A note on optimal mixture and mixture amount designs

Statistica Sinica

Volumn 13, Issue 3, 2003, Pages 709-725

  Heiligers, Berthold ^a   Hilgers, Ralf Dieter ^b  

^a OTTO VON GUERICKE UNIVERSITY   (Germany)

^b RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

A optimality; Admissibility; Approximate design; Becker models; Complete class; D optimality; I optimality; Mixture amount experiments; Mixture experiments; Tic type polynomials; V optimality

Indexed keywords

EID: 0141461505     PISSN: 10170405     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (41)

1. Atwood, C. L. (1969). Optimal and efficient designs of experiments. Ann. Math. Statist. 40, 1570-1602.
2. Atkinson, A. C. and A. N. Donev (1992). Optimum Experimental Designs. 2nd edition. Oxford University Press, Oxford.
3. Becker, N. G. (1968). Models for the response of a mixture. J. Roy. Statist. Soc. Ser. B 30,349-358.
4. Becker, N. G. (1978). Models and designs for experiments with mixtures. Austral, J. Statist.20, 195-208.
5. Cornell, J. A. (1990). Experiments with Mixtures. 2nd edition. Wiley, New York.
6. Cornell, J. A. and German, J. W. (1978). On the detection of an additive blending component in multicomponent mixtures. Biometrics 34, 251-263.
7. Darroch, J. N. and Waller, J. (1995). Additivity and interaction in three-component experiments with mixtures. Biometrien 72, 153-163.
8. Draper, N. R. and Pukelsheim, F. (1998a). Polynomial representations for response surface modeling. In New Developments and Applications in Experimental Design. 34 (Edited by N. Flournoy, W. F. Rosenberger and W. K. Wong), 199-212. IMS Lecture Notes Monogr. Ser.
9. Draper, N. R. and Pukelsheim, F. (1998b). Mixture models based on homogeneous polynomials. J. Statist. Plann. Inference 71, 303-311.
10. Draper, N. R. and Pukelsheim, F. (1999). Kiefer ordering of simplex designs for first- and second-degree mixture models. J. Statist. Plann. Inference 79, 325-348.
11. Draper, N. R., Heiligers, B. and Pukelsheim, F. (1999). Kiefer ordering of simplex designs for second-degree mixture models with four or more ingredients. Proceedings of the JSM99,Baltimore, MD.
12. Draper, N. R., Heiligers, B. and Pukelsheim, F. (2000). Kiefer ordering of simplex designs for second-degree mixture models with four or more ingredients. Ann. Statist. 28, 578-590.
13. Ehrenfeld, S. (1956). Complete class theorems in experimental design. Proc. 3rd Berkely Symp. on Math. Statist. Probab. Vol. I, 57-67.
14. Farrell, R. H., Kiefer, J. C. and Walbran, A. (1967). Optimum multivariate designs. Proc. 5th Berkely Symp. on Math. Statist. Probab. Vol, I, 113-138.
15. Gaffke, N. and Heiligers, B. (1996a). Approximate designs for polynomial regression: Invariance, admissibility, and optimality. In Handbook of Statistics 13 (Edited by S. Ghosh and C.R. Rao), 1149-1199, Elsevier, New York.
16. Gaffke, N. and Heiligers, B. (1996b), Second order methods for solving extremum problems from experimental design. Optimization 36, 41-57.
17. Harville, D. A. (1997). Matrix Algebra from the Statistcan's Perspective. Springer, New York.
18. Heiligers, B. (1991). Admissibility of experimental designs in linear regression with constant term, J. Statist. Plann. Inference 28, 107-123.
19. Heiligers, B. (1992). Admissible experimental designs in multiple polynomial regression, J. Statist. Plann. Inference 31, 219-233.
20. Hilgers, R.-D. (1991). Optimale Overmuch in Mischungs-Mengen-Experimenten (in German). Dissertation, University of Dortmund.
21. Hilgers, R.-D. (1999). Design efficiency and estimation for component amount models. Biometrical J. 41, 783-798.
22. Hilgers, R.-D. (2000). D-optimal designs for Becker's minimum polynomial. Statist. Probab. Lett. 49, 175-179.
23. Hilgers, R.-D. and Bauer, P. (1995). Optimal designs for mixture amount experiments. J. Statist. Plann. Inference 48, 241-246.
24. Karlin, S. and Studden, W. J. (1966). Optimal experimental designs. Ann. Math. Statist. 37, 783-815.
25. Kiefer, J. C. (1961). Optimum designs in regression problems II. Ann. Math. Statist. 32, 298-325.
26. Kiefer, J. C. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2, 849-879.
27. Kiefer, J. C. and Wolfowitz, J. (1960). The equivalence of two extremum problems. Canad. J. Math. 36, 1627-1655.
28. Liu, S. and Neudecker, H. (1995). A V-optimal design for Scheffe's polynomial model. Statist. Probab. Lett. 23, 253-258.
29. Liu, S. and Neudecker, H. (1997). Experiments with mixtures: optimal allocations for Becker's models. Metrika 45, 53-66.
30. Piepel, G. F. (1988). A note on models for mixture-amount experiments when the total amount takes a zero value. Technometrics 30, 449-450.
31. Piepel, G. F. and Cornell, J. A. (1985). Response depends on the total amount. Technometrics 27, 219-227.
32. Piepel, G. F. and Cornell, J. A. (1994). Mixture experiments approaches: Examples, discussion, and recommendations. J. Quality Tech. 26, 177-196.
33. Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York.
34. Pukelsheim, F. and Torsney, B. (1991). Optimal weights for experimental design on linearly independent support points. Ann. Statist. 19, 1614-1625.
35. Quenouille, M. H. (1959). Experiments with mixtures. J. Roy. Statist. Soc. Ser. B 21, 201-202.
36. Scheffé, H. (1958). Experiments with mixtures. J. Roy. Statist. Soc. Ser. B 20, 344-360.
37. Scheffé, H. (1963). The simplex centroid design for experiments with mixtures. J. Roy. Statist. Soc. Ser. B 25, 235-263.
38. Snee, R. D. (1973). Techniques for the analysis of mixture data. Technometrics 15, 517-528.
39. Studden, W. J. (1977). Optimal designs for integrated variance in polynomial regression. In Statistical Decision Theory and Related Topics II. Proceedings of a Symposium, Purdue University 1976 (Ededited by S. S. Gupta and D. S. Moore), 411-420. Academic Press, New York.
40. Uranisi, H. (1964). Optimum design for the special cubic regression on the g-simplex. Math. Rep. College Gen. Educ. Kyushu Univ. 1, 7-12.
41. Zhang, C. and Guan, Y. (1992). Generalized additive mixture model and its D-optimal designs (in Chinese). J. Northeast Univ. Tech. 13, 86-93.