The Coordinate-Free Approach to Linear Models (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 19)

This Book Is About The Coordinate-free, Or Geometric, Approach To The Theory Of Linear Models; More Precisely, Model I Anova And Linear Regression Models With Non-random Predictors In A Finite-dimensional Setting. This Approach Is More Insightful, More Elegant, More Direct, And Simpler Than The More Common Matrix Approach To Linear Regression, Analysis Of Variance, And Analysis Of Covariance Models In Statistics. The Book Discusses The Intuition Behind And Optimal Properties Of Various Methods Of Estimating And Testing Hypotheses About Unknown Parameters In The Models. Topics Covered Range From Linear Algebra, Such As Inner Product Spaces, Orthogonal Projections, Book Orthogonal Spaces, Tjur Experimental Designs, Basic Distribution Theory, The Geometric Version Of The Gauss-markov Theorem, Optimal And Non-optimal Properties Of Gauss-markov, Bayes, And Shrinkage Estimators Under Assumption Of Normality, The Optimal Properties Of F-test, And The Analysis Of Covariance And Missing Observations. Michael J. Wichura. Title From Publisher's Bibliographic System (viewed On 01 Jun 2016). Mode Of Access: World Wide Web. COVER......Page 1 HALF-TITLE......Page 3 SERIES-TITLE......Page 4 TITLE......Page 5 COPYRIGHT......Page 6 DEDICATION......Page 7 CONTENTS......Page 9 PREFACE......Page 13 1. Orientation......Page 17 2. An illustrative example......Page 18 3. Notational conventions......Page 20 Exercise......Page 21 1. Orthogonal projections......Page 22 2A. Characterization of orthogonal projections......Page 28 2B. Differences of orthogonal projections......Page 29 2C. Sums of orthogonal projections......Page 32 2D. Products of orthogonal projections......Page 33 2E. An algebraic form of Cochran’s theorem......Page 35 3. Tjur’s theorem......Page 37 4. Self-adjoint transformations and the spectral theorem......Page 48 5. Representation of linear and bilinear functionals......Page 52 6. Problem set: Cleveland’s identity......Page 56 7. Appendix: Rudiments......Page 57 7A. Vector spaces......Page 58 7D. Linear transformations......Page 59 1. Random vectors taking values in an inner product space......Page 61 2. Expected values......Page 62 3. Covariance operators......Page 63 4. Dispersion operators......Page 65 5. Weak sphericity......Page 67 7. Normality......Page 68 8. The main result......Page 70 9. Problem set: Distribution of quadratic forms......Page 73 1. Linear functionals of Mu......Page 76 2. Estimation of linear functionals of Mu......Page 78 3. Estimation of Mu itself......Page 83 4. Estimation of Sigma2......Page 86 5. Using the wrong inner product......Page 88 6. Invariance of GMEs under linear transformations......Page 90 7. Some additional optimality properties of GMEs......Page 91 8. Estimable parametric functionals......Page 94 9. Problem set: Quantifying the Gauss-Markov theorem......Page 101 1. Maximum likelihood estimation......Page 105 2. Minimum variance unbiased estimation......Page 106 3. Minimaxity of PMY......Page 108 4. James-Stein estimation......Page 113 CHAPTER 6 NORMAL THEORY: TESTING......Page 126 1. The likelihood ratio test......Page 127 2. The F-test......Page 128 3. Monotonicity of the power of the F-test......Page 133 4. An optimal property of the F-test......Page 137 5. Confidence intervals for linear functionals of Mu......Page 143 6. Problem set: Wald’s theorem......Page 152 1. Preliminaries on nonorthogonal projections......Page 157 1B. The adjoint of a projection......Page 158 1D. A formula for PJ;I J;when J is given by a basis......Page 159 1E. A formula for P'J;I when J is given by a basis......Page 161 2. The analysis of covariance framework......Page 162 3. Gauss-Markov estimation......Page 163 4. Variances and covariances of GMEs......Page 166 5. Estimation of Sigma2......Page 168 6. Scheffe intervals for functionals of MuM......Page 169 8. Problem set: The Latin square design......Page 175 1. Framework and Gauss-Markov estimation......Page 180 2A. The consistency equation method......Page 185 2B. The quadratic function method......Page 188 