E-OPTIMAL DESIGNS FOR SECOND-ORDER RESPONSE SURFACE MODELS

E-optimal experimental designs for a second-order response surface model with k >= 1 predictors are investigated. If the design space is the k-dimensional unit cube, Galil and Kiefer [J. Statist. Plann. Inference 1 (1977a) 121-132] 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.