Equivalence theorem of $D$-optimal equal allocation design for multiresponse mixture experiments

Year 2021, , 1212 - 1224, 06.08.2021

Xiaoyuan Zhu , Honghua Hao , Weixia Li , Chongqi Zhang

https://doi.org/10.15672/hujms.777430

Cited By: 2

Abstract

The equivalence theorem is the most important theorem of experimental design. For single response, the D-optimal equivalence theorem of the continuous design and equal allocation design already exist. However, the equivalence theorem of D-optimal equal allocation design for multiresponse mixture experiments has not been investigated. In this paper, we study this problem and find that the maximize of the variance function of the equivalence theorem equal to the number of response. D-optimal designs for multiresponse are illustrated by two examples.

Keywords

multiresponse, mixture experiment, D-optimal, equivalence theorem, equal allocation

Supporting Institution

National Natural Science Foundation of China

Project Number

11671104

Thanks

We would like to thank the Referees and the journal editorial team for providing valuable advice that improved the quality of the original manuscript. This work was supported by the National Nature Sciences Foundation of China (11671104) and the Guangzhou University graduate innovative ability training funding program (2019GDJC-D19).

References

• [1] A.C. Atkinson, A.N. Donev and R.D. Tobias, Optimum Experiments Design, with SAS, Oxford, New York, 2007.
• [2] C.L. Atwood, Optimal and efficient designs of experiments, Ann. Math. Statist. 40 (5), 1570-1602, 1969.
• [3] B. Ceranka and M. Graczyk, Regular A-optimal spring balance weighing designs with correlated errors, Hacet. J. Math. Stat. 44 (6), 1527-1535, 2015.
• [4] S.I. Chang, Some properties of multiresponse D-optimal designs, J. Math. Anal. Appl. 184, 256-262, 1994.
• [5] F. Chang, M.L. Huang, D.K.J. Lin and H. Yang, Optimal designs for dual response polynomial regression models, J. Statist. Plann. Inference 93 (1-2), 309-322, 2001.
• [6] J.A. Cornell, Experiments with Mixtures, Designs, Models, and the Analysis of Mixture Data, 3rd ed, John Wiley and Sons, New York, 2002.
• [7] N.R. Draper and W.G. Hunter, Design of experiments for parameter estimation in multiresponse situations, Biometrika 53 (3), 525-533, 1966.
• [8] V.V. Federov, Theory of Optimal Experiments, Academic Press, New York, 1972.
• [9] P. Goos, B. Jones and U. Syafitri, I-optimal design of mixture experiments, J. Amer. Statist. Assoc. 111 (514), 899-911, 2016.
• [10] H.H. Hao and C.Q. Zhang, Multiresponse Scheffé mixture experiment optimal designs, in Chinese, Math. Prac. Theor. 48 (23), 183-188, 2018.
• [11] W. Hassanein and N. Kilany, DE- and EDPM- compound optimality for the information and probability-based criteria, Hacet. J. Math. Stat. 48 (2), 580-591, 2019.
• [12] L. Imhof, Optimum designs for a multiresponse regression model, J. Multivariate Anal. 72, 120-131, 2000.
• [13] H. Jin and R.X. Yue, D-and A-optimal designs for mixture experiments with multiresponse models, Journal of Shanghai Normal University (Natural Sciences) 37 (2), 124-130, 2008.
• [14] A.I. Khuri and J.A. Cornell, Response Surfaces: Designs and Analyses, Marcel Dekker, New York, 1987.
• [15] J. Kiefer, General equivalence theory for optimal designs (approximate theory), Ann. Statist. 2 (5), 849-879, 1974.
• [16] J. Kiefer, Optimum designs in regression problems, II, Ann. Math. Statist. 32 (1), 298-325, 1961.
• [17] J. Kiefer and J. Wolfowitz, The equivalence of two extremum problems, Canad. J. Math. 12, 363-366, 1960.
• [18] J. Kiefer and J. Wolfowitz, Optimum experimental designs, J. R. Stat. Soc. Ser. B. Stat. Methodol. 21 (2), 272-319, 1959.
• [19] O. Krafft and M. Schaefer, D-optimal designs for a multivariate regression model, J. Multivariate Anal. 42, 130-140, 1992.
• [20] P. Laake, On the optimal allocation of observations in experiments with mixtures, Scand. J. Stat. 2 (3), 153-157, 1975.
• [21] C. Li and C.Q. Zhang, A-optimal designs for quadratic mixture canonical polynomials with spline, J. Statist. Plann. Inference 207, 1-9, 2020.
• [22] G.H. Li and C.Q. Zhang, Random search algorithm for optimal mixture experimental design, Comm. Statist. Theory Methods 47 (6), 1413-1422, 2018.
• [23] X. Liu and R.X. Yue, Design admissibility,invariance,and optimality in multipesponse linear models, Statist. Sinica 29, 2187-2203, 2019.
• [24] X. Liu and R.X. Yue, A note on R-optimal designs for multiresponse models, Metrika 76, 483-493, 2013.
• [25] X. Liu, R.X. Yue and W.K. Wong, D-optimal designs for multi-response linear mixed models, Metrika 82, 87-98, 2019.
• [26] N.K. Mandal and M. Pal, Optimal designs for optimum mixtures in multiesponse experiments, Comm. Statist. Simulation Comput. 42 (5), 1104-1112, 2013.
• [27] M. Nezhad, F. Saredorahi, M. Owlia and M. Zad. Design of economically and statistically optimal sampling plans, Hacet. J. Math. Stat. 47 (3), 685-708. 2018.
• [28] M. Pal and N.K. Mandal, Optimum designs for parameter estimation in a mixture experiment with two correlated responses, Comm. Statist. Simulation Comput. 46 (10), 7698-7709, 2017.
• [29] F. Pukelsheim, Optimal Design of Experiments, John Wiley and Sons, New York, 2006.
• [30] H. Scheffé, Simplex-centroid design for experiments with mixtures, J. R. Stat. Soc. Ser. B. Stat. Methodol. 25 (2), 235-263, 1963.
• [31] H. Scheffé, Experiments with mixtures, J. R. Stat. Soc. Ser. B. Stat. Methodol. 20 (2), 344-360, 1958.
• [32] W.F. Smith and J.A. Cornell, Biplot displays for looking at multiple response data in mixture experiments, Technometrics 35 (4), 337-350, 1993.
• [33] M.C. Spruill and W.J. Studden, A Kiefer-Wolfowitz theorem in a stochastic process setting, Ann. Statist. 7 (6), 1329-1332, 1979.
• [34] M.C. Wijesinha, Design of experiments for multiresponse models, PhD thesis, Department of Statistics, University of Florida, Gainesville, FL, 1984.
• [35] W.K. Wong, Y. Yin and J. Zhou, Optimal designs for multi-response nonlinear regression models with several factors via semidefinite programming, J. Comput. Graph. Statist. 28 (1), 61-73, 2019.
• [36] C.Q. Zhang and H. Peng, D-optimal designs for quadratic mixture canonical polynomials with spline, Statist. Probab. Lett. 82 (6), 1095-1101, 2012.
• [37] Z.B. Zhu, G.H. Li and C.Q. Zhang, A-optimal designs for mixture central polynomial model with qualitative factors, Comm. Statist. Theory Methods 48 (10), 2345-2355, 2019.