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Subsampling. (English) Zbl 0931.62035
Springer Series in Statistics. New York, NY: Springer. xv, 347 p. (1999).
The book concerns one of the resampling methods – subsampling – that has received considerable attention in the nineties. The method relies on recomputing a statistic under consideration over appropriate subsamples of the data set and using these recomputed values to approximate a sampling distribution. The method is closely related to the bootstrap method. It has appeared that subsampling provides consistency results under very weak assumptions even for situations where the usual bootstrap does not. The authors remarkably contributed to the development of the subsampling methodology. The goal of this book is to provide a rigorous foundation for the theory and practice of the subsampling methodology.
The book is divided into two parts. Chapters 1 – 7 provide the basic theory of the bootstrap and subsampling. Chapters 8 – 13 contain extensions of the basic theory and practical issues in implementation and applications. Chapter 1 contains the basic theory of the bootstrap for independent data. It also provides the mathematical tools needed in studying consistency properties of resampling methods. Chapter 2 concerns the subsampling in the case of independent and identically distributed observations, while Chapter 3 is devoted to subsampling for stationary time series. Generalizations to the nonstationary or heteroscedastic case are presented in Chapter 4. Chapter 5 concerns the subsampling for random fields and Chapter 6 does the subsampling for marked point processes. The first part of the book is closed by Chapter 7 where a quite general assertion on subsampling is proved, namely, the parameter space can be quite abstract.
The later chapters are devoted to the following topics: The problem of choice of block size; Higher-order accuracy and interpolation; Subsampling means with heavy tails; Inference for an autoregressive parameter in the possibly integrated case; Application of subsampling to finance, particularly the problem of predictability of stock returns from dividend yields is deeply discussed. The appendices contain results on mixing sequences that are used throughout the book.
The book is intended for graduate students and researchers. Basic knowledge of theoretical statistics is assumed. Chapters 1 – 3 can serve as an introduction to subsampling. The book provides a large number of illustrative (theoretical) examples, however there are no exercises. It contains an index of authors and an index of subjects, however a list of symbols is missing. The book concerns a topic that is still developing quite fast and that has appeared to provide useful approximations to the distributions of various statistics. I think that this book is well written. Many statisticians will find it interesting and useful.
Reviewer: M.Huškova (Praha)

##### MSC:
 62G09 Nonparametric statistical resampling methods 62-02 Research exposition (monographs, survey articles) pertaining to statistics 62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics