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$$U$$-statistics. Theory and practice. (English) Zbl 0771.62001
Statistics: Textbooks and Monographs. 110. New York etc.: Marcel Dekker, Inc. xi, 302 p. (1990).
The study of $$U$$-statistics started more than forty years ago with P. R. Halmos in a paper on unbiased estimation of functionals and with W. Hoeffding, who proved asymptotic normality for this class of statistics. The present monograph aims to survey this theory and to give several examples and applications. The literature list at the end of the book has over 200 items but is (of course) incomplete. The main part of the book is Chapter 3 (about one third of the total number of pages) and is devoted to the classical asymptotic theory of $$U$$-statistics, such as the central limit theorem, the law of large numbers, the law of the iterated logarithm, the invariance principle, rates of convergence, etc. This survey is in fact a slight extension of the already existing excellent survey in Chapter 5 of a book by R. J. Serfling [Approximation theorems of mathematical statistics (1980; Zbl 0538.62002)].
Chapters 1 and 2 give the basic description of $$U$$-statistics: definitions, examples, moments, etc., and the important Hoeffding decomposition. Chapter 4 deals with symmetric statistics, and $$V$$- statistics, and gives a nice survey on incomplete $$U$$-statistics. Chapter 5 gives results on resampling techniques (such as jackknifing and bootstrapping) applied to $$U$$-statistics, while Chapter 6 provides more applications.

##### MSC:
 62-02 Research exposition (monographs, survey articles) pertaining to statistics 62G99 Nonparametric inference 60Fxx Limit theorems in probability theory 62G09 Nonparametric statistical resampling methods 62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics 62E20 Asymptotic distribution theory in statistics