Multivariate Statistical Methods: Going Beyond The Linear

Description

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 Preface Contents 1 Some Introductory Algebra 1.1 Permutations 1.2 Tensor Product, vec Operator, and Commutation 1.2.1 Tensor Product 1.2.2 The vec Operator 1.2.3 Commutation Matrices 1.2.4 Commuting T-Products of Vectors 1.3 Symmetrization and Multilinear Algebra 1.3.1 Symmetrization 1.3.2 Multi-Indexing, Elimination, and Duplication 1.3.2.1 q-Symmetrizing Vectors 1.4 Partitions and Diagrams 1.4.1 Generating all Partitions 1.4.2 The Number of All Partitions 1.4.3 Canonical Partitions 1.4.4 Partitions and Permutations 1.4.5 Partitions with Lattice Structure 1.4.6 Indecomposable Partitions 1.4.7 Alternative Ways of Checking Indecomposability 1.4.8 Diagrams 1.4.8.1 Closed Diagrams Without Loops 1.4.8.2 Closed Diagrams with Arms and No Loops 1.5 Appendix 1.5.1 Proof of Lemma 1.1 1.5.2 Proof of Lemma 1.3 1.5.3 Proof of Lemma 1.5 1.5.4 Star Product 1.6 Exercises Section 1.1 Section 1.2 Section 1.3 Section 1.4 1.7 Bibliographic Notes 2 The Tensor Derivative of Vector Functions 2.1 Derivatives of Composite Functions 2.1.1 Faà di Bruno's Formula 2.1.2 Mixed Higher-Order Derivatives 2.2 T-derivative 2.2.1 Differentials and Derivatives 2.2.2 The Operator of T-derivative 2.2.3 Basic Rules 2.2.4 T-derivative of T-products 2.2.4.1 T-derivative of More Tensor Products 2.2.4.2 T-derivative with Higher Orders 2.2.5 Taylor Series Expansion 2.3 Multi-Variable Faà di Bruno's Formula 2.4 Appendix 2.4.1 Proof of Faà di Bruno's Lemma 2.4.2 Proof of Faà di Bruno's T-formula 2.4.3 Moment Commutators 2.5 Exercises 2.6 Bibliographic Notes 3 T-Moments and T-Cumulants 3.1 Multiple Moments 3.2 Tensor Moments 3.3 Cumulants for Multiple Variables 3.3.1 Definition of Cumulants 3.3.2 Definition of T-cumulants 3.3.3 Basic Properties 3.4 Expressions between Moments and Cumulants 3.4.1 Expression for Cumulants via Moments 3.4.1.1 Expressions for scalar variates 3.4.1.2 Expressions for Vector Variates 3.4.2 Expressions for Moments via Cumulants 3.4.2.1 Expressions for Scalar Variates 3.4.2.2 Expressions for Vector Variates 3.4.3 Expression of the Cumulant of Products via Products of Cumulants 3.4.3.1 Expressions for Scalar Variates 3.4.3.2 Expressions for Vector Variates 3.5 Additional Matters 3.5.1 Expressions of Moments and Cumulants via Preceding Moments and Cumulants 3.5.2 Cumulants and Fourier Transform 3.5.3 Conditional Cumulants 3.5.3.1 Conditional Gaussian Cumulants 3.5.4 Cumulants of the Log-likelihood Function 3.5.4.1 Cumulants of the log-likelihood Function, the Vector Parameter Case 3.6 Appendix 3.6.1 Proof of Lemma 3.6 and Theorem 3.7 3.6.2 A Hint for Proof of Lemma 3.8 3.6.3 Proof of Lemma 3.2 3.6.4 Proof of Lemma 3.5 3.7 Exercises 3.8 Bibliographic Notes 4 Gaussian Systems, T-Hermite Polynomials, Moments,and Cumulants 4.1 Hermite Polynomials in One Variable 4.2 Hermite Polynomials of Several Variables 4.3 Moments and Cumulants for Gaussian Systems 4.3.1 Moments of Gaussian Systems and HermitePolynomials 4.3.2 Cumulants for Product of Gaussian Variates and Hermite Polynomials 4.4 Products of Hermite Polynomials, Linearization 4.5 T-Hermite Polynomials 4.6 Moments, Cumulants, and Linearization 4.6.1 Cumulants for T-Hermite Polynomials 4.6.2 Products for T-Hermite Polynomials 4.7 Gram–Charlier Expansion 4.8 Appendix 4.8.1 Proof of Theorem 4.2 4.8.2 Proof of (4.79) 4.9 Exercises 4.10 Bibliographic Notes 5 Multivariate Skew Distributions 5.1 The Multivariate Skew-Normal Distribution 5.1.1 The Inverse Mill's Ratio and the Central Folded Normal Distribution 5.1.2 Skew-Normal Random Variates 5.1.3 Canonical Fundamental Skew-Normal (CFUSN) Distribution 5.1.3.1 Cumulants of CFUSN Distribution 5.2 Elliptically Symmetric and Skew-Spherical Distributions 5.2.1 Elliptically Contoured Distributions 5.2.1.1 Marginal Moments and Cumulants 5.2.2 Multivariate Moments and Cumulants 5.2.3 Canonical Fundamental Skew-Spherical Distribution 5.3 Multivariate Skew-t Distribution 5.3.1 Multivariate t-Distribution 5.3.2 Skew-t Distribution 5.3.3 Higher-Order Cumulants of Skew-t Distributions 5.4 Scale Mixtures of Skew-Normal Distribution 5.5 Multivariate Skew-Normal-Cauchy Distribution 5.5.1 Moments of h(|Z|) 5.6 Multivariate Laplace 5.7 Appendix 5.7.1 Spherically Symmetric Distribution 5.7.2 T-Derivative of an Inner Product 5.7.3 Proof of (5.44) 5.7.4 Proof of Lemma 5.6 5.8 Exercises 5.9 Bibliographic Notes 6 Multivariate Skewness and Kurtosis 6.1 Multivariate Skewness of Random Vectors 6.2 Multivariate Kurtosis of Random Vectors 6.3 Indices Based on Distinct Elements of Cumulant Vectors 6.4 Testing Multivariate Skewness 6.4.1 Estimation of Skewness 6.4.2 Testing Zero Skewness 6.4.2.1 Testing Gaussianity by Skewness 6.4.2.2 Testing Elliptical Symmetry by Skewness 6.5 Testing Multivariate Kurtosis 6.5.1 Estimation of Kurtosis 6.5.2 Testing Zero Kurtosis 6.5.2.1 Testing Gaussianity 6.5.2.2 Testing Alternate Symmetry 6.6 A Simulation Study 6.7 Appendix 6.7.1 Estimated Hermite Polynomials 6.8 Exercises 6.9 Bibliographic Notes A Formulae A.1 Bell Polynomials A.1.1 Incomplete (Partial) Bell Polynomials A.1.2 Bell Polynomials A.1.2.1 Bell Numbers A.2 Commutators A.2.1 Moment Commutators A.2.2 Commutators Connected to T-Hermite Polynomials A.2.2.1 Mixing Commutator A.2.2.2 H-Commutators A.3 Derivatives of Composite Functions A.4 Moments, Cumulants A.4.1 T-Moments, T-Cumulants A.5 Hermite Polynomials A.5.1 Product of Hermite Polynomials A.5.2 T-Hermite Polynomials A.6 Function G A.6.1 Moments, Cumulants for Skew-t Generator R A.6.2 Moments of Beta Powers A.7 Complementary Error Function A.8 Derivatives of i-Mill's Ratio Notations Notations Solutions Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 References Index