Multivariate Statistical Analysis in the Real and Complex Domains.

• Author: Mathai, Arak M.
• TeilnehmendeR: Provost, Serge B.; Haubold, Hans J.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2022.; ©2022.
• Year of Publication: 2022
• Edition: 1st ed.
• Language: English
• Physical Description: 1 online resource (939 pages)

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.