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The regression estimation problem has a long history. Already in 1632 Galileo Galilei used a procedure which can be interpreted as ?tting a linear relationship to contaminated observed data. Such ?tting of a line through a cloud of points is the classical linear regression problem. A solution of this problem is provided by the famous principle of least squares, which was discovered independently by A. M. Legendre and C. F. Gauss and published in 1805 and 1809, respectively. The principle of least squares can also be applied to construct nonparametric regression estimates, where one does not restrict the class of possible relationships, and will be one of the approaches studied in this book. Linear regression analysis, based on the concept of a regression function, was introduced by F. Galton in 1889, while a probabilistic approach in the context of multivariate normal distributions was already given by A. B- vais in 1846. The ?rst nonparametric regression estimate of local averaging type was proposed by J. W. Tukey in 1947. The partitioning regression - timate he introduced, by analogy to the classical partitioning (histogram) density estimate, can be regarded as a special least squares estimate.
From the reviews: MATHEMATICAL REVIEWS "...this book is written by a highly competent, international team of researchers, who have made important contributions and have had a large impact on the field of nonparametric regression estimation and are still active in different subfields; the monograph is almost self-contained and written in such a way that it is a valuable resource for both the researches and graduate students who are novices in the field. The above is possible thanks to the authors... clear and systematic way of presenting fundamental results and their proofs...this book is a valuable source of mathematical techniques and provides systematic in-depth analysis of nonparametric regression with random design. Although this research monograph reflects recent studies in the field, it can also serve as an encyclopedia of nonparametric regression estimation." SHORT BOOK REVIEWS "The book gives a deep and modern mathematical treatment of nonparametric regression with random design. From the table of contents it is seen that all well-known classes of estimators are dealt with. For each of them, the authors mainly prove results on consistency and on rates of convergence. The book follow the style Theorem-Proof and gives rigorous derivations of all the results. There is a useful mathematical appendix with proofs and exponential type inequalities for sums of independent variables and for sum of martingale differences. Each chapter has a section called "Bibliographic Notes" containing references to the extensive bibliography of more than 400 items. Each chapter ends with a number of problems and exercises, which could be used in a teaching situation."