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Optimal design of experiments. Reprint of the 1993 original. (English) Zbl 1101.62063
Classics in Applied Mathematics 50. Philadelphia, PA: Society for Industrial and Applied Mathematics (ISBN 0-89871-604-7/pbk; 978-0-89871-910-9/ebook). xxix, 454 p. (2006).
[For the review of the original edition from 1993 see Zbl 0834.62068.]
This is an interesting book on design optimality under linear models, and the material presented involves and rests heavily on linear algebra, convex analysis, and statistics. The subject-matter is covered in 15 chapters of which the first six chapters include linear models, Gauss-Markov theorem, information matrices, moment matrices, Loewner optimal designs, optimal designs for one-dimensional parameter systems, and material for developing the concavity properties of prospective optimality criteria. Also, the four classical criteria of optimality (trace criterion, determinant criterion, smallest eigenvalue criterion, and the average variance criterion) are introduced in these chapters. The general equivalence theorem, which provides necessary and sufficient conditions for a design moment matrix to be \(\varphi\)-optimal \((\varphi\) being some information function) for the set of parameters of interest, is covered in Chapter 7. The next chapter deals with obtaining necessary conditions on optimal support points in terms of their number, location, weights, and is also concerned with studying the interrelation between optimal designs and optimal moment matrices. Chapter 9 is on characterizing \(D\)-, \(A\)-, \(E\)-, and \(T\)-optimal polynomial regression designs over the interval \([-1,+ 1]\).
Admissibility of the moments and information matrices of a polynomial regression design are discussed in Chapter 10. Chapter 11 includes material on integrating some prior information in a design setting by using the general equivalence theorem for each special setting such as Bayes setting, designs with bounded weights, mixtures of models, etc. The next chapter includes techniques dealing with how to round the weights of a design with sample size infinity in order for us to obtain designs for finite sample size \(n\). The final three chapters deal with the concept of invariance which is very important in reducing the dimensionally and complexity of general design problems, Kiefer optimality which involves the powerful concept of the Kiefer ordering of moment matrices, and the final topic deals with response surface designs and the concept of rotatability.
Besides providing very brief biographies of C. Loewner, G. Elfving, and J. Kiefer, the author also discusses under “Comments and References” the pertinent literature chapter by chapter including as well some material on further developments.
The techniques presented here could prove useful even for nonlinear cases in obtaining practical designs through the use of local linearization. It has a useful subject index, a bibliography at the end, and every chapter ends with a list of exercises. The interested reader needs to have a solid background in statistics and mathematics. It is a very theoretical book on optimal designs and should appeal to mathematical statisticians, and those mathematicians dealing with matrix optimization. The subject matter has been presented with clarity and it should prove useful for those who deal with planning of statistical experiments. A good addition to the statistical literature.

MSC:
62K05 Optimal statistical designs
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62J05 Linear regression; mixed models
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