The Annals of Statistics

E-optimal designs for second-order response surface models

Holger Dette and Yuri Grigoriev

Full-text: Open access

Abstract

$E$-optimal experimental designs for a second-order response surface model with $k\geq1$ predictors are investigated. If the design space is the $k$-dimensional unit cube, Galil and Kiefer [J. Statist. Plann. Inference 1 (1977a) 121–132] determined optimal designs in a restricted class of designs (defined by the multiplicity of the minimal eigenvalue) and stated their universal optimality as a conjecture. In this paper, we prove this claim and show that these designs are in fact $E$-optimal in the class of all approximate designs. Moreover, if the design space is the unit ball, $E$-optimal designs have not been found so far and we also provide a complete solution to this optimal design problem.

The main difficulty in the construction of $E$-optimal designs for the second-order response surface model consists in the fact that for the multiplicity of the minimum eigenvalue of the “optimal information matrix” is larger than one (in contrast to the case $k=1$) and as a consequence the corresponding optimality criterion is not differentiable at the optimal solution. These difficulties are solved by considering nonlinear Chebyshev approximation problems, which arise from a corresponding equivalence theorem. The extremal polynomials which solve these Chebyshev problems are constructed explicitly leading to a complete solution of the corresponding $E$-optimal design problems.

Article information

Source
Ann. Statist., Volume 42, Number 4 (2014), 1635-1656.

Dates
First available in Project Euclid: 7 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1407420011

Digital Object Identifier
doi:10.1214/14-AOS1241

Mathematical Reviews number (MathSciNet)
MR3262463

Zentralblatt MATH identifier
1310.62097

Subjects
Primary: 62K05: Optimal designs
Secondary: 41A50: Best approximation, Chebyshev systems

Keywords
Response surface models optimal designs $E$-optimality extremal polynomial duality nonlinear Chebyshev approximation

Citation

Dette, Holger; Grigoriev, Yuri. E -optimal designs for second-order response surface models. Ann. Statist. 42 (2014), no. 4, 1635--1656. doi:10.1214/14-AOS1241. https://projecteuclid.org/euclid.aos/1407420011


Export citation

References

  • Anderson-Cook, C. M., Borror, C. M. and Montgomery, D. C. (2009). Response surface design evaluation and comparison. J. Statist. Plann. Inference 139 629–641.
  • Cheng, C.-S. (1987). An application of the Kiefer–Wolfowitz equivalence theorem to a problem in Hadamard transform optics. Ann. Statist. 15 1593–1603.
  • Denisov, V. I. and Popov, A. A. (1976). $A-E$-optimal and orthogonal designs of experiments for polynomial models. Preprint. Scientific Council on a Complex Problem “Cybernetic,” Academy of Science (in Russian). Moscow, Russia.
  • Dette, H. (1993). A note on $E$-optimal designs for weighted polynomial regression. Ann. Statist. 21 767–771.
  • Dette, H. and Haines, L. M. (1994). $E$-optimal designs for linear and nonlinear models with two parameters. Biometrika 81 739–754.
  • Dette, H. and Röder, I. (1997). Optimal discrimination designs for multifactor experiments. Ann. Statist. 25 1161–1175.
  • Dette, H. and Studden, W. J. (1993). Geometry of $E$-optimality. Ann. Statist. 21 416–433.
  • Draper, N. R., Heiligers, B. and Pukelsheim, F. (2000). Kiefer ordering of simplex designs for second-degree mixture models with four or more ingredients. Ann. Statist. 28 578–590.
  • Draper, N. R. and Pukelsheim, F. (2003). Canonical reduction of second-order fitted models subject to linear restrictions. Statist. Probab. Lett. 63 401–410.
  • Farrell, R. H., Kiefer, J. and Walbran, A. (1967). Optimum multivariate designs. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) 113–138. Univ. California Press, Berkeley, CA.
  • Galil, Z. and Kiefer, J. (1977a). Comparison of design for quadratic regression on cubes. J. Statist. Plann. Inference 1 121–132.
  • Galil, Z. and Kiefer, J. (1977b). Comparison of rotatable designs for regression on balls. I. Quadratic. J. Statist. Plann. Inference 1 27–40.
  • Golikova, T. I. and Pantchenko, L. A. (1977). Continuous $A$ and $Q$-optimal second order designs on a cube. In Regression Experiments (Design and Analysis) (in Russian) (V. V. Nalimov, ed.) 71–84. Moscow Univ., Moscow.
  • Jennrich, R. I. (1969). Asymptotic properties of non-linear least squares estimators. Ann. Math. Statist. 40 633–643.
  • Kiefer, J. (1959). Optimum experimental designs. J. Roy. Statist. Soc. Ser. B 21 272–319.
  • Kiefer, J. (1961a). Optimum experimental designs. V. With applications to systematic and rotatable designs. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 381–405. Univ. California Press, Berkeley, CA.
  • Kiefer, J. (1961b). Optimum designs in regression problems. II. Ann. Math. Statist. 32 298–325.
  • Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 849–879.
  • Kiefer, J. and Wolfowitz, J. (1959). Optimum designs in regression problems. Ann. Math. Statist. 30 271–294.
  • Kôno, K. (1962). Optimum design for quadratic regression on $k$-cube. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 16 114–122.
  • Laptev, V. N. (1974). Some problems relating to construction of regression experimental designs by means of computer. Ph.D. thesis, Novosibirsk State Technical Univ., Russia.
  • Lim, Y. B. and Studden, W. J. (1988). Efficient $D_s$-optimal designs for multivariate polynomial regression on the $q$-cube. Ann. Statist. 16 1225–1240.
  • Melas, V. B. (1982). A duality theorem and $E$-optimality (translated from Russian). Industrial Laboratory 48 295–296.
  • Melas, V. B. (2006). Functional Approach to Optimal Experimental Design. Springer, New York.
  • Myers, R. H., Montgomery, D. C. and Anderson-Cook, C. M. (2009). Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 3rd ed. Wiley, Hoboken, NJ.
  • Pázman, A. (1986). Foundations of Optimum Experimental Design. Reidel, Dordrecht.
  • Pesotchinsky, L. L. (1975). $D$-optimum and quasi-$D$-optimum second-order designs on a cube. Biometrika 62 335–340.
  • Pukelsheim, F. (2006). Optimal Design of Experiments. SIAM, Philadelphia, PA.
  • Pukelsheim, F. and Studden, W. J. (1993). $E$-optimal designs for polynomial regression. Ann. Statist. 21 402–415.
  • Rafajlowicz, E. and Myszka, W. (1988). Optimum experimental design for a regression on a hypercube-generalization of Hoel’s result. Ann. Inst. Statist. Math. 40 821–827.
  • Silvey, S. D. (1980). Optimal Design. Chapman & Hall, London.