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Mathematische Statistik I: Parametrische Verfahren bei festem Stichprobenumfang. (German) Zbl 0581.62001

Stuttgart: B. G. Teubner. XVIII, 538 S. DM 125.00 (1985).
In 1966 the first edition of ”Mathematische Statistik” by the author, see Zbl 0149.142, was published. Together with the volume ”Angewandte Mathematische Statistik” (1970; Zbl 0212.209) by G. Nölle and the author it has become one of the most popular German textbooks on mathematical statistics.
In the meantime the author has decided to rewrite and strongly enlarge both books. The first of the intended two volumes has now appeared. Its content roughly corresponds to that of ”Mathematische Statistik” from 1966. However, it has been enlarged considerably, increasing from 223 pages of the old book to 538 in the present version. Besides being more detailed, the most important change is that invariance theory and the theory of linear models has been included, which originally was touched upon in ”Angewandte mathematische Statistik”.
The first chapter contains basic notions of mathematical statistics. Besides those parts which are known from the earlier version, one can find a discussion of the game theoretic aspects of statistics, basic notions of optimization theory and a first introduction to linear models.
The second chapter deals with the theory of testing and estimation as far as it can be treated by tools from optimization theory, in particular i.e. the Neyman-Pearson theory of one-parametric families. New is the treatment of locally optimal tests, robust tests in the sense of Neyman- Pearson tests between least favorable pairs for contamination models, and some results concerning estimators.
The third chapter corresponds to the last two sections of the old textbook. It contains the reduction of testing problems by means of sufficiency and the theory of testing for multidimensional exponential families. Additionally, it now presents some invariance theory, the theorem of Hunt and Stein in the testing set-up and the theorem of Girshick an Savage on the minimaxity of Pitman estimators.
The fourth and last chapter is new. Its discussion of linear models with fixed covariance structure is mainly taken from ”Angewandte mathematische Statistik”. The presentation of variance component models, which in its present form has not appeared in textbooks yet, deserves special attention. The chapter is closed by a short introduction to multivariate analysis.
A characteristic feature of this monograph is that the mathematical frame of statistics is developed with full rigor and, simultaneously, every general result is illustrated and discussed at hand of several important examples. Discussing the examples the mathematical details are elaborated as far as necessary for an easy understanding.
The book reveals the connection between the mathematical theory and the applied statistical problems which led to its development. Maybe, that due to the enlargement the book lost some of its perspicuity for the beginner which the earlier version certainly had. Nevertheless, it should find a wide readership, since it shows up a bridge between applied statistics and its mathematical theory.
Reviewer: H.Strasser

MSC:

62-02 Research exposition (monographs, survey articles) pertaining to statistics
62-XX Statistics