Marginal Densities in the Real Matrix-variate Gaussian Case
  • 4.4a Marginal Densities in the Complex Matrix-variate Gaussian Case
  • 4.5 Conditional Densities in the Real Matrix-variate Gaussian Case
  • 4.5a Conditional Densities in the Matrix-variate Complex Gaussian Case
  • 4.5.1 Re-examination of the case q=1.
  • 4.6 Sampling from a Real Matrix-variate Gaussian Density
  • 4.6.1 The distribution of the sample sum of products matrix, real case
  • 4.6.2 Linear functions of sample vectors
  • 4.6.3 The general real matrix-variate case
  • 4.6a The General Complex Matrix-variate Case
  • 4.7 The Singular Matrix-variate Gaussian Distribution
  • References
  • 5 Matrix-Variate Gamma and Beta Distributions
  • 5.1 Introduction
  • 5.1a The Complex Matrix-variate Gamma
  • 5.2 The Real Matrix-variate Gamma Density
  • 5.2.1 The mgf of the real matrix-variate gammadistribution
  • 5.2a The Matrix-variate Gamma Function and Density,Complex Case
  • 5.2a.1 The mgf of the complex matrix-variate gamma distribution
  • 5.3 Matrix-variate Type-1 Beta and Type-2 Beta Densities,Real Case
  • 5.3.1 Some properties of real matrix-variate type-1 and type-2 beta densities
  • 5.3a Matrix-variate Type-1 and Type-2 Beta Densities, Complex Case
  • 5.3.2 Explicit evaluation of type-1 matrix-variate beta integrals, real case
  • 5.3a.1 Evaluation of matrix-variate type-1 beta integrals, complex case
  • 5.3.3 General partitions, real case
  • 5.3.4 Methods avoiding integration over the Stiefel manifold
  • 5.3.5 Arbitrary moments of the determinants, real gamma and beta matrices
  • 5.3a.2 Arbitrary moments of the determinants in the complex case
  • 5.4 The Densities of Some General Structures
  • 5.4.1 The G-function
  • 5.4.2 Some special cases of the G-function
  • 5.4.3 The H-function
  • 5.4.4 Some special cases of the H-function
  • 5.5, 5.5a The Wishart Density
  • 5.5.1 Explicit evaluations of the matrix-variate gamma integral, real case
  • 5.5a.1 Evaluation of matrix-variate gamma integrals in the complex case
  • 5.5.2 Triangularization of the Wishart matrixin the real case
  • 5.5a.2 Triangularization of the Wishart matrix in the complex domain.
  • 5.5.3 Samples from a p-variate Gaussian population and the Wishart density
  • 5.5a.3 Sample from a complex Gaussian population and the Wishart density
  • 5.5.4 Some properties of the Wishart distribution, real case
  • 5.5.5 The generalized variance
  • 5.5.6 Inverse Wishart distribution
  • 5.5.7 Marginal distributions of a Wishart matrix
  • 5.5.8 Connections to geometrical probability problems
  • 5.6 The Distribution of the Sample Correlation Coefficient
  • 5.6.1 The special case ρ=0
  • 5.6.2 The multiple and partial correlation coefficients
  • 5.6.3 Different derivations of ρ1.(2…p)
  • 5.6.4 Distributional aspects of the sample multiple correlation coefficient
  • 5.6.5 The partial correlation coefficient
  • 5.7 Distributions of Products and Ratios of Matrix-variate Random Variables
  • 5.7.1 The density of a product of real matrices
  • 5.7.2 M-convolution and fractional integralof the second kind
  • 5.7.3 A pathway extension of fractional integrals
  • 5.7.4 The density of a ratio of real matrices
  • 5.7.5 A pathway extension of first kind integrals, real matrix-variate case
  • 5.7a Density of a Product and Integrals of the Second Kind
  • 5.7a.1 Density of a product and fractional integral of the second kind, complex case
  • 5.7a.2 Density of a ratio and fractional integrals of the first kind, complex case
  • 5.8 Densities Involving Several Matrix-variate Random Variables, Real Case
  • 5.8.1 The type-1 Dirichlet density, real scalar case
  • 5.8.2 The type-2 Dirichlet density, real scalar case
  • 5.8.3 Some properties of Dirichlet densities in the real scalar case
  • 5.8.4 Some generalizations of the Dirichlet models
  • 5.8.5 A pseudo Dirichlet model
  • 5.8.6 The type-1 Dirichlet model in real matrix-variate case
  • 5.8.7 The type-2 Dirichlet model in the real matrix-variate case
  • 5.8.8 A pseudo Dirichlet model.
  • 5.8a Dirichlet Models in the Complex Domain.

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