Elfving's Theorem for $D$-Optimality

Abstract

We consider a model robust version of the $c$-optimality criterion minimizing a weighted product with factors corresponding to the variances of the least squares estimates for linear combinations of the parameters in different models. A generalization of Elfving's theorem is proved for the optimal designs with respect to this criterion by an application of an equivalence theorem for mixtures of optimality criteria. As a special case an Elfving theorem for the $D$-optimal design problem is obtained. In the case of identical models the connection between the $A$-optimality criterion and the model robust criterion is investigated. The geometric characterizations of the optimal designs are illustrated by a couple of examples.