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\(U\)-statistics, \(M_m\)-estimators and resampling. (English) Zbl 1408.62008

Texts and Readings in Mathematics 75. Singapore: Springer; New Delhi: Hindustan Book Agency (ISBN 978-981-13-2247-1/hbk; 978-981-13-2248-8/ebook). xv, 171 p. (2018).
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.

MSC:

62-02 Research exposition (monographs, survey articles) pertaining to statistics
62G30 Order statistics; empirical distribution functions
62G09 Nonparametric statistical resampling methods
62H10 Multivariate distribution of statistics
62F40 Bootstrap, jackknife and other resampling methods

Software:

R; UStatBookABSC
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