The coordinate-free approach to linear models

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