Cover Page

Series Editor
Nikolaos Limnios

Interpolation and Extrapolation Optimal Designs 1

Polynomial Regression and Approximation Theory

Giorgio Celant

Michel Broniatowski

images

Preface

This book is the first of a series of three which cover a part of the field called optimal designs, in the context of interpolation and extrapolation. This area has been studied extensively, due to its numerous applications in engineering, physics, chemistry, and more generally, in all fields where experiments can be planned according to some expected accuracy on the resulting conclusions, under operational constraints.

The context of the present volume, which has been considered, is very specific by choice. Indeed reducing the model to the case where the observations are real numbers, hence unidimensional, and the environmental variable being unidimensional too, most of the concepts gain in clarity; also the tools leading to optimal solutions are quite accessible, although they require some technicalities. This choice will help the reader to consider more general models, keeping in mind the basic ingredients which are developed in more involved situations, at the cost of some additional regularity assumptions. This is a well-known way to proceed in science.

The focus of this book is statistics. The contents of the book are mostly real analysis and approximation of functions. This duality in the arguments is not surprising, and is a constant in most advanced fields in statistics, and also in various other disciplines. It happens, and this is a major fact in the present field of statistics, that optimal designs are obtained as special problems in the theory of functional approximation. So, those two fields, namely statistics and numerical analysis, meet in the present setting.

The framework of the present book is thus one of the classical real analysis, of the basic tools in algebra together with standard basic tools in statistics. The reader may also be interested in the chronological aspect (or historical aspect) of the development, as the authors have been. Although of statistical concern, the main arguments used in this book stem from the theory of the uniform approximation of functions. This argument has a long and interesting history, from the pioneering works of Lagrange and Legendre, followed by the contributions of Chebyshev and Markov, continuing through famous results by Lebesgue; the results obtained by Bernstein and Vitali provide sharp and interesting insights into the properties of polynomials, and are of interest in the accuracy of the approximation of functions. Borel provided a final description to the approximation results due to Chebyshev. Erdös gave a strong improvement to Bernstein’s contribution in the rates of approximation. The reader will find those elements in the core of the present volume and in the Appendices.

Most optimal designs do not result as analytic solutions for approximation problems. Algorithmic solutions have been developed over the years: the optimal designs are obtained through algorithms which were developed in the field of the theory of the uniform approximation of functions, and are nowadays, important tools in numerical analysis. Henceforth, those algorithms have also been studied by statisticians; such is the case for the Remez algorithm, and to its extension by de Boor and Rice to constrained cases. This volume presents these tools in the statistical context.

The choice of the statistical context is restricted to the regular one, in the sense that all random variables which describe the variability of the inputs are supposed to be independent, essentially with the same distribution, with finite variance, hence allowing the least mean square paradigm in the field of linear models. This is the basic framework. The companion volumes consider nonlinear models, heteroscedastic models, models with dependence in the errors, etc.

This book results from our teaching, both in the University of Padova and in University Pierre and Marie Curie (Sorbonne University) in Paris. This corresponds to a one semester course in statistics or in applied mathematics.

The authors express their gratitude to their families and friends, who provided inestimable support during the completion of this work.

Giorgio CELANT, Padova

Michel BRONIATOWSKI, Paris

February 2016