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
5.8a.1 A type-2 Dirichlet model in the complex domain
References
6 Hypothesis Testing and Null Distributions
6.1 Introduction
6.2 Testing Ho: μ=μ0 (Given) When Σ is Known, the Real Np(μ, Σ) Case
6.2.1 Paired variables and linear functions
6.2.2 Independent Gaussian populations
6.2a Testing Ho: μ=μo (given) When Σ is Known, Complex Gaussian Case
6.2.3 Test involving a subvector of a mean value vector when Σ is known
6.2.4 Testing μ1=@汥瑀瑯步渠=μp, with Σ known, real Gaussian case
6.2.5 Likelihood ratio criterion for testing Ho:μ1=@汥瑀瑯步渠=μp, Σ known
6.3 Testing Ho: μ=μo (given) When Σ is Unknown, RealGaussian Case
6.3.1 The distribution of the test statistic
6.3.2 Paired values or linear functions when Σ is unknown
6.3.3 Independent Gaussian populations
6.3.4 Testing μ1=@汥瑀瑯步渠=μp when Σ is unknown in the real Gaussian case
6.3.5 Likelihood ratio test for testing Ho:μ1=@汥瑀瑯步渠=μp when Σ is unknown
6.4 Testing Hypotheses on the Population Covariance Matrix
6.4.1 Arbitrary moments of λ
6.4.2 The asymptotic distribution of -2lnλ when testing Ho:Σ=Σo
6.4.3 Tests on Wilks' concept of generalized variance
6.5 The Sphericity Test or Testing if Ho :Σ=σ2 I, Given a Np(μ,Σ) Sample
6.6 Testing the Hypothesis that the Covariance Matrix is Diagonal
6.7 Equality of Diagonal Elements, Given that Σ is Diagonal, Real Case
6.8 Hypothesis that the Covariance Matrix is Block Diagonal, Real Case
6.8.1 Special case: k=2
6.8.2 General case: k=2
6.9 Hypothesis that the Mean Value and Covariance Matrixare Given
6.10 Testing Hypotheses on Linear Regression Models or Linear Hypotheses
6.10.1 A simple linear model
6.10.2 Hypotheses on individual parameters
6.11 Problem Involving Two or More Independent GaussianPopulations
6.11.1 Equal but unknown covariance matrices
6.12 Equality of Covariance Matrices in Independent Gaussian Populations
6.12.1 Asymptotic behavior
6.13 Testing the Hypothesis that k Independent p-variate Real Gaussian Populations are Identical and Multivariate Analysis of Variance
6.13.1 Conditional and marginal hypotheses
6.13.2 Arbitrary moments of λ1
6.13.3 The asymptotic distribution of -2lnλ1
References
7 Rectangular Matrix-Variate Distributions
7.1 Introduction
7.2 Rectangular Matrix-Variate Gamma Density, Real Case
7.2.1 Extension of the gamma density to the real rectangular matrix-variate case
7.2.2 Multivariate gamma and Maxwell-Boltzmann densities, real case
7.2.3 Some properties of the rectangular matrix-variate gamma density
7.2.4 Connection to the volume of a random parallelotope
7.2.5 Pathway to real matrix-variate gamma and Maxwell-Boltzmann densities
7.2.6 Multivariate gamma and Maxwell-Boltzmann densities, pathway model
7.2.7 Concluding remarks
7.2a Complex Matrix-Variate Gamma and Maxwell-Boltzmann Densities
7.2a.1 Extension of the Matrix-Variate Gamma Density in the Complex Domain
7.2a.2 The multivariate gamma density in the complex matrix-variate case
7.2a.3 Arbitrary moments, complex case
7.2a.4 A pathway extension in the complex case
7.2a.5 The Maxwell-Boltzmann and Raleigh cases in the complex domain
7.3 Real Rectangular Matrix-Variate Type-1 and Type-2 Beta Densities
7.3.1 Arbitrary moments
7.3.2 Special case: p=1
7.3a Rectangular Matrix-Variate Type-1 Beta Density, Complex Case
7.3a.1 Different versions of the type-1 beta density, the complex case
7.3a.2 Multivariate type-1 beta density, the complex case
7.3a.3 Arbitrary moments in the complex case
7.4 The Real Rectangular Matrix-Variate Type-2 Beta Density
7.4.1 The real type-2 beta density in the multivariate case
7.4.2 Moments in the real rectangular matrix-variate type-2 beta density
7.4.3 A pathway extension in the real case
7.4a Complex Rectangular Matrix-Variate Type-2 Beta Density
7.4a.1 Multivariate complex type-2 beta density
7.4a.2 Arbitrary moments in the complex type-2 beta density
7.4a.3 A pathway version of the complex rectangular matrix-variate type-1 beta density
