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.