IDEAS home Printed from https://ideas.repec.org/a/spr/stpapr/v64y2023i4d10.1007_s00362-023-01431-2.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00362-023-01431-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00362-023-01431-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Pepelyshev, Andrey & Melas, Viatcheslav B. & Strigul, Nikolay & Dette, Holger, 2004. "Design of experiments for the Monod model : robust and efficient designs," Technical Reports 2004,36, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    2. Wong, Weng Kee & Melas, Viatcheslav B. & Dette, Holger, 2004. "Optimal design for goodness-of-fit of the Michaelis-Menten enzyme kinetic function," Technical Reports 2004,24, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    3. Masoudi, Ehsan & Holling, Heinz & Wong, Weng Kee, 2017. "Application of imperialist competitive algorithm to find minimax and standardized maximin optimal designs," Computational Statistics & Data Analysis, Elsevier, vol. 113(C), pages 330-345.
    4. Lenka Filová & Mária Trnovská & Radoslav Harman, 2012. "Computing maximin efficient experimental designs using the methods of semidefinite programming," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(5), pages 709-719, July.
    5. Braess, Dietrich & Dette, Holger, 2004. "On the number of support points of maximin and Bayesian D-optimal designs in nonlinear regression models," Technical Reports 2004,78, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    6. Dette, Holger & Biedermann, Stefanie & Pepelyshev, Andrey, 2004. "Some robust design strategies for percentile estimation in binary response models," Technical Reports 2004,19, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    7. Ulrike Graßhoff & Heinz Holling & Rainer Schwabe, 2012. "Optimal Designs for the Rasch Model," Psychometrika, Springer;The Psychometric Society, vol. 77(4), pages 710-723, October.
    8. Biedermann, Stefanie & Dette, Holger & Pepelyshev, Andrey, 2005. "Optimal Discrimination Designs for Exponential Regression Models," Technical Reports 2005,22, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    9. Lei He & Rong-Xian Yue, 2022. "$$I_L$$ I L -optimal designs for regression models under the second-order least squares estimator," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(1), pages 53-66, January.
    10. Dette, Holger & Pepelyshev, Andrey, 2005. "Efficient experimental designs for sigmoidal growth models," Technical Reports 2005,13, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    11. Abebe, Haftom T. & Tan, Frans E.S. & Van Breukelen, Gerard J.P. & Berger, Martijn P.F., 2014. "Bayesian D-optimal designs for the two parameter logistic mixed effects model," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 1066-1076.
    12. Mahmood Ul Hassan & Frank Miller, 2019. "Optimal Item Calibration for Computerized Achievement Tests," Psychometrika, Springer;The Psychometric Society, vol. 84(4), pages 1101-1128, December.
    13. Dennis Schmidt & Rainer Schwabe, 2015. "On optimal designs for censored data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(3), pages 237-257, April.
    14. Dette, Holger & Kiss, Christine, 2007. "Optimal experimental designs for inverse quadratic regression models," Technical Reports 2007,37, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    15. Lei He & Rong-Xian Yue, 2020. "R-optimal designs for trigonometric regression models," Statistical Papers, Springer, vol. 61(5), pages 1997-2013, October.
    16. Dette, Holger & O'Brien, Timothy E., 2003. "Efficient experimental design for the Behrens-Fisher problem with application to bioassay," Technical Reports 2003,21, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    17. Wim Linden & Hao Ren, 2015. "Optimal Bayesian Adaptive Design for Test-Item Calibration," Psychometrika, Springer;The Psychometric Society, vol. 80(2), pages 263-288, June.
    18. Yuan-chin Chang & Hung-Yi Lu, 2010. "Online Calibration Via Variable Length Computerized Adaptive Testing," Psychometrika, Springer;The Psychometric Society, vol. 75(1), pages 140-157, March.
    19. 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.
    20. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:stpapr:v:64:y:2023:i:4:d:10.1007_s00362-023-01431-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.