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Computing c-optimal experimental designs using the simplex method of linear programming

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  • Harman, Radoslav
  • Jurík, Tomás

Abstract

An experimental design is said to be c-optimal if it minimizes the variance of the best linear unbiased estimator of , where c is a given vector of coefficients, and [beta] is an unknown vector parameter of the model in consideration. For a linear regression model with uncorrelated observations and a finite experimental domain, the problem of approximate c-optimality is equivalent to a specific linear programming problem. The most important consequence of the linear programming characterization is that it is possible to base the calculation of c-optimal designs on well-understood computational methods. In particular, the simplex algorithm of linear programming applied to the problem of c-optimality reduces to an exchange algorithm with different pivot rules corresponding to specific techniques of selecting design points for exchange. The algorithm can also be applied to "difficult" problems with singular c-optimal designs and relatively high dimension of [beta]. Moreover, the algorithm facilitates identification of the set of all the points that can support some c-optimal design. As an example, optimal designs for estimating the individual parameters of the trigonometric regression on a partial circle are computed.

Suggested Citation

  • Harman, Radoslav & Jurík, Tomás, 2008. "Computing c-optimal experimental designs using the simplex method of linear programming," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 247-254, December.
    • Handle: RePEc:eee:csdana:v:53:y:2008:i:2:p:247-254
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    References listed on IDEAS

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    1. Dette, Holger & Melas, Viatcheslav B. & Biedermann, Stefanie, 2002. "A functional-algebraic determination of D-optimal designs for trigonometric regression models on a partial circle," Statistics & Probability Letters, Elsevier, vol. 58(4), pages 389-397, July.
    2. Holger Dette & Viatcheslav Melas & Andrey Pepelyshev, 2002. "D-Optimal Designs for Trigonometric Regression Models on a Partial Circle," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(4), pages 945-959, December.
    3. Holger Dette, 1997. "Designing Experiments with Respect to ‘Standardized’ Optimality Criteria," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(1), pages 97-110.
    4. Holger Dette & Viatcheslav Melas & Piter Shpilev, 2007. "Optimal designs for estimating the coefficients of the lower frequencies in trigonometric regression models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(4), pages 655-673, December.
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    Cited by:

    1. Peng, Cheng & Kouri, Drew P. & Uryasev, Stan, 2024. "Efficient and robust optimal design for quantile regression based on linear programming," Computational Statistics & Data Analysis, Elsevier, vol. 192(C).
    2. Duarte, Belmiro P.M. & Sagnol, Guillaume & Wong, Weng Kee, 2018. "An algorithm based on semidefinite programming for finding minimax optimal designs," Computational Statistics & Data Analysis, Elsevier, vol. 119(C), pages 99-117.
    3. Michal Černý & Milan Hladík, 2012. "Two complexity results on c-optimality in experimental design," Computational Optimization and Applications, Springer, vol. 51(3), pages 1397-1408, April.
    4. Rodríguez-Díaz, Juan M., 2017. "Computation of c-optimal designs for models with correlated observations," Computational Statistics & Data Analysis, Elsevier, vol. 113(C), pages 287-296.
    5. Katarína Burclová & Andrej Pázman, 2016. "Optimal design of experiments via linear programming," Statistical Papers, Springer, vol. 57(4), pages 893-910, December.
    6. Duarte, Belmiro P.M. & Atkinson, Anthony C. & Granjo, Jose F.O & Oliveira, Nuno M.C, 2019. "Optimal design of experiments for liquid–liquid equilibria characterization via semidefinite programming," LSE Research Online Documents on Economics 102500, London School of Economics and Political Science, LSE Library.
    7. Elham Yousefi & Werner G. Müller, 2023. "Impact of the Error Structure on the Design and Analysis of Enzyme Kinetic Models," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 15(1), pages 31-56, April.
    8. Duarte, Belmiro P.M. & Atkinson, Anthony C. & Granjo, Jose F.O & Oliveira, Nuno M.C, 2022. "Optimal design of experiments for implicit models," LSE Research Online Documents on Economics 107584, London School of Economics and Political Science, LSE Library.
    9. Belmiro P. M. Duarte, 2023. "Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming," Mathematics, MDPI, vol. 11(4), pages 1-17, February.

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