Minimax Designs in Linear Regression Models

Abstract

In the usual linear regression model we investigate the geometric structure of a class of minimax optimality criteria containing Elfving's minimax and Kiefer's $\phi_p$-criteria as special cases. It is shown that the optimal designs with respect to these criteria are also optimal for $A'\theta$, where $A$ is any inball vector (in an appropriate norm) of a generalized Elfving set. The results explain the particular role of the $A$- and $E$-optimality criterion and are applied for determining the optimal design with respect to Elfving's minimax criterion in polynomial regression up to degree 9.