A-optimal design for cubic model without a 3-way effect for mixture experiment

Year 2024, , 1759 - 1773, 28.12.2024

Mahesh Kumar Panda

https://doi.org/10.15672/hujms.1374974

Abstract

In this article, we obtain a saturated A-optimal design for the cubic model without a 3-way effect for mixture experiment and get a general formula of the weights. The necessary and sufficient condition of the A-optimality criterion is confirmed by using the corresponding equivalence theorem.

Keywords

Saturated design, A-optimal design, cubic model, mixture experiment, equivalence theorem

References

• [1] M.L. Aggrawal, P. Singh and M.K. Panda, A-optimal designs for an additive cubic model, Stat. Probab. Lett. 81 (2), 259-266, 2011.
• [2] J.A. Cornell, Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data, John Wiley & Sons, New York, 2002.
• [3] R.H. Farrell, J. Kiefer and J. Walbran, Optimum multivariate designs, Proc. 5th Berkeley Symp. Math. Stat. Probab. 1, 113-139, University of California Press, 1967.
• [4] V.V. Fedorov, Design of experiments for linear optimality criteria, Theory Probab. Appl. 16 (1), 189-195, 1971.
• [5] P. Goos and M. Vandebroek, D-optimal response surface designs in the presence of random block effects, Comput. Stat. Data Anal. 37 (4), 433-453, 2001.
• [6] J.W. Gorman and J. Hinman, E Simplex lattice designs for multicomponent systems, Technometrics 4 (4), 463-487, 1962.
• [7] J. Kiefer, Optimal designs in regression problems II, Ann. Math. Stat. 32 (1), 298-325, 1961.
• [8] J. Kiefer, General equivalence theory for optimum designs (approximate theory), Ann. Statist. 2, 849-879, 1974.
• [9] T. Klein, Invariant symmetric block matrices for the design of mixture experiments, Linear Algebra Appl. 388, 261-278, 2004.
• [10] Y.B. Lim, D-optimal design for cubic polynomial regression on the q-simplex, J. Stat. Plan. Infer. 25 (2), 141-152, 1990.
• [11] F. Mikaeili, D-optimum design for cubic without 3-way effect on the simplex, J. Stat. Plan. Infer. 21 (1), 107-115, 1989.
• [12] F. Mikaeili, D-optimum design for full cubic on q-simplex, J. Stat. Plan. Infer. 35 (1), 121-130, 1993.
• [13] M. Pal and N.K. Mandal, Optimum designs for parameter estimation in mixture experiments with group synergism, Commun. Stat. Theory Methods 50 (9), 2001- 2014, 2021.
• [14] M.K. Panda, R-optimal designs for canonical polynomial models with mixture experiments, Calcutta Stat. Assoc. Bull. 73 (2), 146-161, 2021.
• [15] M.K. Panda and R.P. Sahoo, D-optimal designs for Scheffès quadratic mixture model with spline involving two insensitive components, Int. J. Stat. Reliab. Eng. 9 (1), 108-117, 2022a.
• [16] M.K. Panda and R.P. Sahoo, A-optimal designs for cubic polynomial models with mixture experiments in three components, Stat. Appl. 20 (2), 41-55, 2022b.
• [17] M.K. Panda and R.P. Sahoo, R-optimal designs for linear log contrast model with mixture experiments, Commun. Stat. Theory Methods 53 (7), 2355-2368, 2024.
• [18] H. Scheffé, Experiments with mixtures, J. R. Stat. Soc. Ser. B. 20 (2), 344-360, 1958.
• [19] P. Singh and M.K. Panda, Optimal design for second degree K-model for mixture experiments based on weighted simplex centroid design, Metron 69 (3), 251-263, 2011.
• [20] X. Zhu and H. Hao, A-optimal design for the special cubic mixture model, Commun. Stat. Theory Methods 53 (3), 1081-1090, 2024.