\(U\)-statistics, \(M_m\)-estimators and resampling
type: Item
instance of: scholarly article
title: \(U\)-statistics, \(M_m\)-estimators and resampling
zbMATH Open document ID: 1408.62008
DOI: 10.1007/978-981-13-2248-8
author: Arup Bose
author: Snigdhansu Chatterjee
published in: Texts and Readings in Mathematics
publication date: 2018-10-02T00:00:00Z
review text: The book gives an introduction to U-statistics, generalized M-estimators, bootstrap and the software R. In the first chpater, U-statistics and how to derive their asymptotic behavior by means of the Hoeffding decomposition is discussed. In the second chapter, \(M_m\)-estimators are introduced, which are a generalization of M-estimators. The law of large numbers for such estimators is proved, and the linearization (a generalization of the Bahadur representation) is discussed. The third chapter gives an overview over different resampling techniques (Jackknife, Efron's bootstrap, wild bootstrap) and also discusses the generalized weighted bootstrap as a concept to integrate the different bootstrapping methods. In Chapter 4, the bootstrap methods are applied to U-statistics and \(M_m\)-estimators. In the final chapter, R is presented, with special emphasis on multivariate medians and bootstrap. The book is a very accessible introduction to the topic mentioned above, it does not demand to much previous knowledge from the reader. Many examples are discussed, starting from trivial ones (like the sample mean as a special case of a U-statistics), but also discussing more sophisticated examples like the Oja median. Every chapter ends with several exercises. The last chapter about R also starts with elementary properties of this software like data structures. So in my opinion this book is very suitable as a material for teaching students. For researchers and practitioners, however, the book might be good to have a first look into the topic, but it does not cover much of the recent developments in the area. There are only few references from the last ten years, and topics of practical relevance like time series are not discussed.
reviewed by: Martin Wendler
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 62G09
Mathematics Subject Classification ID: 62H10
Mathematics Subject Classification ID: 62F40
zbMATH DE Number: 6946621
zbMATH Keywords: U-statistics
zbMATH Keywords: M-estimators
zbMATH Keywords: bootstrap
zbMATH Keywords: Hoeffding decomposition
zbMATH Keywords: Oja median
describes a project that uses: R
describes a project that uses: UStatBookABSC
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-981-13-2248-8
OpenAlex ID: W2888791327
description: scientific article; zbMATH DE number 6946621