Introduction to nonparametric estimation
This is a revised and extended version of the French original edition, `Introduction à l'estimation non-paramétrique.' (2004; Zbl 1029.62034). The book is based on courses taught by the author. After discussing methods of construction of nonparametric estimators having optimal rates of convergence in a minimax sense for different classes of functions and several distances defining the risk, this book concentrates on the study of optimality of the estimators. The problem of adaptive estimation in Gaussian sequence models is also analyzed. The monograph consists of three chapters. In the first chapter, the nonparametric estimation problem is introduced through probability density estimation, nonparametric regression and the Gaussian white noise model. Kernel density estimators are then introduced with a study of mean squared errors, variance, bias, mean squared and integrated mean squared risks, construction of kernels, Fourier analysis of kernel density estimators and inadmissibility of the Epanechnikov kernel. The other topics in the first chapter include unbiased risk estimation, cross-validation density estimators, the Nadaraya-Watson estimator, local polynomial estimators, projection estimators, oracles and unbiased risk estimation for regression. The second chapter, `Lower bounds on the minimax risk.', includes a general reduction scheme for obtaining lower bounds, lower bounds based on two hypotheses, a discussion on distances between probability measures, lower bounds on the risk of regression estimators at a point, and tools for minimax lower bounds. The third chapter consists of a detailed proof of Pinsker's theorem, and after a discussion of the Stein phnomenon in a parametric setting, unbiased estimation of the risk in Gaussian sequence models, oracle inequalities, minimax adaptivity and inadmissibility of the Pinsker estimator are dealt with. Each chapter ends with Notes giving remarks and historical notes on the topics in the chapter, followed by an extensive list of Exercises which complement the theory discussed. The book ends with a list of references and a subject index. As the author says in the Preface to the original French edition, the book is meant to be an introduction to the rich theory of nonparametric estimation through some simple models and examples. The detailed proofs given in the book will help the interested reader to understand the subject better. This well written book will be welcomed by all those interested in learning the presented concepts. The author should be complimented for a good treatise with detailed proofs of several important results in nonparametric estimation theory.
scientific article; zbMATH DE number 5358920