On optimal third order rotatable designs

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 % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaadchaaeqaaaaa!38C6!\[\phi _p \] (which include the often used E-, A-, and D-criteria). The E-optimal design attains a particularly simple, explicit form.