$$E$$-optimal designs for second-order response surface models.(English)Zbl 1310.62097

Summary: $$E$$-optimal experimental designs for a second-order response surface model with $$k\geq 1$$ predictors are investigated. If the design space is the $$k$$-dimensional unit cube, Z. Galil and J. Kiefer [J. Stat. Plann. Inference 1, 121–132 (1977; Zbl 0381.62062)] determined optimal designs in a restricted class of designs (defined by the multiplicity of the minimal eigenvalue) and stated their universal optimality as a conjecture. In this paper, we prove this claim and show that these designs are in fact $$E$$-optimal in the class of all approximate designs. Moreover, if the design space is the unit ball, $$E$$-optimal designs have not been found so far and we also provide a complete solution to this optimal design problem. {
} The main difficulty in the construction of $$E$$-optimal designs for the second-order response surface model consists in the fact that for the multiplicity of the minimum eigenvalue of the “optimal information matrix” is larger than one (in contrast to the case $$k=1$$) and as a consequence the corresponding optimality criterion is not differentiable at the optimal solution. These difficulties are solved by considering nonlinear Chebyshev approximation problems, which arise from a corresponding equivalence theorem. The extremal polynomials which solve these Chebyshev problems are constructed explicitly leading to a complete solution of the corresponding $$E$$-optimal design problems.

MSC:

 62K20 Response surface designs 62K05 Optimal statistical designs 41A50 Best approximation, Chebyshev systems

Zbl 0381.62062
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