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 % 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Box, G. E. P. and Draper, N. R. (1987). Empirical Model-Building and Response Surfaces, Wiley, New York.
Draper, N. R. and Pukelsheim, F. (1994). On third order rotatability, Metrika, 41, 137–161.
Draper, N. R., Gaffke, N. and Pukelsheim, F. (1991). First and second order rotatability of experimental designs, moment matrices, and information surfaces, Metrika, 38, 129–161.
Gaffke, N. and Heiligers, B. (1995a). Optimal and robust invariant designs for cubic multiple regression, Metrika, 42, 29–48.
Gaffke, N. and Heiligers, B. (1995b). Computing optimal approximate invariant designs for cubic regression on multidimensional balls and cubes, J. Statist. Plann. Inference, 47, 347–376.
Galil, Z. and Kiefer, J. C. (1979). Extrapolation designs and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaadchaaeqaaaaa!38C6!\[\phi _p \] designs for cubic regression on the q-ball, J. Statist. Plann. Inference, 3, 27–38.
Heiligers, B. and Schneider, K. (1992). Invariant admissible and optimal designs in cubic regression on the ν-ball, J. Statist. Plann. Inference, 31, 113–125.
Karlin, S. and Studden, W. J.. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics, Wiley Interscience, New York.
Kiefer, J. C. (1961). Optimum experimental designs V, with applications to systematic and rotatable designs, Proc. Fourth Berkeley Symp. on Math. Statist. Prob. (ed. J.Neyman), Vol. 1, 381–405, University of California, Berkeley.
Kiefer, J. C. (1974). General equivalence theory for optimum designs (approximate theory), Ann. Statist., 2, 849–879.
Pukelsheim, F. (1993). Optimal Design of Experiments, Wiley, New York.
Author information
Authors and Affiliations
Additional information
N. R. D. is grateful for the partial support from the Scientific and Environmental Affairs Division of the North Atlantic Treaty Organization, and from the National Security Agency through Grant MDA904-95-H-1020.
About this article
Cite this article
Draper, N.R., Heiligers, B. & Pukelsheim, F. On optimal third order rotatable designs. Ann Inst Stat Math 48, 395–402 (1996). https://doi.org/10.1007/BF00054798
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00054798