7.5,7.5a Ratios Involving Rectangular Matrix-Variate Random Variables
7.5.1 Multivariate F, Student-t and Cauchy densities
7.5a.1 A complex multivariate Student-t having ν degrees of freedom
7.5a.2 A complex multivariate Cauchy density
7.6 Rectangular Matrix-Variate Dirichlet Density, Real Case
7.6.1 Certain properties, real rectangular matrix-variate type-1 Dirichlet density
7.6.2 A multivariate version of the real matrix-variate type-1 Dirichlet density
7.7 Generalizations of the Real Rectangular Dirichlet Models
References
8 The Distributions of Eigenvalues and Eigenvectors
8.1 Introduction
8.1.1 Matrix-variate gamma and beta densities, real case
8.1a Matrix-variate Gamma and Beta Densities,Complex Case
8.1.2 Real Wishart matrices
8.2 Some Eigenvalues and Eigenvectors, Real Case
8.2a The Distributions of Eigenvalues in the Complex Case
8.2.1 Eigenvalues of matrix-variate gamma and Wishart matrices, real case
8.2a.1 Eigenvalues of complex matrix-variate gamma and Wishart matrices
8.2.2 An alternative procedure in the real case
8.2.3 The joint density of the eigenvectors in the real case
8.2a.2 An alternative procedure in the complex case
8.3 The Singular Real Case
8.3.1 Singular Wishart and matrix-variate gamma distributions, real case
8.3.2 A direct evaluation as an eigenvalue problem
8.3a The Singular Complex Case
8.3a.1 Singular gamma or singular Gaussian distribution, complex case
8.3a.2 A direct method of evaluation in the complex case
8.4 The Case of One Wishart or Gamma Matrix in the Real Domain
8.4a The Case of One Wishart or Gamma Matrix, Complex Domain
References
9 Principal Component Analysis
9.1 Introduction
9.2 Principal Components
9.3 Issues to Be Mindful of when Constructing Principal Components
9.4 The Vector of Principal Components
9.4.1 Principal components viewed from differingperspectives
9.5 Sample Principal Components
9.5.1 Estimation and evaluation of the principal components
9.5.2 L1 and L2 norm principal components
9.6 Distributional Aspects of Eigenvalues and Eigenvectors
9.6.1 The distributions of the largest and smallesteigenvalues
9.6.2 Simplification of the factor i<j(λi-λj)
9.6.3 The distributions of the eigenvalues
9.6.4 Density of the smallest eigenvalue λp in the real matrix-variate gamma case
9.6.5 Density of the largest eigenvalue λ1 in the real matrix-variate gamma case
9.6.6 Density of the largest eigenvalue λ1 in the generalreal case
9.6.7 Density of the smallest eigenvalue λp in the generalreal case
References
10 Canonical Correlation Analysis
10.1 Introduction
10.1.1 An invariance property
10.2 Pairs of Canonical Variables
10.3 Estimation of the Canonical Correlations and CanonicalVariables
10.3.1 An iterative procedure
10.4 The Sampling Distribution of the Canonical Correlation Matrix
10.4.1 The joint density of the eigenvalues and eigenvectors
10.4.2 Testing whether the population canonical correlations equal zero
10.5 The General Sampling Distribution of the Canonical Correlation Matrix
10.5.1 The sampling distribution of the multiple correlation coefficient
References
11 Factor Analysis
11.1 Introduction
11.2 Linear Models from Different Disciplines
11.2.1 A linear regression model
11.2.2 A design of experiment model
11.3 A General Linear Model for Factor Analysis
11.3.1 The identification problem
11.3.2 Scaling or units of measurement
11.4 Maximum Likelihood Estimators of the Parameters
11.4.1 Maximum likelihood estimators under identification conditions
