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Optimal design of experiments. (English) Zbl 0834.62068
New York, NY: Wiley. xxiii, 454 p. (1993).
The aim of the book is to present the “Optimality theory of experimental designs in linear models”. “The design problems originate from statistics, but are solved using special tools from linear algebra and convex analysis” (all from the preface). Consequently, a matrix oriented approach is chosen which uses information functionals rather than functionals of the covariance matrices, and the optimization problems are described in terms of subgradients and polars. By its high theoretical level this textbook is complementary to the series of so far existing monographs on optimal designs, e.g. by V. V. Fedorov [Theory of optimal experiments. New York etc.: Academic Press (1972); translation of the original Russian edition. Moscow: Nauka (1971; Zbl 0261.62002)], H. Bandemer [Theorie und Anwendung der optimalen Versuchsplanung. I. Handbuch zur Theorie. Berlin: Akademie-Verlag (1977; Zbl 0391.62001); II. Handbuch zur Anwendung. Berlin: Akademie-Verlag (1980; Zbl 0486.62067)], S. D. Silvey [Optimal design. London etc.: Chapman and Hall (1980; Zbl 0468.62070)], A. Pázman [Foundations of optimum experimental design. Dordrecht etc.: D. Reidel (1986; Zbl 0588.62117)] or A. C. Atkinson and A. N. Donev [Optimum experimental designs. Oxford: Clarendon Press (1992; Zbl 0829.62070)].
The present book is a comprehensive source of background information on the optimality of designs rather than a collection of recipes for the construction. Its material is arranged into fifteen sections including a matrix oriented summary of linear models, a problem description illustrated by one-dimensional optimization tasks emphasizing the importance of the Elfving set, optimality criteria based on the information matrices like uniform (Loewner) optimality and matrix means, optimality results centered around the corresponding general equivalence theorems, and further topics like admissibility, Bayesian designs, efficient approximations for finite sample sizes and invariance considerations leading to the concept of universal (Kiefer) optimality.
Applications are mainly devoted to polynomial regression models. The text is augmented by comments on the literature, biographical notes on Loewner, Elfving and Kiefer, an extensive bibliography and an exhaustive subject index.

62K05 Optimal statistical designs
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics