Geometry of E-Optimality

In the usual linear model y=θ'f(x) we consider the E-optimal design problem. A sequence of generalized Elfving sets Rₖ⊆\ mathbbRn × k (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 desing for A'θ, where An × n is any in-ball vector of a generalized Elfving set Rₙn × n. The minimum eigenvalue of the E-optimal design can be identified as the corresponding squared in-ball radius of Rₙ. 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 Rₙ. The results presented allow the determination of E-optimal designs by an investigation of the geometric properties of a convex symmetric subset Rₙ of Rn × n 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.