Geometry of E-Optimality

Abstract

In the usual linear model $y=\theta'f(x)$ we consider the E-optimal design problem. A sequence of generalized Elfving sets $\mathscr{R}_k\subseteq\ mathbb{R}^{n \times k}$ (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 $A'\theta$, where $A\in\mathbb{R}^{n \times n}$ is any in-ball vector of a generalized Elfving set $\mathscr{R}_n\subseteq\mathbb{R}^{n \times n}$. The minimum eigenvalue of the E-optimal design can be identified as the corresponding squared in-ball radius of $\mathscr{R}_n$. 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 $\mathscr{R}_n$. The results presented allow the determination of E-optimal designs by an investigation of the geometric properties of a convex symmetric subset $\mathscr{R}_n$ of $\mathbb{R}^{n \times n}$ 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.