Asymptotic statistics.(English)Zbl 0910.62001

Cambridge Series in Statistical and Probabilistic Mathematics, 3. Cambridge: Cambridge Univ. Press. xv, 443 p. (1998).
The text covers an unusually broad range of asymptotic statistics: Classical and modern estimation theory, so as moment estimators, $$M$$-estimators and efficiency of estimators. Also contiguity, local asymptotic normality, and limits of experiments are dealt with. There are chapters on Bayes procedures, projections, $$U$$-statistics, rank, sign and permutation statistics, efficiency of tests, likelihood ratio tests, and chi square tests. Empirical processes, the functional delta method, quantiles and order statistics, and $$L$$-statistics are described. Modern topics covered are the bootstrap, nonparametric density estimation and semiparametric models.
The text grew out of courses that the author gave for students at various places. Therefore, it addresses to students who know about technical details of measure theory and probability, but little about statistics, and vice versa. So, for the benefit of the reader, brief explanations of statistical methodology are given and similar excursions to introduce mathematical details. Measure theory can at most places be avoided by skipping proofs and ignoring the word “measurable”.
A unifying theme is approximation by a limit experiment. The full theory is not developed but the material is limited to the weak topology on experiments. Altogether the book can be seen as exemplification of this theme by the case of smooth parameters of the distribution of i.i.d. observations. Overall, this is a valuable addition to the statistical literature, and I recommend it to any student or statistician who would like to expand their theoretical basis or to have a text as basic reference of this field.

MSC:

 62-02 Research exposition (monographs, survey articles) pertaining to statistics 62F12 Asymptotic properties of parametric estimators 62G20 Asymptotic properties of nonparametric inference 62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics 62F05 Asymptotic properties of parametric tests
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