Interpolation and Extrapolation Optimal Designs 2: Finite Dimensional General Models [Hardcover]

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. Elfvings 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

1.2.1. Splines 6

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

1.4. Optimality with respect to a norm 12

1.4.1. Existence of an optimal solution 12

1.4.2. Uniqueness of the optimal solution 13

1.4.3. Examples 15

1.5. Characterizing the optimal solution 18

1.5.1. The Hilbertian case 18

1.5.2. The non-Hilbertian case 21

1.5.3. Optimization, L_p norms and robustness 24

Chapter 2 Chebyshev Systems 27

2.1. Introduction 27&