Interpolation and Extrapolation Optimal Designs 2 (eBook)

G Celant, Associate Professor of Statistics, University of Padova, Italy. Michel Broniatowski, Professor of Statistics, Universitei Pierre et Marie Curie, Paris, France.

Preface ix

Introduction xi

Chapter 1 Approximation of Continuous Functions in Normed Spaces 1

1.1 Introduction 1

1.2 Some remarks on the meaning of the word "simple" Choosing the approximation 2

1.3 The choice of the norm in order to specify the error 8

1.4 Optimality with respect to a norm 12

1.5 Characterizing the optimal solution.18

Chapter 2 Chebyshev Systems 27

2.1 Introduction 27

2.2 From the classical polynomials to the generalized ones 28

2.3 Properties of a Chebyshev system 34

Chapter 3 Uniform Approximations in a Normed Space 45

3.1 Introduction 45

3.2 Characterization of the best uniform approximation in a normed space 46

Chapter 4 Calculation of the Best Uniform Approximation in a Chebyshev System 69

4.1 Some preliminary results 69

4.2 Functional continuity of the approximation scheme 71

4.3 Property of the uniform approximation on a finite collection of points in [a, b] 74

4.4 Algorithm of de la Vallée Poussin80

4.5 Algorithm of Remez 80

Chapter 5 Optimal Extrapolation Design for the Chebyshev Regression 85

5.1 Introduction 85

5.2 The model and Gauss-Markov estimator 87

5.3 An expression of the extrapolated value through an orthogonalization procedure 91

5.4 The Gauss-Markov estimator of the extrapolated value 93

5.5 The Optimal extrapolation design for the Chebyshev regression 97

Chapter 6 Optimal Design for Linear Forms of the Parameters in a Chebyshev Regression 107

6.1 Outlook and notations 107

6.2 Matrix of moments 113

6.3 Estimable forms 118

6.4 Matrix of moments and Gauss-Markov estimators of a linear form 119

6.5 Geometric interpretation of estimability: Elfving set 133

6.6 Elfving theorem 148

6.7 An intuitive approach to Elfving theorem 154

6.8 Extension of Hoel-Levine result: optimal design for a linear c-form 160

Chapter 7 Special Topics and Extensions 169

7.1 Introduction 169

7.2 The Gauss-Markov theorem in various contexts 170

7.3 Criterions for optimal designs 178

7.4 G-optimal interpolation and extrapolation designs for the Chebyshev regression 188

7.5 Some questions pertaining to the model 209

7.6 Hypotheses pertaining to the regressor 225

7.7 A few questions pertaining to the support of the optimal design for extrapolation 229

7.8 The proofs of some technical results 239

Chapter 8 Multivariate Models and Algorithms 249

8.1 Introduction 249

8.2 Multivariate models 250

8.3 Optimality criterions and some optimal designs 257

8.4 Algorithms 266

Bibliography 289

Index 295