D-optimal axial designs for quadratic and cubic additive mixture models

Abstract

The paper discusses D-optimal axial designs for the additive quadratic and cubic mixture models Σ 1≤i≤q(β ix i + β iix i 2) and Σ 1≤i≤q(β ix i + β iix i 2 + β iiix i 3), where x i ≥ 0, x 1 + ⋯ + x q = 1. For the quadratic model, a saturated symmetric axial design is used, in which support points are of the form (x 1, ⋯ , x q) = [1 -(q - 1)δ i, δ i, ⋯ , δ i], where i = 1, 2 and 0 ≤ δ 2 < δ 1 ≤ 1/(q - 1). It is proved that when 3 ≤ q ≤ 6, the above design is D-optimal if δ 2 = 0 and & δ 1 = 1/(q - 1), and when q ≥ 7 it is D-optimal if δ 2 = 0 and δ 1 = [5q - 1 - (9q 2 - 10q + 1) 1/2]/(4q 2). Similar results exist for the cubic model, with support points of the form (x 1,⋯, x q) = [1 - (q - 1)δ i, δ i,⋯, δ i], where i. = 1, 2, 3 and 0 = δ 3 < δ 2 < δ 1 ≤ 1/(q -1). The saturated D-optimal axial design and D-optimal design for the quadratic model are compared in terms of their efficiency and uniformity. © Australian Statistical Publishing Association Inc. 1998. Published by Blackwell Publishers Ltd.