Optimal Characteristic Designs for Polynomial Models

  • J. M. Rodríguez-Díaz
  • J. López-Fidalgo
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 51)


Using the characteristic polynomial coefficients of the inverse of the information matrix, design criteria can be defined between A- and D-optimality (López-Fidalgo and Rodríguez-Díaz, 1998). With a slight modification of the classical algorithms, the gradient expression allows us to find some optimal characteristic designs for polynomial regression. We observe that these designs are a smooth transition from A- to D-optimal designs. Moreover, for some of these optimal designs, the efficiencies for both criteria, A- and D-optimality, are quite good.

Nice relationships emerge when plotting the support points of these optimal designs against the number of parameters of the model. In particular, following the ideas developed by Pukelsheim and Torsney (1991), we have considered A-optimality. Another mathematical expression can be given for finding A-optimal support points using nonlinear regression. This could be very useful for obtaining optimal designs for the other characteristic criteria.


A-optimality characteristic criteria D-optimality polynomial regression 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bellhouse, R. and Herzberg, A. M. (1984). Equally spaced design points in polynomial regression: a comparison of systematic sampling methods with the optimal design of experiments. Canadian Journal of Statistics 12, 77–90.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Dette, H. (1994). Discrimination designs for polynomial regression on compact intervals. Annals of Statistics 22, 890–903.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Dette, H. and Studden, W. J. (1994). Optimal designs with respect to Elfving’s partial minimax criterion in polynomial regression. Ann. Inst. Statist. Math. 46, 389–403.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Dette, H. and Wong, W.K. (1995). On G-efficiency calculation for polynomial models. Annals of Statistics 23, 2081–2101.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Dette, H. and Wong, W.K. (1996). Robust optimal extrapolation designs. Biometrika 83, 667–680.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Fedorov, V.V. (1972). Theory of Optimal Experiments. New York: Academic Press.Google Scholar
  7. Hoel, P. G. and Levine, A. (1964). Optimal spacing and weighting in polynomial prediction. Ann. Math. Statist. 35, 1553–1560.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Kiefer, J. and Studden, W. J. (1976). Optimal designs for large degree polynomial regression. Annals of Statistics 4, 1113–1123.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Kiefer, J. and Wolfowitz, J. (1959). Optimum designs in regression problems. Ann. Math. Statist. 30, 271–294.MathSciNetzbMATHCrossRefGoogle Scholar
  10. López-Fidalgo, J. and Rodríguez-Díaz, J.M. (1998). Characteristic Polynomial Criteria in Optimal Experimental Design. In MODA 5-Advances in Model-Oriented Data Analysis and Experimental Design Eds A.C. Atkinson, L. Pronzato and H.P. Wynn, pp. 31–38. Heidelberg: Physica-Verlag.CrossRefGoogle Scholar
  11. Pázman, A. (1986). Foundations of Optimum Experimental Design. Dordrecht: Reidel.zbMATHGoogle Scholar
  12. Pukelsheim, F. and Studden, W. J. (1993). E-optimal designs for polynomial regression. Annals of Statistics 21, 402–415.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Pukelsheim, F. and Torsney, B. (1991). Optimal weights for experimental designs on linearly independent support points. Annals of Statistics 19, 1614–1625.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Studden, W. J. (1980). D s-optimal designs for polynomial regression using continued fractions. Annals of Statistics 8, 1132–1141.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • J. M. Rodríguez-Díaz
    • 1
  • J. López-Fidalgo
    • 1
  1. 1.Department of Statistics of the University of SalamancaSalamancaSpain

Personalised recommendations