The Annals of Statistics

On the number of support points of maximin and Bayesian optimal designs

Dietrich Braess and Holger Dette

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We consider maximin and Bayesian D-optimal designs for nonlinear regression models. The maximin criterion requires the specification of a region for the nonlinear parameters in the model, while the Bayesian optimality criterion assumes that a prior for these parameters is available. On interval parameter spaces, it was observed empirically by many authors that an increase of uncertainty in the prior information (i.e., a larger range for the parameter space in the maximin criterion or a larger variance of the prior in the Bayesian criterion) yields a larger number of support points of the corresponding optimal designs. In this paper, we present analytic tools which are used to prove this phenomenon in concrete situations. The proposed methodology can be used to explain many empirically observed results in the literature. Moreover, it explains why maximin D-optimal designs are usually supported at more points than Bayesian D-optimal designs.

Article information

Ann. Statist. Volume 35, Number 2 (2007), 772-792.

First available in Project Euclid: 5 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62K05: Optimal designs

Bayesian optimal design maximin optimal design nonlinear models


Braess, Dietrich; Dette, Holger. On the number of support points of maximin and Bayesian optimal designs. Ann. Statist. 35 (2007), no. 2, 772--792. doi:10.1214/009053606000001307.

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  • Braess, D. and Dette, H. (2005). On the number of support points of maximin and Bayesian $D$-optimal designs in nonlinear regression models. Technical report. Available at
  • Chaloner, K. and Larntz, K. (1989). Optimal Bayesian design applied to logistic regression experiments. J. Statist. Plann. Inference 21 191--208.
  • Dette, H. (1997). Designing experiments with respect to ``standardized'' optimality criteria. J. Roy. Statist. Soc. Ser. B 59 97--110.
  • Dette, H. and Biedermann, S. (2003). Robust and efficient designs for the Michaelis--Menten model. J. Amer. Statist. Assoc. 98 679--686.
  • Dette, H. and Neugebauer, H.-M. (1996). Bayesian optimal one point designs for one parameter nonlinear models. J. Statist. Plann. Inference 52 17--31.
  • Dette, H. and Neugebauer, H.-M. (1997). Bayesian $D$-optimal designs for exponential regression models. J. Statist. Plann. Inference 60 331--349.
  • Ford, I., Torsney, B. and Wu, C.-F. J. (1992). The use of a canonical form in the construction of locally optimal designs for non-linear problems. J. Roy. Statist. Soc. Ser. B 54 569--583.
  • Han, C. and Chaloner, K. (2003). $D$- and $c$-optimal designs for exponential regression models used in viral dynamics and other applications. J. Statist. Plann. Inference 115 585--601.
  • Imhof, L. A. (2001). Maximin designs for exponential growth models and heteroscedastic polynomial models. Ann. Statist. 29 561--576.
  • Karlin, S. and Studden, W. J. (1966). Tchebycheff Systems. With Applications in Analysis and Statistics. Wiley, New York.
  • Kiefer, J. C. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 849--879.
  • Läuter, E. (1974). Methode der Planung eines Experiments für den Fall nichtlinearer Parametrisierung. Math. Operationsforsch. Stat. 5 625--636. (In Russian.)
  • Liebig, H.-P. (1988). Temperature integration by kohlrabi growth. Acta Horticulturae 230 371--380.
  • Lindsay, B. G. (1983). The geometry of mixture likelihoods: A general theory. Ann. Statist. 11 86--94.
  • Mukhopadhyay, S. and Haines, L. M. (1995). Bayesian $D$-optimal designs for the exponential growth model. J. Statist. Plann. Inference 44 385--397.
  • Müller, Ch. H. (1995). Maximin efficient designs for estimating nonlinear aspects in linear models. J. Statist. Plann. Inference 44 117--132.
  • Pronzato, L. and Walter, E. (1985). Robust experiment design via stochastic approximation. Math. Biosci. 75 103--120.
  • Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York.
  • Wong, W. K. (1992). A unified approach to the construction of minimax designs. Biometrika 79 611--619.