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 5 Preface to the French Edition......Page 6 Notation......Page 8 Contents......Page 10 Examples of nonparametric models and problems......Page 12 Kernel density estimators......Page 13 Mean squared error of kernel estimators......Page 15 Construction of a kernel of order......Page 21 Integrated squared risk of kernel estimators......Page 23 Lack of asymptotic optimality for fixed density......Page 27 Fourier analysis of kernel density estimators......Page 30 Unbiased risk estimation. Cross-validation density estimators......Page 38 Nonparametric regression. The Nadaraya--Watson estimator......Page 42 Local polynomial estimators......Page 45 Pointwise and integrated risk of local polynomial estimators......Page 48 Convergence in the sup-norm......Page 53 Projection estimators......Page 57 Sobolev classes and ellipsoids......Page 60 Integrated squared risk of projection estimators......Page 62 Generalizations......Page 68 Oracles......Page 70 Unbiased risk estimation for regression......Page 72 Three Gaussian models......Page 76 Notes......Page 80 Exercises......Page 83 Introduction......Page 88 A general reduction scheme......Page 90 Lower bounds based on two hypotheses......Page 92 Distances between probability measures......Page 94 Inequalities for distances......Page 97 Bounds based on distances......Page 101 Lower bounds on the risk of regression estimators at a point......Page 102 Lower bounds based on many hypotheses......Page 106 Lower bounds in L2......Page 113 Lower bounds in the sup-norm......Page 119 Fano's lemma......Page 121 Assouad's lemma......Page 127 The van Trees inequality......Page 131 The method of two fuzzy hypotheses......Page 136 Lower bounds for estimators of a quadratic functional......Page 139 Notes......Page 142 Exercises......Page 144 Pinsker's theorem......Page 147 Linear minimax lemma......Page 150 Upper bound on the risk......Page 156 Lower bound on the minimax risk......Page 157 Stein's phenomenon......Page 165 Stein's shrinkage and the James--Stein estimator......Page 167 Other shrinkage estimators......Page 172 Superefficiency......Page 175 Unbiased estimation of the risk......Page 176 Oracle inequalities......Page 184 Minimax adaptivity......Page 189 Inadmissibility of the Pinsker estimator......Page 190 Notes......Page 195 Exercises......Page 197 Appendix......Page 201 Bibliography......Page 212 Index......Page 219 "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.