Khuri Linear Model Methodology

1. Auflage 2010
ISBN: 978-1-4200-1044-2
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 562 Seiten

Given the importance of linear models in statistical theory and experimental research, a good understanding of their fundamental principles and theory is essential. Supported by a large number of examples, Linear Model Methodology provides a strong foundation in the theory of linear models and explores the latest developments in data analysis.

After presenting the historical evolution of certain methods and techniques used in linear models, the book reviews vector spaces and linear transformations and discusses the basic concepts and results of matrix algebra that are relevant to the study of linear models. Although mainly focused on classical linear models, the next several chapters also explore recent techniques for solving well-known problems that pertain to the distribution and independence of quadratic forms, the analysis of estimable linear functions and contrasts, and the general treatment of balanced random and mixed-effects models. The author then covers more contemporary topics in linear models, including the adequacy of Satterthwaite’s approximation, unbalanced fixed- and mixed-effects models, heteroscedastic linear models, response surface models with random effects, and linear multiresponse models. The final chapter introduces generalized linear models, which represent an extension of classical linear models.

Linear models provide the groundwork for analysis of variance, regression analysis, response surface methodology, variance components analysis, and more, making it necessary to understand the theory behind linear modeling. Reflecting advances made in the last thirty years, this book offers a rigorous development of the theory underlying linear models.

Zielgruppe

Researchers and graduate students in statistics and applied mathematics.

Autoren/Hrsg.

Khuri, Andre I.

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

Linear Models: Some Historical Perspectives
The Invention of Least Squares
The Gauss–Markov Theorem
Estimability
Maximum Likelihood Estimation
Analysis of Variance (ANOVA)
Quadratic Forms and Craig’s Theorem
The Role of Matrix Algebra
The Geometric Approach

Basic Elements of Linear Algebra
Introduction
Vector Spaces
Vector Subspaces
Bases and Dimensions of Vector Spaces
Linear Transformations

Basic Concepts in Matrix Algebra
Introduction and Notation
Some Particular Types of Matrices
Basic Matrix Operations
Partitioned Matrices
Determinants
The Rank of a Matrix
The Inverse of a Matrix
Eigenvalues and Eigenvectors
Idempotent and Orthogonal Matrices
Quadratic Forms
Decomposition Theorems
Some Matrix Inequalities
Function of Matrices
Matrix Differentiation

The Multivariate Normal Distribution
History of the Normal Distribution
The Univariate Normal Distribution
The Multivariate Normal Distribution
The Moment Generating Function
Conditional Distribution
The Singular Multivariate Normal Distribution
Related Distributions
Examples and Additional Results

Quadratic Forms in Normal Variables
The Moment Generating Function
Distribution of Quadratic Forms
Independence of Quadratic Forms
Independence of Linear and Quadratic Forms
Independence and Chi-Squaredness of Several Quadratic Forms
Computing the Distribution of Quadratic Forms
Appendix

Full-Rank Linear Models
Least-Squares Estimation
Properties of Ordinary Least-Squares Estimation
Generalized Least-Squares Estimation
Least-Squares Estimation under Linear Restrictions on ß
Maximum Likelihood Estimation
Inference Concerning ß
Examples and Applications
Less-Than-Full-Rank Linear Models
Parameter Estimation
Some Distributional Properties
Reparameterized Model
Estimable Linear Functions
Simultaneous Confidence Intervals on Estimable Linear Functions
Simultaneous Confidence Intervals on All Contrasts among the Means with Heterogeneous Group Variances
Further Results Concerning Contrasts and Estimable Linear Functions

Balanced Linear Models
Notation and Definitions
The General Balanced Linear Model
Properties of Balanced Models
Balanced Mixed Models
Complete and Sufficient Statistics
ANOVA Estimation of Variance Components
Confidence Intervals on Continuous Functions of the Variance Components
Confidence Intervals on Ratios of Variance Components

The Adequacy of Satterthwaite’s Approximation
Satterthwaite’s Approximation
Adequacy of Satterthwaite’s Approximation
Measuring the Closeness of Satterthwaite’s Approximation
Examples
Appendix

Unbalanced Fixed-Effects Models
The R-Notation
Two-Way Models without Interaction
Two-Way Models with Interaction
Higher-Order Models
A Numerical Example
The Method of Unweighted Means

Unbalanced Random and Mixed Models
Estimation of Variance Components
Estimation of Estimable Linear Functions
Inference Concerning the Random One-Way Model
Inference Concerning the Random Two-Way Model
Exact Tests for Random Higher-Order Models
Inference Concerning the Mixed Two-Way Model
Inference Concerning the Random Two-Fold Nested Model
Inference Concerning the Mixed Two-Fold Nested Model
Inference Concerning the General Mixed Linear Model
Appendix

Additional Topics in Linear Models
Heteroscedastic Linear Models
The Random One-Way Model with Heterogeneous Error Variances
A Mixed Two-Fold Nested Model with Heteroscedastic Random Effects
Response Surface Models
Response Surface Models with Random Block Effects
Linear Multiresponse Models

Generalized Linear Models
Introduction
The Exponential Family
Estimation of Parameters
Goodness of Fit
Hypothesis Testing
Confidence Intervals
Gamma-Distributed Response
Bibliography
Index
Exercises appear at the end of each chapter, except for Chapter 1.

Über Autor(innen)

André I. Khuri is a Professor Emeritus in the Department of Statistics at the University of Florida in Gainesville.