J. Kiefer [4] showed that the optimum designs for quadratic regression
k-dimensional cube for
k≤5 exist in a certain class of designs but his class do not contain the optimum designs for
k≥6. We prove that the optimum designs in such settings for all
k exist in a certain class of the second order symmetric designs, and show the method of their construction. Those obtained in our way are equal to J. Kiefer's design when
k≤2 and have less than points of support when 3≤
k≤5.
In Section 1 we introduce the main definitions and notations used in this paper which are essentially those of J. Kiefer [2] and G. E. P. Box and J. S. Hunter [1], but we take λ
3 in place of λ
4, and mention the results of authers [3], [4], [5], [6] needed in this paper.
In Section 2 we prove the results. Firstly we show that the design which maximizes its generalized variance in the class of the second symmetric design on
k-cube X = {
x; - 1≤
xi≤1, 1≤
i≤
k} is supported on
E={
x|
xi| = 0 or 1,1≤
i≤
k}. Secondly we show, by using the equivalence theorem of J. Kiefer and J. Wolfowitz [6], that such design maximizing its generalized variance is optimal.
In Section 3 we mention the method of construction for above optimum designs and of selecting designs supported on less than points in
E. However the designs obtained in such way do not perhaps those having the minimum number of points of support.
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