Study of the excess property of the L-optimal design for the Laible model
DOI:
https://doi.org/10.21638/spbu01.2022.310Abstract
For a rational two-dimensional nonlinear in parameters Laible model used in analytical chemistry, the problem of constructing L-optimal designs is investigated. It is shown that there are two types of optimal designs for this model: saturated (i. e., designs with the number of support points equal to the number of model parameters) and excess (i. e., designs with the number of support points greater than the number of model parameters) and that with some homothetic transformations of the design space, locally L-optimal designs can change the type from saturated to excess and vice versa. An analytical solution to the problem of finding the dependence between the number of the optimal design support points and the values of the model parameters based on the application of a functional approach is proposed. The L-efficiency of D-optimal designs is investigated.Keywords:
L-optimal designs, L-efficiency, optimal designs for estimating the individual coefficients, rational regression models, Laible model
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References
Литература
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References
1. Fedorov V.V. Theory of Optimal Experiment. New York, Academic Press (1972).
2. Pukelsheim F. Optimal Design of Experiments. Philadelphia, SIAM (2006).
3. Atkinson A.C., Donev A.N., Tobias R.D. Optimum Experimental Designs. Oxford, Oxford University Press (2007).
4. Garza A. de la. Spacing of information in polynomial regression. Ann. Math. Statist. 25, 123-130 (1954).
5. Grigoriev Yu.D., Melas V.B., Shpilev P.V. Excess of locally D-optimal designs and homothetic transformations. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 4 (62), iss. 4, 552-562 (2017). https://doi.org/10.21638/11701/spbu01.2017.403 (In Russian) [Eng. transl.: Vestnik St Petersb. Univ. Math. 50, 329-336 (2017). https://doi.org/10.3103/S1063454117040082].
6. Yang M., Stufken J. Support points of locally optimal designs for nonlinear models with two parameters. Annals of Statistics 37, 518-541 (2009).
7. Yang M., Stufken J. Identifying locally optimal designs for nonlinear models: a simple extension with profound consequences. Annals of Statistics 40 (3), 1665-1681 (2012).
8. Yang M. On the de la Garza phenomenon. Annals of Statistics 38, 2499-2524 (2010).
9. Dette H., Melas B. A note on the de la Garza phenomenon for locally optimal designs. Annals of Statistics 39 (2), 1266-1281 (2011).
10. Grigoriev Y.D., Melas V.B., Shpilev P.V. Excess of locally D-optimal designs for Cobb - Douglas model. Statistical Papers 59 (4), 1425-1439 (2018).
11. Grigoriev Y.D., Melas V.B., Shpilev P.V. Excess and saturated D-optimal designs for the rational model. Statistical Papers 62 (3), 1387-1405 (2021).
12. Seber G.A.F., Wild C.J. Nonlinear Regression. New York, John Wiley & Sons (1989).
13. Kiefer J. General equivalence theory for optimum designs (approximate theory). Annals of Statistics 2, 849-879 (1974).
14. Rao C. Linear statistical inference and its applications. New York, Wiley (1968).
15. Dette H., Melas V.B., Shpilev P. Optimal designs for trigonometric regression models. Journal of Statistical Planning and Inference 141 (3), 1343-1353 (2011).
16. Shpilev P.V. Equivalence theorem for singular L-optimal designs. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 2 (60), iss. 1, 68-74 (2015). (In Russian) [Eng. transl.: Vestnik St Petersb. Univ. Math. 48, 29-34 (2015). https://doi.org/10.3103/S1063454115010094].
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18. Laible J.R. The kinetics of the catalytic dehydration of certain tertiary and long chain primary alcohols. Microfilmed Ph. D. Thesis. Madison, University of Wisconsin (1959).
19. Ayen R., Peters M.S. Catalytic reduction of nitric oxide. Ind. Eng. Chem. Process Des. 1 (3), 204-207 (1962).
20. Melas V.B. Functional Approach to Optimal Experimental Design. New York, Springer Science + Business Media (2006).
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Published
2022-10-10
How to Cite
Melas, V. B., & Shpilev, P. V. (2022). Study of the excess property of the L-optimal design for the Laible model. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9(3), 495–505. https://doi.org/10.21638/spbu01.2022.310
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
