Advertisement

Optimal Designs for the Evaluation of an Extremum Point

  • R. C. H. Cheng
  • V. B. Melas
  • A. N. Pepelyshev
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 51)

Abstract

This paper studies the optimal experimental design for the evaluation of an extremum point of a quadratic regression function of one or several variables. Experimental designs which are locally optimal for arbitrary dimension k among all approximate designs are constructed (although for k > 1 an explicit form proves to be available only under a restriction on the location of the extremum point). The result obtained can be considered as an improvement of the last step of the well-known Box-Wilson procedure

Keywords

Optimal Design Extremum Point Regression Function Quadratic Regression Optimal Experimental Design 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Box, G.E.P. and Wilson, K.B. (1951). On the experimental attainment of optimum conditions. J. Royal Statistical Soc. B 13, 1–38.MathSciNetzbMATHGoogle Scholar
  2. Buonaccorsi, J.P. and Iyer, Y.K. (1986). Optimal designs for ratios of linear combinations in the general linear model. JSPI 13, 345–356.MathSciNetzbMATHGoogle Scholar
  3. Chaloner, K. (1989). Optimal Bayesian experimental design for estimation of the turning point of a quadratic regression. Communications in Statistics, Theory and Methods 18, 1385–1400.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Chatterjee, S.K. and Mandai, N.K. (1981). Response surface designs for estimating the optimal point. Bull. Calcutta Statist. Ass. 30, 145–169.zbMATHGoogle Scholar
  5. Fedorov, V.V. and Müller, W.G. (1997). Another view on optimal design for estimating the point of extremum in quadratic regression. Metrika 46, 147–157.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Jennrich, R.J. (1969). Asymptotic properties of non-linear least squares estimators. Ann. Math. Stat. 40, 633–643.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Karlin, S. and Studden, W. (1966). Tchebysheff Systems: with Application in Analysis and Statistics. New York: Wiley.Google Scholar
  8. Mandai, N.K. and Heiligers, B. (1992). Minimax designs for estimating the optimum point in a quadratic response surface. JSPI 31, 235–244.Google Scholar
  9. Mandai, N.K. (1978). On estimation of the maximal point of single factor quadratic response function. Bull. Calcutta Statist. Assoc. 27, 119–125.Google Scholar
  10. Müller, W.G. and Pötscher, B.M. (1992). Batch sequential design for a nonlinear estimation problem. In Model-Oriented Data Analysis 2 Eds V.V. Fedorov, W.G. Müller and I. Vuchkov, pp. 77–87. Heidelberg: Physica-Verlag.Google Scholar
  11. Müller, Ch.H. (1995). Maximin efficient designs for estimating nonlinear aspects in linear models. JSPI 44, 117–132.zbMATHGoogle Scholar
  12. Müller, Ch.H. and Pazman, A. (1998). Applications of necessary and sufficient conditions for maximin efficient designs. Metrika 48, 1–19.MathSciNetzbMATHGoogle Scholar
  13. Pronzato, L. and Walter, E. (1993). Experimental design for estimating the optimum point in a response surface. Acta Applic. Mathemat., 33, 45–68.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • R. C. H. Cheng
    • 1
  • V. B. Melas
    • 2
  • A. N. Pepelyshev
    • 2
  1. 1.Department of MathematicsUniversity of SouthamptonHighfield, SouthamptonUK
  2. 2.Faculty of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations