GEOMETRY OF E-OPTIMALITY

In the usual linear model y = theta'f(x) we consider the E-optimal design problem. A sequence of generalized Elfving sets R(k) subset-or-equal-to R(nxk) (where n is the number of regression functions) is introduced and the corresponding in-ball radii are investigated. It is shown that the E-optimal design is an optimal design for A'theta, where A is-an-element-of R(nxn) is any in-ball vector of a generalized Elfving set R(n) subset-or-equal-to R(nxn). The minimum eigenvalue of the E-optimal design can be identified as the corresponding squared in-ball radius of R(n). A necessary condition for the support points of the E-optimal design is given by a consideration of the supporting hyperplanes corresponding to the in-ball vectors of R(n). The results presented allow the determination of E-optimal designs by an investigation of the geometric properties of a convex symmetric subset R(n) of R(nxn) without using any equivalence theorems. The application is demonstrated in several examples solving elementary geometric problems for the determination of the E-optimal design. In particular we give a new proof of the E-optimal spring balance and chemical balance weighing (approximate) designs.