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1998-34

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

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


Series: 1998-34, Preprints

MSC:
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 Scheffé 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; Scheffé canonical polynomials; Weighted centroid designs.