D-minimax Second-order Designs over Hypercubes for Extrapolation and Restricted Interpolation Regions

The D-minimax criterion for estimating slopes of a response surface involving k factors is considered for situations where the experimental region χ and the region of interest are co-centered cubes but not necessarily identical. Taking χ = [-1, 1]^k and = [-R, R]^k, optimal designs under the criterion for the full second-order model are derived for various values of R and their relative performances investigated. The asymptotically optimal design as R → ∞ is also derived and investigated. In addition, the optimal designs within the class of product designs are obtained. In the asymptotic case it is found that the optimal product design is given by a solution of a cubic equation that reduces to a quadratic equation for k = 3 and 6. Relative performances of various designs obtained are examined. In particular, the optimal asymptotic product design and the traditional D-optimal design are compared and it is found that the former performs very well.