2C. The analysis of covariance method......Page 189 3. Estimation of Sigma2......Page 192 4. F-testing......Page 193 5. Estimation of linear functionals......Page 197 6. Problem set: Extra observations......Page 200 REFERENCES......Page 204 INDEX......Page 206 COVER 1 HALF-TITLE 3 SERIES-TITLE 4 TITLE 5 COPYRIGHT 6 DEDICATION 7 CONTENTS 9 PREFACE 13 CHAPTER 1 INTRODUCTION 17 1. Orientation 17 2. An illustrative example 18 3. Notational conventions 20 Exercise 21 CHAPTER 2 TOPICS IN LINEAR ALGEBRA 22 1. Orthogonal projections 22 2. Properties of orthogonal projections 28 2A. Characterization of orthogonal projections 28 2B. Differences of orthogonal projections 29 2C. Sums of orthogonal projections 32 2D. Products of orthogonal projections 33 2E. An algebraic form of Cochran’s theorem 35 3. Tjur’s theorem 37 4. Self-adjoint transformations and the spectral theorem 48 5. Representation of linear and bilinear functionals 52 6. Problem set: Cleveland’s identity 56 7. Appendix: Rudiments 57 7A. Vector spaces 58 7B. Subspaces 59 7C. Linear functionals 59 7D. Linear transformations 59 CHAPTER 3 RANDOM VECTORS 61 1. Random vectors taking values in an inner product space 61 2. Expected values 62 3. Covariance operators 63 4. Dispersion operators 65 5. Weak sphericity 67 6. Getting to weak sphericity 68 7. Normality 68 8. The main result 70 9. Problem set: Distribution of quadratic forms 73 CHAPTER 4 GAUSS-MARKOV ESTIMATION 76 1. Linear functionals of Mu 76 2. Estimation of linear functionals of Mu 78 3. Estimation of Mu itself 83 4. Estimation of Sigma2 86 5. Using the wrong inner product 88 6. Invariance of GMEs under linear transformations 90 7. Some additional optimality properties of GMEs 91 8. Estimable parametric functionals 94 9. Problem set: Quantifying the Gauss-Markov theorem 101 CHAPTER 5 NORMAL THEORY: ESTIMATION 105 1. Maximum likelihood estimation 105 2. Minimum variance unbiased estimation 106 3. Minimaxity of PMY 108 4. James-Stein estimation 113 CHAPTER 6 NORMAL THEORY: TESTING 126 1. The likelihood ratio test 127 2. The F-test 128 3. Monotonicity of the power of the F-test 133 4. An optimal property of the F-test 137 5. Confidence intervals for linear functionals of Mu 143 6. Problem set: Wald’s theorem 152 CHAPTER 7 ANALYSIS OF COVARIANCE 157 1. Preliminaries on nonorthogonal projections 157 1A. Characterization of projections 158 1B. The adjoint of a projection 158 1C. An isomorphism between J and I⊥ 159 1E. A formula for P'J;I when J is given by a basis 161 1D. A formula for PJ;I J;when J is given by a basis 159 2. The analysis of covariance framework 162 3. Gauss-Markov estimation 163 4. Variances and covariances of GMEs 166 5. Estimation of Sigma2 168 6. Scheffe intervals for functionals of MuM 169 8. Problem set: The Latin square design 175 CHAPTER 8 MISSING OBSERVATIONS 180 1. Framework and Gauss-Markov estimation 180 2. Obtaining Mu 185 2A. The consistency equation method 185 2B. The quadratic function method 188 2C. The analysis of covariance method 189 3. Estimation of Sigma2 192 4. F-testing 193 5. Estimation of linear functionals 197 6. Problem set: Extra observations 200 REFERENCES 204 INDEX 206 This book is about the coordinate-free, or geometric, approach to the theory of linear models; more precisely, Model I ANOVA and linear regression models with nonrandom predictors in a finite-dimensional setting. This approach is more insightful, more elegant, more direct, and simpler than the more common matrix approach to linear regression, analysis of variance, and analysis of covariance models in statistics. The book discusses the intuition behind and optimal properties of various methods of estimating and testing hypotheses about unknown parameters in the models. Topics covered include inner product spaces, orthogonal projections, book orthogonal spaces, Tjur experimental designs, basic distribution theory, the geometric version of the Gauss-Markov theorem, optimal and nonoptimal properties of Gauss-Markov, Bayes, and shrinkage estimators under the assumption of normality, the optimal properties of F-tests, and the analysis of covariance and missing observations.