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CONTENTS INTRODUCTION 1. Introduction . . . . . . . . . . . . . . . . . . . . . 1 1. Orientation . . . . . . . . . . . . . . . . . . . . . 1 2. An illustrative example . . . . . . . . . . . . . . . . 2 3. Notational conventions . . . . . . . . . . . . . . . . 4 2. Topics in Linear Algebra . . . . . . . . . . . . . . . . 6 1. Orthogonal projections . . . . . . . . . . . . . . . . 6 2. Properties of orthogonal projections . . . . . . . . . . . . 12 2A. Characterization of orthogonal projections . . . . . . . 12 2B. Di(r)erences of orthogonal projections . . . . . . . . . 13 2C. Sums of orthogonal projections . . . . . . . . . . . 16 2D. Products of orthogonal projections . . . . . . . . . . 17 2E. An algebraic form of Cochran's theorem . . . . . . . . 19 3. Tjur's theorem . . . . . . . . . . . . . . . . . . . 21 4. Self-adjoint transformations and the spectral theorem . . . . . 32 5. Representation of linear and bilinear functionals . . . . . . . 36 6. Problem set: Cleveland's identity . . . . . . . . . . . . . 40 7. Appendix: Rudiments . . . . . . . . . . . . . . . . . 41 7A. Vector spaces . . . . . . . . . . . . . . . . . . 42 7B. Subspaces . . . . . . . . . . . . . . . . . . . 43 7C. Linear functionals . . . . . . . . . . . . . . . . 43 7D. Linear transformations . . . . . . . . . . . . . . 43 3. Random Vectors . . . . . . . . . . . . . . . . . . . 45 1. Random vectors taking values in an inner product space . . . . 45 2. Expected values . . . . . . . . . . . . . . . . . . . 46 3. Covariance operators . . . . . . . . . . . . . . . . . 47 4. Dispersion operators . . . . . . . . . . . . . . . . . 49 5. Weak sphericity . . . . . . . . . . . . . . . . . . . 51 6. Getting to weak sphericity . . . . . . . . . . . . . . . 52 7. Normality . . . . . . . . . . . . . . . . . . . . . 52 8. The main result . . . . . . . . . . . . . . . . . . . 54 9. Problem set: Distribution of quadratic forms . . . . . . . . 57 4. Gauss-Markov Estimation . . . . . . . . . . . . . . . 60 1. Linear functionals of µ . . . . . . . . . . . . . . . . . 60 2. Estimation of linear functionals of µ . . . . . . . . . . . . 62 3. Estimation of µ itself . . . . . . . . . . . . . . . . . 67 4. Estimation of 3/42 . . . . . . . . . . . . . . . . . . . 70 5. Using the wrong inner product . . . . . . . . . . . . . . 72 6. Invariance of GMEs under linear transformations . . . . . . . 74 7. Some additional optimality properties of GMEs . . . . . . . 75 8. Estimable parametric functionals . . . . . . . . . . . . . 78 9. Problem set: quantifying the Gauss-Markov theorem . . . . . . 85 5. Normal Theory: Estimation . . . . . . . . . . . . . . . 89 1. Maximum likelihood estimation . . . . . . . . . . . . . 89 2. Minimum variance unbiased estimation . . . . . . . . . . . 90 3. Minimaxity of PMY . . . . . . . . . . . . . . . . . 92 4. James-Stein estimation . . . . . . . . . . . . . . . . 97 5. Problem set: Admissible minimax estimation of µ . . . . . . . 104 6. Normal Theory: Testing . . . . . . . . . . . . . . . . 110 1. The likelihood ratio test . . . . . . . . . . . . . . . . 111 2. The F-test . . . . . . . . . . . . . . . . . . . . . 112 3. Monotonicity of the power of the F-test . . . . . . . . . . 117 4. An optimal property of the F-test . . . . . . . . . . . . 121 5. Confidence intervals for linear functionals of µ . . . . . . . . 127 6. Problem set: Wald's theorem . . . . . . . . . . . . . . 136 7. Analysis of Covariance . . . . . . . . . . . . . . . . . 141 1. Preliminaries on non-orthogonal projections . . . . . . . . . 141 1A. Characterization of projections . . . . . . . . . . . 142 1B. The adjoint of a projection . . . . . . . . . . . . . 142 1C. An isomorphism between J and I? . . . . . . . . . . 143 1D. A formula for PJ;I when J is given by a basis . . . . . . 143 1E. A formula for P0J ;I when J is given by a basis . . . . . . 145 2. The analysis of covariance framework . . . . . . . . . . . 146 3. Gauss-Markov estimation . . . . . . . . . . . . . . . . 147 4. Variances and covariances of GMEs . . . . . . . . . . . . 150 5. Estimation of 3/42 . . . . . . . . . . . . . . . . . . . 152 6. Sche(r)´e intervals for functionals of µM . . . . . . . . . . . 153 7. F-Testing . . . . . . . . . . . . . . . . . . . . . 155 8. Problem set: The Latin square design . . . . . . . . . . . 159 8. Missing Observations . . . . . . . . . . . . . . . . . 164 1. Framework; Gauss-Markov estimation . . . . . . . . . . . 164 2. Obtaining ¿µ . . . . . . . . . . . . . . . . . . . . 169 2A. The consistency equation method . . . . . . . . . . 169 2B. The quadratic function method . . . . . . . . . . . 172 2C. The analysis of covariance method . . . . . . . . . . 173 3. Estimation of 3/42 . . . . . . . . . . . . . . . . . . . 176 4. F-testing . . . . . . . . . . . . . . . . . . . . . 177 5. Estimation of linear functionals . . . . . . . . . . . . . 181 6. Problem set: Extra observations . . . . . . . . . . . . . 184 References . . . . . . . . . . . . . . . . . . . . . . 188 Index . . . . . . . . . . . . . . . . . . . . . . . . 190
Library of Congress Subject Headings for this publication:
Linear models (Statistics).
Analysis of variance.
Regression analysis.
Analysis of covariance.