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Comparison of statistical experiments. (English) Zbl 0732.62009

Encyclopedia of Mathematics and its Applications, 36. Cambridge etc.: Cambridge University Press. xx, 675 p. £60.00; $ 99.50 (1991).
The monograph presents the high level of mostly finite sample results concerning the comparison of the information of different statistical experiments. The main tools are based on the work of A. Wald (decision theory) and L. LeCam, which is actually the core of mathematical statistics. The present work is an excellent score within this field. It is mostly selfcontained. It may serve as a textbook for the mathematical background of statistics but it is also of great importance for research. The unified principles can be applied to a lot of different fields in statistics.
Sections 1-3 give some basic tools. Two sections about statistical decision problems and vector lattices prepare the definition of defiency (section 6). It is based on the questions when does one given experiment yield more information than another and how can we measure the difference of information. The same structure appears in game theory in connection with two person zero-sum games and for the comparison of Lorenz ordering of income distributions in economics.
Section 7 deals with conical measures, sufficiency, completeness and extremeness of experiments. Here one also finds the remarkable asymptotic minimax risk theorem (7.4.19) which is fundamental for asymptotic statistics. As example one gets the Hájek-LeCam convolution theorem (7.4.24) since weak convergence of procedures never increases information (7.4.22). This is the most satisfactory point of view for the convolution theorem.
The concept is applied to linear models in section 8. Here one can find a lot of explicit results. Section 9 is devoted to majorization and local comparison of experiments which is related to Schur convexity. The local comparison can be motivated via the slope of one-sided test problems. For translation families with strongly unimodal densities the local deficiency and the deficiency coincide.
Section 10 includes further material and comments. Recent developments show that the ideas are of high significance also in different practical fields: time series, goodness of fit tests, adaptive procedures, survival analysis and sampling plans. Hopefully, the monograph will inspire a lot of statisticians to use the presented methods.
Reviewer: A.Janssen (Siegen)

MSC:

62B15 Theory of statistical experiments
62C05 General considerations in statistical decision theory
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62B05 Sufficient statistics and fields
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