Interpolation and Extrapolation Optimal Designs 2Finite Dimensional General Models
This book considers various extensions of the topics treated in the first volume of this series, in relation to the class of models and the type of criterion for optimality. The regressors are supposed to belong to a generic finite dimensional Haar linear space, which substitutes for the classical polynomial case. The estimation pertains to a general linear form of the coefficients of the model, extending the interpolation and extrapolation framework; the errors in the model may be correlated, and the model may be heteroscedastic. Non-linear models, as well as multivariate ones, are briefly discussed. The book focuses to a large extent on criteria for optimality, and an entire chapter presents algorithms leading to optimal designs in multivariate models. Elfving’s theory and the theorem of equivalence are presented extensively. The volume presents an account of the theory of the approximation of real valued functions, which makes it self-consistent.
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
G Celant, Associate Professor of Statistics, University of Padova, Italy. Michel Broniatowski, Professor of Statistics, Universitei Pierre et Marie Curie, Paris, France.