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An equivalence theorem for design optimality with respect to a multi-objective criterion

Author

Listed:
  • Chiara Tommasi

    (University of Milan)

  • Juan M. Rodríguez-Díaz

    (University of Salamanca)

  • Jesús F. López-Fidalgo

    (University of Navarre
    University of Navarre)

Abstract

Maxi-min efficiency criteria are a kind of multi-objective criteria, since they enable us to take into consideration several tasks expressed by different component-wise criteria. However, they are difficult to manage because of their lack of differentiability. As a consequence, maxi-min efficiency designs are frequently built through heuristic and ad hoc algorithms, without the possibility of checking for their optimality. The main contribution of this study is to prove that the maxi-min efficiency optimality is equivalent to a Bayesian criterion, which is differentiable. In addition, we provide an analytic method to find the prior probability associated with a maxi-min efficient design, making feasible the application of the equivalence theorem. Two illustrative examples show how the proposed theory works.

Suggested Citation

  • Chiara Tommasi & Juan M. Rodríguez-Díaz & Jesús F. López-Fidalgo, 2023. "An equivalence theorem for design optimality with respect to a multi-objective criterion," Statistical Papers, Springer, vol. 64(4), pages 1041-1056, August.
    • Handle: RePEc:spr:stpapr:v:64:y:2023:i:4:d:10.1007_s00362-023-01431-2
    DOI: 10.1007/s00362-023-01431-2
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    References listed on IDEAS

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    1. Tommasi, C. & López-Fidalgo, J., 2010. "Bayesian optimum designs for discriminating between models with any distribution," Computational Statistics & Data Analysis, Elsevier, vol. 54(1), pages 143-150, January.
    2. Lopez-Fidalgo, Jesus & Tommasi, Chiara, 2004. "Construction of MV- and SMV-optimum designs for binary response models," Computational Statistics & Data Analysis, Elsevier, vol. 44(3), pages 465-475, January.
    3. Ray-Bing Chen & Ping-Yang Chen & Cheng-Lin Hsu & Weng Kee Wong, 2020. "Hybrid algorithms for generating optimal designs for discriminating multiple nonlinear models under various error distributional assumptions," PLOS ONE, Public Library of Science, vol. 15(10), pages 1-30, October.
    4. Holger Dette & Viatcheslav B. Melas & Andrey Pepelyshev & Nikolai Strigul, 2003. "Efficient design of experiments in the Monod model," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(3), pages 725-742, August.
    5. Martijn Berger & C. Joy King & Weng Wong, 2000. "Minimax d-optimal designs for item response theory models," Psychometrika, Springer;The Psychometric Society, vol. 65(3), pages 377-390, September.
    6. 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.
    7. Duarte, Belmiro P.M. & Wong, Weng Kee & Atkinson, Anthony C., 2015. "A Semi-Infinite Programming based algorithm for determining T-optimum designs for model discrimination," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 11-24.
    8. Dette, Holger & Biedermann, Stefanie, 2003. "Robust and Efficient Designs for the Michaelis-Menten Model," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 679-686, January.
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