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

The paper discusses D-optimal axial designs for the additive quadratic and cubic mixture models Sigma(1 less than or equal to i less than or equal to q)(beta(i)x(i) + beta(ii)x(i)(2)) and Sigma(1 less than or equal to i less than or equal to q)(beta(i)x(i) + beta(ii)x(i)(2) + beta(iii)x(i)(3)), where x(i) greater than or equal to 0, x(1) +...+ x(q) = 1. Fbr 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)delta(i), delta(i),...,delta(i)], where i = 1,2 and 0 less than or equal to sigma(2) < delta(1) less than or equal to 1/(q - 1). It is proved that when 3 less than or equal to q less than or equal to 6, the above design is D-optimal if delta(2) = 0 and delta(1) = 1/(q - 1), and when q greater than or equal to 7 it is D-optimal if delta(2) = 0 and delta(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)delta(i), delta(i),..., delta(i)], where i = 1, 2, 3 and 0 = delta(3) < delta(2) < delta(1) less than or equal to 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.