Multivariate, Multilinear and Mixed Linear Models (Contributions to Statistics)

This book presents the latest findings on statistical inference in multivariate, multilinear and mixed linear models, providing a holistic presentation of the subject. It contains pioneering and carefully selected review contributions by experts in the field and guides the reader through topics related to estimation and testing of multivariate and mixed linear model parameters. Starting with the theory of multivariate distributions, covering identification and testing of covariance structures and means under various multivariate models, it goes on to discuss estimation in mixed linear models and their transformations. The results presented originate from the work of the research group Multivariate and Mixed Linear Models and their meetings held at the Mathematical Research and Conference Center in Będlewo, Poland, over the last 10 years. Featuring an extensive bibliography of related publications, the book is intended for PhD students and researchers in modern statistical science who are interested in multivariate and mixed linear models. Preface Contents Contributors 1 Holonomic Gradient Method for Multivariate Distribution Theory 1.1 Introduction 1.2 Origin of HGM 1.3 Definition and Properties of Holonomic Functions 1.4 HGM for the Hypergeometric Function of a Matrix Argument 1.5 HGM for Evaluation of Probability of Some Regions Under Multivariate Normality 1.6 Application to Problems in Wireless Communication 1.7 Some Other Applications and Developments of HGM References 2 From Normality to Skewed Multivariate Distributions: A Personal View 2.1 Introduction 2.2 Elliptical Distributions 2.3 Multivariate Edgeworth Type Expansions 2.4 Skew Elliptical Distributions 2.5 Asymmetric Laplace Distribution 2.6 Copulas 2.7 Conclusion References 3 Multivariate Moments in Multivariate Analysis 3.1 Introduction 3.2 Mathematical Background 3.2.1 The vec-Operator and the Kronecker Product 3.2.2 Matrix Derivatives 3.2.3 Transforms 3.2.4 Definition of Moments 3.2.5 Moments of the Spectral Distribution Based on Freeness 3.2.6 Common Multivariate Distributions 3.3 Moment Expressions 3.3.1 Matrix Normal Distribution 3.3.2 Moments for Rotationally Invariant Symmetric Matrices 3.3.3 Moments for the Wishart Distribution 3.3.4 Spectral Moments for Wishart Matrices 3.3.5 Inverse Wishart Moments 3.3.6 Moments for Multivariate β-type Distributions References 4 Regularized Estimation of Covariance Structure Through Quadratic Loss Function 4.1 Introduction 4.2 Main Results 4.3 Numerical Experiments 4.3.1 Simulation Studies 4.3.2 Real Data Analysis 4.4 Conclusions References 5 Separable Covariance Structure Identification for Doubly Multivariate Data 5.1 Introduction 5.2 Models and Discrepancy Functions 5.3 Minimum of Discrepancy Functions 5.3.1 Approximation via the Frobenius Norm 5.3.2 Approximation via the Entropy Loss Function 5.4 Simulation Studies 5.5 Real Data Example 5.6 Conclusions References 6 Estimation and Testing of the Covariance Structure of Doubly Multivariate Data 6.1 Introduction 6.2 Model for Doubly Multivariate Data 6.3 Covariance Structures and Maximum Likelihood Estimators 6.3.1 Covariance Structures 6.3.2 Maximum Likelihood Estimators 6.4 Hypotheses and Tests 6.4.1 Likelihood Ratio Test 6.4.2 Rao Score Test 6.5 Comparison of Tests 6.5.1 Distribution of Test Statistics Versus True Values of Parameters 6.5.2 Simulation Studies 6.6 Real Data Applications 6.7 Conclusions References 7 Testing Equality of Mean Vectors with Block-Circular and Block Compound-Symmetric Covariance Matrices 7.1 Introduction 7.2 The Likelihood Ratio Test and Its Statistic 7.3 The Distribution of the Likelihood Ratio Statistic 7.3.1 The Exact Distribution of Λ for Odd q or Even r 7.3.2 Near-Exact Distributions for Λ for the Case When q Is Even and r Is Odd 7.4 Numerical Studies for the Near-Exact Distributions 7.5 The Case of Block Compound-Symmetric Matrices 7.5.1 The Exact Distribution of Λ in (7.36) for Odd q or Even r 7.5.2 Near-Exact Distributions for Λ in (7.36) for Odd r and Even q 7.6 Conclusions References 8 Estimation and Testing Hypotheses in Two-Level and Three-Level Multivariate Data with Block Compound Symmetric Covariance Structure 8.1 Introduction 8.2 Two-Level Multivariate Data 8.2.1 Block Compound Symmetric Covariance Structure 8.2.2 Estimation in Model with Unstructured Mean Vector 8.2.3 Estimation in Model with Structured Mean Vector 8.2.4 Comparison of BUE in Two Models 8.2.5 A Real Data Example 8.2.6 Testing Hypotheses in BCS Models 8.3 Three-Level Multivariate Data 8.3.1 Double Block Compound Symmetry Covariance Structure 8.3.2 Estimation in Model with Unstructured Mean Vector 8.3.3 Estimation in Model with Structured Mean Vector 8.3.4 Comparison of BUE in Two Models 8.3.5 A Real Data Example 8.4 Conclusions References 9 Testing of Multivariate Repeated Measures Data with Block Exchangeable Covariance Structure 9.1 Introduction 9.2 Preliminaries 9.3 One-Sample Test 9.3.1 Orthogonal Decomposition Solution 9.3.2 Canonical Transformation Solution 9.3.3 Exchangeable Mean Structure 9.4 Paired Samples Test 9.5 Two-Sample Test 9.5.1 Exchangeable Means Structure 9.6 Simulation Study 9.6.1 Comparisons for Fixed Type of Alternative 9.6.2 Comparisons for Individual Tests 9.7 Concluding Remarks References 10 On a Simplified Approach to Estimation in Experiments with Orthogonal Block Structure 10.1 Introduction 10.2 Three Basic Experiments 10.3 Various Representations of the BLUE 10.4 Estimation of Variance Components 10.5 Some Final Comments and Conclusions References 11 A Review of the Linear Sufficiency and Linear Prediction Sufficiency in the Linear Model with New Observations 11.1 Preliminaries and Introduction to the Models 11.2 BLUEs and BLUPs 11.3 Conditions for Linear Sufficiency 11.4 The Transformed Model 11.5 Relative Linear Sufficiency 11.6 The ``Vice Versa'' Problem 11.7 Partitioned Linear Model 11.8 Mutual Relations of Linear Sufficiencies 11.9 Mixed Linear Model 11.10 Linear Sufficiency in the Misspecified Linear Model 11.11 Conclusions References 12 Linear Mixed-Effects Model Using Penalized Spline Based on Data Transformation Methods 12.1 Introduction 12.2 Data Transformation Methods 12.2.1 KNN Imputation Technique 12.2.2 Kaplan-Meier Weights 12.3 Penalized Spline as LMEM 12.4 Properties of Penalized Estimators Based on KMW and kNN 12.5 Evaluation Criteria 12.6 Simulation Study 12.7 Real Data Examples 12.8 Discussion References Appendix MMLM Meetings—List of Publications Index