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A geometric characterization of c-optimal designs for heteroscedastic regression

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  • Dette, Holger
  • Holland-Letz, Tim

Abstract

We consider the common nonlinear regression model where the variance as well as the mean is a parametric function of the explanatory variables. The c-optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest. A geometric characterization of c-optimal designs in this context is presented, which generalizes the classical result of Elfving (1952) for c-optimal designs. As in Elfving's famous characterization c-optimal designs can be described as representations of boundary points of a convex set. However, in the case where there appear parameters of interest in the variance, the structure of the Elfving set is different. Roughly speaking the Elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space. The c-optimal designs are characterized as representations of the points where the line in direction of the vector c intersects the boundary of the new Elfving set. The theory is illustrated in several examples including pharmacokinetic models with random effects.

Suggested Citation

  • Dette, Holger & Holland-Letz, Tim, 2008. "A geometric characterization of c-optimal designs for heteroscedastic regression," Technical Reports 2008,26, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    • Handle: RePEc:zbw:sfb475:200826
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    References listed on IDEAS

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    1. Isabel Ortiz & Carmelo Rodríguez, 1998. "Optimal designs with respect to Elfving's partial minimax criterion for heteroscedastic polynomial regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 7(2), pages 347-360, December.
    2. Dette, Holger & Pepelyshev, Andrey & Wong, Weng Kee, 2008. "Optimal designs for dose finding experiments in toxicity studies," Technical Reports 2008,09, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    3. Dette, Holger & Bretz, Frank & Pepelyshev, Andrey & Pinheiro, José, 2008. "Optimal Designs for Dose-Finding Studies," Journal of the American Statistical Association, American Statistical Association, vol. 103(483), pages 1225-1237.
    4. Grace Montepiedra & Weng Wong, 2001. "A New Design Criterion When Heteroscedasticity is Ignored," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(2), pages 418-426, June.
    5. Biedermann, Stefanie & Dette, Holger & Zhu, Wei, 2006. "Optimal Designs for DoseResponse Models With Restricted Design Spaces," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 747-759, June.
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