Optimal Kiefer Ordering of Simplex Designs for Third-Degree Mixture Kronecker-Models with three ingredients.

Abstract

This paper investigates the Kiefer optimality in the third-degree Kronecker model for mixture experiments. For mixture models on the simplex, a better design is obtained, by matrix majorization that yields a larger moment matrix due to increase of symmetry and Loewner ordering. The two criteria together constitute the Kiefer design ordering and any such criteria single out one or a few designs that are Kiefer optimal. For the third-degree mixture models with three ingredients, an exchangeable moment matrix was constructed by use of Kronecker product algebra. These moment matrices are symmetrical, balanced, invariant and have homogenous regression entries which are good and have desirable properties for an optimal design. Then, the necessary and sufficient conditions for two exchangeable third-degree K-moment matrices to be comparable in the Loewner matrix ordering were set up. The weights obtained from the original design were used in the construction of the weighted centroid designs. Based on the results obtained, it was shown that the set of the weighted centroid designs constitutes a minimal complete class designs for the Kiefer design ordering and that any design that is not weighted centroid design can be improved upon by convex combination of an appropriate elementary designs.