Contents vii §8.4. Smoothing Kernel Estimator 106 Exercises 112 Chapter 9. Local Polynomial Approximation of the Regression Function 115 §9.1. Preliminary Results and Definition 115 §9.2. Polynomial Approximation and Regularity of Design 119 §9.3. Asymptotically Minimax Lower Bound 122 §9.4. Proofs of Auxiliary Results 126 Exercises 130 Chapter 10. Estimation of Regression in Global Norms 131 §10.1. Regressogram 131 §10.2. Integral L2-Norm Risk for the Regressogram 133 §10.3. Estimation in the Sup-Norm 136 §10.4. Projection on Span-Space and Discrete MISE 138 §10.5. Orthogonal Series Regression Estimator 141 Exercises 148 Chapter 11. Estimation by Splines 151 §11.1. In Search of Smooth Approximation 151 §11.2. Standard B-splines 152 §11.3. Shifted B-splines and Power Splines 155 §11.4. Estimation of Regression by Splines 158 §11.5. Proofs of Technical Lemmas 161 Exercises 166 Chapter 12. Asymptotic Optimality in Global Norms 167 §12.1. Lower Bound in the Sup-Norm 167 §12.2. Bound in L2-Norm. Assouad’s Lemma 171 §12.3. General Lower Bound 174 §12.4. Examples and Extensions 177 Exercises 182 Part 3. Estimation in Nonparametric Models Chapter 13. Estimation of Functionals 185 §13.1. Linear Integral Functionals 185 §13.2. Non-Linear Functionals 188
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