Computing optimal approximate invariant designs for cubic regression on multidimensional balls and cubes

Abstract

We consider the problem of computing numerically optimal approximate designs for cubic multiple regression in v ≥ 2 variables under a given convex and differentiable optimality criterion, where the experimental region is either a ball or a symmetric cube, both centered at zero. We restrict to designs which are invariant w.r.t. the group of permutations and sign changes acting on the experimental region. For many optimality criteria this is justified by the fact that within the class of all designs there exists an optimal one which is invariant. In particular, this holds true for any convex and orthogonally invariant criterion, including Kiefer's ф p -criteria, but also for integrated variance criteria ( I-criteria) and for mixtures thereof. Based on the mathematical tools developed by Gaffke and Heiligers (Metrika 42 (1995) 29–48), we show how the algorithms of Gaffke and Mathar (Optimization 24 (1992) 91–126) can be applied to compute (nearly) optimal designs. As examples, we present numerical results for (mixtures of) Kiefer's ф p -criteria under a fixed regression model, which show that the proposed methods work well. For the case that the experimental region is a ball (centered at zero) we also apply the algorithms to the subclass of (third order) rotatable designs, and evaluate the efficiency loss of optimal rotatable designs. Finally, we examine several strategies for obtaining exact designs from optimal approximate ones by changing the weights to integer multiples of 1 n .