A Non-Least Squares Approach to Linear Models

Description

This book provides a unifying framework which can be used to apply many types of linear models used in applications to the analysis of data generated by scientific experiments. While other texts on linear models use least squares as the basis for developing linear estimation theory, this book uses a non-least squares approach for developing the same theory. The benefits of the approach used here are that it allows for the initial consideration of more complex models and it simplifies the proofs of many of the main results given, thus making the book easier to read. In addition, through the use of a concept called “correspondence”, the text provides the first formal systematic approach for exploring relationships between different representations for models used to describe the same expectation space assumed for a given data set. Table of Contents Preface Chapter 1 1.1 Random Vectors and Matrices 1.2 Expectation Vectors and Matrices 1.3 Covariance Matrices 1.4 The Multivariate Normal Distribution 1.5 The Chi-Squared Distribution 1.6 Quadratic Forms in Normal Random Vectors 1.7 Other Distributions of Interest 1.8 Problems for Chapter 1 Chapter 2 2.1 Introduction 2.2 The Basic Linear Model 2.3 Preliminary Notions 2.4 Identifiability and Estimability of Parametric Vectors 2.5 Best Linear Unbiased Estimation when cov(Y) = o2V 2.6 The Gauss-Markov Property 2.7 Least Squares, Gauss Markov, Residuals and Maximum Likelihood Estimation when cov(Y) = o2In 2.8 Generalized Least Squares, Gauss-markov, Residuals and Maximum Likelihood Estimation when cov(Y) = o2V 2.9 Models with Nonhomogeneous Constraints 2.10 Sampling Distributions of Estimators 2.11 Problems for Chapter 2 Chapter 3 3.1 Introduction 3.2 Correspondence 3.3 Correspondence and Full Rank Parameterizations 3.4 Problems for Chapter 3 Chapter 4 4.1 Introduction 4.2 Some Preliminaries 4.3 An Intuitive Approach to Testing 4.4 The Likelihood Ratio Test 4.5 Formulating Linear Hypotheses 4.6 ANOVA Tables for Testing Linear Hypotheses 4.7 An Alternative Test Statistic for Testing Parametric Vectors 4.8 Tests of Hypotheses in Models with Nonhomogeneous Constraints 4.9 Testing Parametric Vectors in Models with Nonhomogeneous constraints 4.10 Problems for Chapter 4 Chapter 5 5.1 Introduction and Preliminaries 5.2 A Single Covariance Matrix 5.3 Some Further Results on Estimation 5.4 More on Estimation 5.5 Problems for Chapter 5 Appendices Appendix A1 Appendix A2 Appendix A3 Appendix A4 Appendix A5 Appendix A6 Appendix A7 Appendix A8 Appendix A9 Appendix A10 Appendix A11 Appendix A12 Appendix A13 Appendix A14 Appendix A15 References Index