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

✍ Scribed by Katarzyna Filipiak (editor), Augustyn Markiewicz (editor), Dietrich von Rosen (editor)

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 357
• Category: Library

✦ Synopsis

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 statisticalscience who are interested in multivariate and mixed linear models.

✦ Table of Contents

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