Multivariate statistical methods : going beyond the linear : vector-moments and vector-cumulants, nonlinear statistics of normal multivariates, testing skewness and kurtosis

This book presents a general method for deriving higher-order statistics of multivariate distributions with simple algorithms that allow for actual calculations. Multivariate nonlinear statistical models require the study of higher-order moments and cumulants. The main tool used for the definitions is the tensor derivative, leading to several useful expressions concerning Hermite polynomials, moments, cumulants, skewness, and kurtosis. A general test of multivariate skewness and kurtosis is obtained from this treatment. Exercises are provided for each chapter to help the readers understand the methods. Lastly, the book includes a comprehensive list of references, equipping readers to explore further on their own. Foreword 7 Preface 8 Contents 10 1 Some Introductory Algebra 14 1.1 Permutations 14 1.2 Tensor Product, vec Operator, and Commutation 18 1.2.1 Tensor Product 18 1.2.2 The vec Operator 19 1.2.3 Commutation Matrices 20 1.2.4 Commuting T-Products of Vectors 23 1.3 Symmetrization and Multilinear Algebra 26 1.3.1 Symmetrization 26 1.3.2 Multi-Indexing, Elimination, and Duplication 29 1.3.2.1 q-Symmetrizing Vectors 33 1.4 Partitions and Diagrams 39 1.4.1 Generating all Partitions 42 1.4.2 The Number of All Partitions 43 1.4.3 Canonical Partitions 47 1.4.4 Partitions and Permutations 48 1.4.5 Partitions with Lattice Structure 50 1.4.6 Indecomposable Partitions 51 1.4.7 Alternative Ways of Checking Indecomposability 53 1.4.8 Diagrams 56 1.4.8.1 Closed Diagrams Without Loops 57 1.4.8.2 Closed Diagrams with Arms and No Loops 60 1.5 Appendix 62 1.5.1 Proof of Lemma 1.1 62 1.5.2 Proof of Lemma 1.3 63 1.5.3 Proof of Lemma 1.5 63 1.5.4 Star Product 64 1.6 Exercises 65 Section 1.1 65 Section 1.2 65 Section 1.3 70 Section 1.4 71 1.7 Bibliographic Notes 72 2 The Tensor Derivative of Vector Functions 73 2.1 Derivatives of Composite Functions 73 2.1.1 Faà di Bruno's Formula 75 2.1.2 Mixed Higher-Order Derivatives 79 2.2 T-derivative 82 2.2.1 Differentials and Derivatives 82 2.2.2 The Operator of T-derivative 84 2.2.3 Basic Rules 86 2.2.4 T-derivative of T-products 89 2.2.4.1 T-derivative of More Tensor Products 92 2.2.4.2 T-derivative with Higher Orders 95 2.2.5 Taylor Series Expansion 100 2.3 Multi-Variable Faà di Bruno's Formula 102 2.4 Appendix 110 2.4.1 Proof of Faà di Bruno's Lemma 110 2.4.2 Proof of Faà di Bruno's T-formula 111 2.4.3 Moment Commutators 112 2.5 Exercises 113 2.6 Bibliographic Notes 118 3 T-Moments and T-Cumulants 119 3.1 Multiple Moments 119 3.2 Tensor Moments 122 3.3 Cumulants for Multiple Variables 130 3.3.1 Definition of Cumulants 130 3.3.2 Definition of T-cumulants 132 3.3.3 Basic Properties 136 3.4 Expressions between Moments and Cumulants 138 3.4.1 Expression for Cumulants via Moments 138 3.4.1.1 Expressions for scalar variates 138 3.4.1.2 Expressions for Vector Variates 144 3.4.2 Expressions for Moments via Cumulants 150 3.4.2.1 Expressions for Scalar Variates 150 3.4.2.2 Expressions for Vector Variates 153 3.4.3 Expression of the Cumulant of Products via Products of Cumulants 155 3.4.3.1 Expressions for Scalar Variates 155 3.4.3.2 Expressions for Vector Variates 158 3.5 Additional Matters 161 3.5.1 Expressions of Moments and Cumulants via Preceding Moments and Cumulants 161 3.5.2 Cumulants and Fourier Transform 163 3.5.3 Conditional Cumulants 166 3.5.3.1 Conditional Gaussian Cumulants 172 3.5.4 Cumulants of the Log-likelihood Function 173 3.5.4.1 Cumulants of the