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In order to present results on asymptotic theory to readers or students, a decision must be made on the nature and level of mathematical background to provide and/or assume. One possibility is to essentially avoid discussion of regularity conditions, in favor of obtaining by formal manipulations as many useful results as possible. This is the approach taken in [O. E. Barndorff-Nielsen and D. R. Cox, op. cit., 1989; op. cit., 1994], to some extent in [P. G. Hall, The bootstrap and Edgeworth expansion, Springer, New York, 1992; MR 93h:62029], and in a good number of papers appearing in the literature. Another choice is to provide a detailed discussion of regularity conditions, in a form accessible to readers whose mathematical background extends only to multi-variable calculus. This approach will typically involve Taylor series with study of remainder terms, and will rarely lead to the most general results, but it does give the reader/student an exposure to the detailed study needed to verify that plausible asymptotic results are indeed true. This is the approach taken in [E. L. Lehmann, op. cit.], and that book would be accessible I think to most graduate students in statistics.

The present book aims to present a relatively sophisticated discussion of asymptotic theory, with careful attention to regularity conditions and with the aim of establishing very general results; i.e., weakening the regularity conditions as much as possible. The approach taken is to develop from the ground up the theory of asymptotics pioneered by Le Cam, providing much of the needed background in probability theory along the way. While the book does not explicitly assume a strong background in probability theory or in theoretical statistics, a reader who had not studied both could probably not learn the material only from this book. However, a mathematically interested student who has completed typical North American Ph.D. level courses in probability and mathematical statistics would find a great deal to learn in this book.

Thus one way of describing this book is "Le Cam explained", and in this it succeeds beautifully. There have been a number of attempts to present the theory of contiguity and limits of experiments as the basis for most general results in the asymptotic theory of statistics, but this is by good measure the most successful. It is very elegantly written, the theorems are preceded in most instances by an heuristic explanation, and the proofs are clear, if concise.

This is just one way of describing the book, because it is extremely wide-ranging in scope and ambition, although a fundamental goal is to show that approximation by a limit experiment is a very general and flexible way to develop the asymptotic theory of statistics. Attention is restricted mainly to a first-order asymptotic theory, i.e., to obtaining limiting distributions.

Beyond that there are several different ways to consider the book, and the author helpfully provides a (rather complicated) flowchart in the preface to guide the reader or teacher in this. The first part of the book, roughly Chapters 2 through 8, covers the basic asymptotic theory that is usually treated in a standard Ph.D. course in mathematical statistics, but at a fairly advanced mathematical level. There is an excellent, if very concise, review of limit theorems, an introduction to the delta method, and a brief discussion of moment estimators and $M$-estimators from a classical point of view. The notions of contiguity and local asymptotic normality are introduced in Chapters 6 and 7 and used in Chapter 8 to study the efficiency of estimators in a more general framework than is possible in the earlier chapters. Chapter 9 discusses limits of experiments more generally ("local asymptotic nonnormality") and Chapter 10 investigates the convergence of Bayes procedures, mainly Bayes point estimators, so these two chapters could be viewed as a natural extension of the basic material.

Chapters 11 to 14 almost start another book: the topics covered here are the Hajek projection and Hoeffding decomposition of a statistic, the theory of $U$-statistics, asymptotic theory for rank, signed rank and permutation tests, and Pitman and Bahadur efficiency. This material is approximately a modern updating of [J. Hajek and Z. Sidak, Theory of rank tests, Academic Press, New York, 1967; MR 37 #4925]. In Chapter 15 we return to a discussion of asymptotic experiments, with particular application to asymptotic power of test statistics, and the rank tests of Chapter 13 appear as examples. Chapters 16 and 17 consider asymptotic theory for likelihood ratio tests and $\chi\sp 2$-type tests. Brief mention is given in Chapter 16 of Bartlett correction for likelihood ratio tests, but this is a rare reference to higher-order asymptotic theory. Higher-order approximations are discussed in detail in [O. E. Barndorff-Nielsen and D. R. Cox, op. cit., 1989; op. cit., 1994], with emphasis on approximations to null distributions of test statistics. The Le Cam approach permits through contiguity a discussion of non-null asymptotics, leading to results on power of tests and efficiency of estimators. Higher-order asymptotics in this vein is discussed in [J. Pfanzagl, Asymptotic expansions for general statistical models, Lecture Notes in Statist., 31, Springer, Berlin, 1985; MR 87i:62004].

Chapters 18 through 25 present yet another body of material, a brief introduction to the theory of empirical processes and its use in establishing asymptotic results for quantiles, order statistics, $L$-statistics, bootstrap confidence intervals and nonparametric density estimates. The book by G. R. Shorack and J. A. Wellner [Empirical processes with applications to statistics, Wiley, New York, 1986; MR 88e:60002] provides a more expanded treatment, and the books by D. B. Pollard [Empirical processes: theory and applications, Inst. Math. Statist., Hayward, CA, 1990; MR 93e:60046] and van der Vaart and J. A. Wellner [Weak convergence and empirical processes, Springer, New York, 1996; MR 97g:60035] a more abstract treatment. From one point of view this section is a generalization of that presented in Chapters 2 through 9, and some sections of the earlier chapters refer ahead to results in Chapters 18 and 19. The decision of the author to present a more elementary account first is quite sensible, but it would complicate teaching a first (graduate) course in asymptotic theory from this book. The final chapter, Chapter 25, is the longest in the book: more than 60 pages on semiparametric models, and an excellent survey of work mainly only available in research papers.

Throughout, the writing style is clear and concise, and the explanatory material is excellent. There are exercises for every chapter, most of them rather difficult. Each chapter has a very helpful abstract and set of bibliographic notes. There are several courses that could be taught using this book as a guide: a first course in asymptotic theory, a course on the use of empirical processes in statistics, or part of a course on nonparametric theory. A research seminar could also be built around Chapter 25. Inevitably with a book of this scope some important topics get short shrift, and most instructors would want to supplement the book with background material. This book would also be an excellent reference for researchers familiar with some of the material, but wanting an overview of modern rigorous treatments of asymptotic theory.

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