Group invariant orderings and experimental designs

Abstract

Recent work by Giovagnoli and Wynn and by Eaton develops the theory of G-majorization with application to matrix orderings. Using this theory much of the work begun by Kiefer on ‘universally’ optimal designs of experiments can be better understood. The technique is to combine a group ordering (G-majorization) with another invariant ordering, such as the Loewner ordering, to define upper weak G-majorization on the information matrices of the experiments. Using an idea from previous work of Giovagnoli and Wynn combined with work by Pukelsheim and Styan on the matrix concavity of information matrices a general theory of weak G-majorization for linear models is developed which includes orderings for subsets of estimable functions.