log-likelihood Function, the Vector Parameter Case 178 3.6 Appendix 179 3.6.1 Proof of Lemma 3.6 and Theorem 3.7 179 3.6.2 A Hint for Proof of Lemma 3.8 182 3.6.3 Proof of Lemma 3.2 184 3.6.4 Proof of Lemma 3.5 185 3.7 Exercises 186 3.8 Bibliographic Notes 192 4 Gaussian Systems, T-Hermite Polynomials, Moments,and Cumulants 194 4.1 Hermite Polynomials in One Variable 194 4.2 Hermite Polynomials of Several Variables 196 4.3 Moments and Cumulants for Gaussian Systems 201 4.3.1 Moments of Gaussian Systems and HermitePolynomials 201 4.3.2 Cumulants for Product of Gaussian Variates and Hermite Polynomials 204 4.4 Products of Hermite Polynomials, Linearization 208 4.5 T-Hermite Polynomials 213 4.6 Moments, Cumulants, and Linearization 227 4.6.1 Cumulants for T-Hermite Polynomials 231 4.6.2 Products for T-Hermite Polynomials 234 4.7 Gram–Charlier Expansion 237 4.8 Appendix 243 4.8.1 Proof of Theorem 4.2 243 4.8.2 Proof of (4.79) 244 4.9 Exercises 245 4.10 Bibliographic Notes 249 5 Multivariate Skew Distributions 251 5.1 The Multivariate Skew-Normal Distribution 251 5.1.1 The Inverse Mill's Ratio and the Central Folded Normal Distribution 252 5.1.2 Skew-Normal Random Variates 254 5.1.3 Canonical Fundamental Skew-Normal (CFUSN) Distribution 257 5.1.3.1 Cumulants of CFUSN Distribution 258 5.2 Elliptically Symmetric and Skew-Spherical Distributions 262 5.2.1 Elliptically Contoured Distributions 263 5.2.1.1 Marginal Moments and Cumulants 264 5.2.2 Multivariate Moments and Cumulants 271 5.2.3 Canonical Fundamental Skew-Spherical Distribution 276 5.3 Multivariate Skew-t Distribution 285 5.3.1 Multivariate t-Distribution 285 5.3.2 Skew-t Distribution 287 5.3.3 Higher-Order Cumulants of Skew-t Distributions 288 5.4 Scale Mixtures of Skew-Normal Distribution 295 5.5 Multivariate Skew-Normal-Cauchy Distribution 297 5.5.1 Moments of h(|Z|) 302 5.6 Multivariate Laplace 304 5.7 Appendix 307 5.7.1 Spherically Symmetric Distribution 307 5.7.2 T-Derivative of an Inner Product 313 5.7.3 Proof of (5.44) 314 5.7.4 Proof of Lemma 5.6 316 5.8 Exercises 317 5.9 Bibliographic Notes 320 6 Multivariate Skewness and Kurtosis 322 6.1 Multivariate Skewness of Random Vectors 322 6.2 Multivariate Kurtosis of Random Vectors 330 6.3 Indices Based on Distinct Elements of Cumulant Vectors 336 6.4 Testing Multivariate Skewness 337 6.4.1 Estimation of Skewness 338 6.4.2 Testing Zero Skewness 342 6.4.2.1 Testing Gaussianity by Skewness 343 6.4.2.2 Testing Elliptical Symmetry by Skewness 345 6.5 Testing Multivariate Kurtosis 346 6.5.1 Estimation of Kurtosis 347 6.5.2 Testing Zero Kurtosis 350 6.5.2.1 Testing Gaussianity 350 6.5.2.2 Testing Alternate Symmetry 352 6.6 A Simulation Study 352 6.7 Appendix 355 6.7.1 Estimated Hermite Polynomials 355 6.8 Exercises 356 6.9 Bibliographic Notes 357 A Formulae 359 A.1 Bell Polynomials 359 A.1.1 Incomplete (Partial) Bell Polynomials 359 A.1.2 Bell Polynomials 360 A.1.2.1 Bell Numbers 361 A.2 Commutators 361 A.2.1 Moment Commutators 361 A.2.2 Commutators Connected to T-Hermite Polynomials 364 A.2.2.1 Mixing Commutator 364 A.2.2.2 H-Commutators 365 A.3 Derivatives of Composite Functions 367 A.4 Moments, Cumulants 369 A.4.1 T-Moments, T-Cumulants 369 A.5 Hermite Polynomials 371 A.5.1 Product of Hermite Polynomials 371 A.5.2 T-Hermite Polynomials 373 A.6 Function G 376 A.6.1 Moments, Cumulants for Skew-t Generator R 378 A.6.2 Moments of Beta Powers 382 A.7 Complementary Error Function 384 A.8 Derivatives of i-Mill's Ratio 386 Notations 388 Notations 388 Solutions 392 Chapter 1 392 Chapter 2 395 Chapter 3 399 Chapter 4 404 Chapter 5 409 Chapter 6 413 References 415 Index 423