ON OPTIMAL THIRD ORDER ROTATABLE DESIGNS

NORMAN R. DRAPER^{1}, BERTHOLD HEILIGERS^{2} AND FRIEDRICH PUKELSHEIM^{3}

1Department of Statistics, University of Wisconsin-Madison, Madison, WI 53706, U.S.A.

2Fakultät für Mathematik, Universität Magdeburg, D-39016 Magdeburg, Germany 3Institut für Mathematik, Universität Augsburg, D-86135 Augsburg, Germany

(Received November 7, 1994; revised May 15, 1995)

Abstract. We obtain results for choosing optimal third order rotatable designs for the fitting of a third order polynomial response surface model, for m ≥ 3 factors. By representing the surface in terms of Kronecker algebra, it can be established that the two parameter family of boundary nucleus designs forms a complete class, under the Loewner matrix ordering. In this paper, we first narrow the class further to a smaller complete class, under the componentwise eigenvalue ordering. We then calculate specific optimal designs under Kiefer’s ϕ_{p}-criteria (which include the often used E-, A-, and D-criteria). The E-optimal design attains a particularly simple, explicit form.

Key words and phrases: Complete classes of designs, design efficiency, E-, A-, D-optimal designs, response surface designs, third order models.