11.4.2 Simplifications of |Σ| and tr(Σ-1S)
11.4.3 Special case Ψ=σ2 Ip
11.4.4 Maximum value of the exponent
11.5 General Case
11.5.1 The exponent in the likelihood function
11.6 Tests of Hypotheses
11.6.1 Asymptotic distribution of the likelihood ratio statistic
11.6.2 How to decide on the number r of main factors?
References
12 Classification Problems
12.1 Introduction
12.2 Probabilities of Classification
12.3 Two Populations with Known Distributions
12.3.1 Best procedure
12.4 Linear Discriminant Function
12.5 Classification When the Population Parameters are Unknown
12.5.1 Some asymptotic results
12.5.2 Another method
12.5.3 A new sample from π1 or π2
12.6 Maximum Likelihood Method of Classification
12.7 Classification Involving k Populations
12.7.1 Classification when the populations are real Gaussian
12.7.2 Some distributional aspects
12.7.3 Classification when the population parameters are unknown
12.8 The Maximum Likelihood Method when the Population Covariances Are Equal
12.9 Maximum Likelihood Method and Unequal Covariance Matrices
13 Multivariate Analysis of Variation
13.1 Introduction
13.2 Multivariate Case of One-Way Classification Data Analysis
13.2.1 Some properties
13.3 The Likelihood Ratio Criterion
13.3.1 Arbitrary moments of the likelihood ratio criterion
13.3.2 Structural representation of the likelihood ratiocriterion
13.3.3 Some special cases
13.3.4 Asymptotic distribution of the λ-criterion
13.3.5 MANOVA and testing the equality of population mean values
13.3.6 When Ho is rejected
13.4 MANOVA for Two-Way Classification Data
13.4.1 The model in a two-way classification
13.4.2 Estimation of parameters in a two-way classification
13.5 Multivariate Extension of the Two-Way Layout
13.5.1 Likelihood ratio test for multivariate two-way layout
13.5.2 Asymptotic distribution of λ in the MANOVA two-way layout
13.5.3 Exact densities of w in some special cases
References
14 Profile Analysis and Growth Curves
14.1 Introduction
14.1.1 Profiles
14.2 Parallel Profiles
14.3 Coincident Profiles
14.3.1 Conditional hypothesis for coincident profiles
14.4 Level Profiles
14.4.1 Level profile, two independent Gaussian populations
14.5 Generalizations to k Populations
14.6 Growth Curves or Repeated Measures Designs
14.6.1 Growth curve model
14.6.2 Maximum likelihood estimation
14.6.3 Some special cases
14.6.4 Asymptotic distribution of the likelihood ratio criterion λ
14.6.5 A general structure
References
15 Cluster Analysis and Correspondence Analysis
15.1 Introduction
15.1.1 Clusters
15.1.2 Distance measures
15.2 Different Methods of Clustering
15.2.1 Optimization or partitioning
15.3 Hierarchical Methods of Clustering
15.3.1 Single linkage or nearest neighbor method
15.3.2 Average linking as a modified distance measure
15.3.3 The centroid method
15.3.4 The median method
15.3.5 The residual sum of products method
15.3.6 Other criteria for partitioning or optimization
15.4 Correspondence Analysis
15.4.1 Two-way contingency table
15.4.2 Some general computations
15.5 Various Representations of Pearson's χ2 Statistic
15.5.1 Testing the hypothesis of no association in a two-way contingency table
15.6 Plot of Row and Column Profiles
15.7 Correspondence Analysis in a Multi-way Contingency Table
Correction to: Multivariate Statistical Analysis in the Real and Complex Domains
Tables of Percentage Points
Some Additional Reading Materials
Author Index
Subject Index