Introduction to nonparametric estimation
type: Item
instance of: scholarly article
title: Introduction to nonparametric estimation
zbMATH Open document ID: 1176.62032
author: Alexandre B. Tsybakov
published in: Springer Series in Statistics
publication date: 2008-10-29T00:00:00Z
review text: 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 exte…
reviewed by: Sreenivasan Ravi
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62G08
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62-01
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62-02
zbMATH DE Number: 5358920
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5358920