Theory and Methods: 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 σ1≤i≤q(βixi + βiix2i) and σ1≤i≤q(βixi + βiix2i + βiiix3i), where xi≥ 0, x1 + . . . + xq = 1. For the quadratic model, a saturated symmetric axial design is used, in which support points are of the form (x1, . . . , xq) = [1 − (q−1)δi, δi, . . . , δi], where i = 1, 2 and 0 ≤δ2 <δ1 ≤ 1/(q −1). It is proved that when 3 ≤q≤ 6, the above design is D‐optimal if δ2 = 0 and δ1 = 1/(q−1), and when q≥ 7 it is D‐optimal if δ2 = 0 and δ1 = [5q−1 − (9q2−10q + 1)1/2]/(4q2). Similar results exist for the cubic model, with support points of the form (x1, . . . , xq) = [1 − (q−1)δi, δi, . . . , δi], where i = 1, 2, 3 and 0 = δ3 <δ2 < δ1 ≤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.