Advanced Multivariate Statistics with Matrices（SPINGER出版社）

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Advanced Multivariate Statistics with Matrices（SPINGER出版社）

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关键词：Multivariate multivariat Statistics statistic Matrices Multivariate Advanced Statistics With

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TABLE OF CONTENTS
PREFACE xi
INTRODUCTION xiii
CHAPTER 1. BASIC MATRIX THEORY AND LINEAR ALGEBRA 1
1.1. Matrix algebra 2
1.1.1.   Operations and notations 2
1.1.2.   Determinant, inverse 7
1.1.3.   Rank, trace 8
1.1.4.   Positive definite matrices 12
1.1.5.   Factorizations 12
1.1.6.   Generalized inverse 15
1.1.7.   Problems 19
1.2. Algebra of subspaces 20
1.2.1.   Introduction 20
1.2.2.   Lattices and algebra of subspaces 21
1.2.3.   Disjointness, orthogonality and commutativity 25
1.2.4.   Range spaces 34
1.2.5.   Tensor spaces 40
1.2.6.   Matrix representation of linear operators in vector spaces 45
1.2.7.   Column vector spaces 48
1.2.8.   Eigenvalues and eigenvectors 51
1.2.9.   Eigenstructure of normal matrices 57
1.2.10. Eigenvalue-based factorizations 64
1.2.11. Problems 71
1.3. Partitioned matrices 72
1.3.1.   Basic notation and relations 72
1.3.2.   The commutation matrix 79
1.3.3.   Direct product 80
1.3.4.   vec-operator 88
1.3.5.   Linear equations 91
1.3.6.   Patterned matrices 97
1.3.7.   Vectorization operators 114
1.3.8.   Problems 119
1.4. Matrix derivatives 121
1.4.1.   Introduction 121
1.4.2.   Fr´echet derivative and its matrix representation 122
1.4.3.   Matrix derivatives, properties 126
1.4.4.   Derivatives of patterned matrices 135
1.4.5.   Higher order derivatives 137
1.4.6.   Higher order derivatives and patterned matrices 138
1.4.7.   Differentiating symmetric matrices using an alternative derivative 140viii
1.4.8.   Minimal derivatives 146
1.4.9.   Tables of derivatives 147
1.4.10. Taylor expansion 150
1.4.11. Integration by parts and orthogonal polynomials 152
1.4.12. Jacobians 155
1.4.13. Problems 169
CHAPTER 2. MULTIVARIATE DISTRIBUTIONS 171
2.1. Moments and cumulants 172
2.1.1.   Introduction 172
2.1.2.   Basic statistical functions 174
2.1.3.   Moments and central moments 175
2.1.4.   Cumulants 181
2.1.5.   Moments and cumulants of patterned matrices 182
2.1.6.   Minimal moments and cumulants 184
2.1.7.   Relations between moments and cumulants 187
2.1.8.   Problems 190
2.2. The normal distribution 191
2.2.1.   Introduction and definition 191
2.2.2.   Some properties of the matrix normal distribution 193
2.2.3.   Moments of the matrix normal distribution 200
2.2.4.   Hermite polynomials 210
2.2.5.   Multilinear normal distribution 215
2.2.6.   Problems 219
2.3. Elliptical distributions 221
2.3.1.   Introduction, spherical distributions 221
2.3.2.   Elliptical distributions: definition and basic relations 224
2.3.3.   Moments and cumulants 226
2.3.4.   Density 229
2.3.5.   Elliptical matrix distributions 231
2.3.6.   Moments and cumulants of matrix elliptical distributions 234
2.3.7.   Problems 236
2.4. The Wishart distribution 237
2.4.1.   Definition and basic properties 237
2.4.2.   Characteristic and density functions 244
2.4.3.   Multivariate beta distributions 248
2.4.4.   Partitioned Wishart matrices 253
2.4.5.   Moments of the Wishart distribution 256
2.4.6.   Cumulants of the Wishart matrix 266
2.4.7.   Derivatives of the Wishart density 270
2.4.8.   Centered Wishart distribution 272
2.4.9.   Problems 275
CHAPTER 3. DISTRIBUTION EXPANSIONS 277
3.1. Asymptotic normality 277
3.1.1.   Taylor series of a random vector 277
3.1.2.   Asymptotic normality of functions of random vectors 283ix
3.1.3.   Asymptotic distribution of statistics with functionally dependent
arguments 287
3.1.4.   Asymptotic distribution of the sample correlation matrix 289
3.1.5.   Asymptotics of eigenvalues and eigenvectors of a symmetric
matrix 292
3.1.6.   Asymptotic normality of eigenvalues and eigenvectors of S 298
3.1.7.   Asymptotic normality of eigenvalues and eigenvectors of R 302
3.1.8.   Asymptotic distribution of eigenprojectors of S and R 305
3.1.9.   Asymptotic normality of the MANOVA matrix 309
3.1.10. Asymptotics of Hotelling T 2 -statistic 312
3.1.11. Problems 316
3.2. Multivariate formal density expansions in Rp 317
3.2.1.   Introduction 317
3.2.2.   General relation between densities in Rp 317
3.2.3.   Multivariate Edgeworth type expansions 321
3.2.4.   Wishart expansions 323
3.2.5.   Problems 327
3.3. General multivariate expansions 329
3.3.1.   General relation between two densities 329
3.3.2.   Normal expansions of densities of different dimensions 335
3.3.3.   Wishart expansions for different dimensional densities 341
3.3.4.   Density expansions of R 346
3.3.5.   Problems 353
CHAPTER 4. MULTIVARIATE LINEAR MODELS 355
4.1. The Growth Curve model and extensions 355
4.1.1.   Introduction 355
4.1.2.   Maximum likelihood estimators 358
4.1.3.   The Growth Curve model with a singular dispersion matrix          366
4.1.4.   Extensions of the Growth Curve model 372
4.1.5.   When are the maximum likelihood estimators unique? 388
4.1.6.   Restrictions on B in the Growth Curve model 397
4.1.7.   Problems 409
4.2. Moments and multivariate linear models 410
4.2.1.   Moments of the mean estimator of the Growth Curve model         410
4.2.2.   E[ ˆΣ] and D[ ˆΣ] for the Growth Curve model 417
4.2.3.   Moments of estimators for the MLNM(ABC + B 2 C 2 ) 427
4.2.4.   Moments of estimators for the MLNM(
3
i=1
A i B i C i ) 429
4.2.5.   Problems 448
4.3. Approximations in multivariate linear models 449
4.3.1.   Introduction 449
4.3.2.   Approximation of the density of ˆB in the Growth Curve model    450
4.3.3.   Approximation of the distribution of  ˆΣ in the Growth Curve
model 457
4.3.4.   Approximation of the distribution of the mean parameter
estimators in the MLNM(ABC + B 2 C 2 ) 464x
4.3.5.   Approximation of the distribution of the mean parameter
estimators in the MLNM(
3
i=1
A i B i C i ) 465
4.3.6.   Problems 472
REFERENCES 473
SUBJECT INDEX 485

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