Introduction to Nonparametric Estimation (Springer Series in Statistics)

This is a concise text developed from lecture notes and ready to be used for a course on the graduate level. The main idea is to introduce the fundamental concepts of the theory while maintaining the exposition suitable for a first approach in the field. Therefore, the results are not always given in the most general form but rather under assumptions that lead to shorter or more elegant proofs. The book has three chapters. Chapter 1 presents basic nonparametric regression and density estimators and analyzes their properties. Chapter 2 is devoted to a detailed treatment of minimax lower bounds. Chapter 3 develops more advanced topics: Pinsker’s theorem, oracle inequalities, Stein shrinkage, and sharp minimax adaptivity. This book will be useful for researchers and grad students interested in theoretical aspects of smoothing techniques. Many important and useful results on optimal and adaptive estimation are provided. As one of the leading mathematical statisticians working in nonparametrics, the author is an authority on the subject. Preface to the English Edition......Page 4 Preface to the French Edition......Page 5 Notation......Page 7 Contents......Page 9 Examples of nonparametric models and problems......Page 11 Kernel density estimators......Page 12 Mean squared error of kernel estimators......Page 14 Construction of a kernel of order......Page 20 Integrated squared risk of kernel estimators......Page 22 Lack of asymptotic optimality for fixed density......Page 26 Fourier analysis of kernel density estimators......Page 29 Unbiased risk estimation. Cross-validation density estimators......Page 37 Nonparametric regression. The Nadaraya--Watson estimator......Page 41 Local polynomial estimators......Page 44 Pointwise and integrated risk of local polynomial estimators......Page 47 Convergence in the sup-norm......Page 52 Projection estimators......Page 56 Sobolev classes and ellipsoids......Page 59 Integrated squared risk of projection estimators......Page 61 Generalizations......Page 67 Oracles......Page 69 Unbiased risk estimation for regression......Page 71 Three Gaussian models......Page 75 Notes......Page 79 Exercises......Page 82 Introduction......Page 87 A general reduction scheme......Page 89 Lower bounds based on two hypotheses......Page 91 Distances between probability measures......Page 93 Inequalities for distances......Page 96 Bounds based on distances......Page 100 Lower bounds on the risk of regression estimators at a point......Page 101 Lower bounds based on many hypotheses......Page 105 Lower bounds in L2......Page 112 Lower bounds in the sup-norm......Page 118 Fano's lemma......Page 120 Assouad's lemma......Page 126 The van Trees inequality......Page 130 The method of two fuzzy hypotheses......Page 135 Lower bounds for estimators of a quadratic functional......Page 138 Notes......Page 141 Exercises......Page 143 Pinsker's theorem......Page 146 Linear minimax lemma......Page 149 Upper bound on the risk......Page 155 Lower bound on the minimax risk......Page 156 Stein's phenomenon......Page 164 Stein's shrinkage and the James--Stein estimator......Page 166 Other shrinkage estimators......Page 171 Superefficiency......Page 174 Unbiased estimation of the risk......Page 175 Oracle inequalities......Page 183 Minimax adaptivity......Page 188 Inadmissibility of the Pinsker estimator......Page 189 Notes......Page 194 Exercises......Page 196 Appendix......Page 200 Bibliography......Page 211 Index......Page 218 "Methods of nonparametric estimation are located at the core of modern statistical science. The aim of this book is to give a short but mathematically self-contained introduction to the theory of nonparametric estimation. The emphasis is on the construction of optimal estimators; therefore the concepts of minimax optimality and adaptivity, as well as the oracle approach, occupy the central place in the book." "This is a concise text developed from lecture notes and ready to be used for a course on the graduate level. The main idea is to introduce the fundamental concepts of the theory while maintaining the exposition suitable for a first approach in the field. Therefore, the results are not always given in the most general form but rather under assumptions that lead to shorter or more elegant proofs." "The book has three chapters. Chapter 1 presents basic nonparametric regression and density estimators and analyzes their properties. Chapter 2 is devoted to a detailed treatment of minimax lower bounds. Chapter 3 develops more advanced topics: Pinsker's theorem, oracle inequalities, Stein shrinkage, and sharp minimax adaptivity."--Jacket

This is a concise text developed from lecture notes and ready to be used for a course on the graduate level. The main idea is to introduce the fundamental concepts of the theory while maintaining the exposition suitable for a first approach in the field. Therefore, the results are not always given in the most general form but rather under assumptions that lead to shorter or more elegant proofs.

The book has three chapters. Chapter 1 presents basic nonparametric regression and density estimators and analyzes their properties. Chapter 2 is devoted to a detailed treatment of minimax lower bounds. Chapter 3 develops more advanced topics: Pinsker’s theorem, oracle inequalities, Stein shrinkage, and sharp minimax adaptivity.

This book will be useful for researchers and grad students interested in theoretical aspects of smoothing techniques. Many important and useful results on optimal and adaptive estimation are provided. As one of the leading mathematical statisticians working in nonparametrics, the author is an authority on the subject.

Developed from lecture notes and ready to be used for a course on the graduate level, this concise text aims to introduce the fundamental concepts of nonparametric estimation theory while maintaining the exposition suitable for a first approach in the field. Presents basic nonparametric regression and density estimators and analyzes their properties. This book covers minimax lower bounds, and develops advanced topics such as: Pinsker's theorem, oracle inequalities, Stein shrinkage, and sharp minimax adaptivity This book will be a valuable reference for researchers in the eare of nonparametrics.