Kiefer Ordering Of Simplex Designs For Second-Degree Mixture Models With Four Or More Ingredients

by Draper, N. R.; Heiligers, B.; Pukelsheim, F.


Preprint series: 98-34, Preprints

62K99 None of the above but in this section
62J05 Linear regression
15A69 Multilinear algebra, tensor products
15A45 Miscellaneous inequalities involving matrices


Abstract: For mixture models on the simplex, we discuss the improvement of a given design in terms of increasing symmetry, as well as obtaining a larger moment matrix under the Loewner ordering. The two criteria together define the Kiefer design ordering. The Kiefer ordering can be discussed in the usual Scheff\xe9 model algebra, or in the alternative Kronecker product algebra. We employ the Kronecker algebra which better reflects the symmetries of the simplex experiment region. For the second-degree mixture model, we show that the setof weighted centroid designs constitutes a convex complete class for the Kiefer ordering. For four ingredients, the class is minimal complete. Of essential importance for the derivation is a certain moment polytope, which is discussed in detail.

Keywords: Complete class results for the Kiefer design ordering; Exchangeable designs; Kronecker product; Loewner matrix ordering; Matrix majorization; Moment matrices; Moment polytope; Permutation invariant designs; Scheff\xe9 canonical polynomials; Weighted centroid designs.

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