The Annals of Statistics

Optimal designs for a class of nonlinear regression models

Holger Dette, Viatcheslav B. Melas, and Andrey Pepelyshev

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For a broad class of nonlinear regression models we investigate the local E- and c-optimal design problem. It is demonstrated that in many cases the optimal designs with respect to these optimality criteria are supported at the Chebyshev points, which are the local extrema of the equi-oscillating best approximation of the function f00 by a normalized linear combination of the regression functions in the corresponding linearized model. The class of models includes rational, logistic and exponential models and for the rational regression models the E- and c-optimal design problem is solved explicitly in many cases.

Article information

Ann. Statist., Volume 32, Number 5 (2004), 2142-2167.

First available in Project Euclid: 27 October 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

E-optimal design c-optimal design rational models local optimal designs Chebyshev systems


Dette, Holger; Melas, Viatcheslav B.; Pepelyshev, Andrey. Optimal designs for a class of nonlinear regression models. Ann. Statist. 32 (2004), no. 5, 2142--2167. doi:10.1214/009053604000000382.

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  • Becka, M., Bolt, H. M. and Urfer, W. (1993). Statistical evalutation of toxicokinetic data. Environmetrics 4 311--322.
  • Becka, M. and Urfer, W. (1996). Statistical aspects of inhalation toxicokinetics. Environ. Ecol. Stat. 3 51--64.
  • Chaloner, K. and Larntz, K. (1989). Optimal Bayesian design applied to logistic regression experiments. J. Statist. Plann. Inference 21 191--208.
  • Chaloner, K. and Verdinelli, I. (1995). Bayesian experimental design: A review. Statist. Sci. 10 273--304.
  • Chernoff, H. (1953). Locally optimal designs for estimating parameters. Ann. Math. Statist. 24 586--602.
  • Dette, H. and Haines, L. (1994). $E$-optimal designs for linear and nonlinear models with two parameters. Biometrika 81 739--754.
  • Dette, H., Haines, L. and Imhof, L. A. (1999). Optimal designs for rational models and weighted polynomial regression. Ann. Statist. 27 1272--1293.
  • Dette, H., Melas, V. B. and Pepelyshev, A. (2002). Optimal designs for a class of nonlinear regression models. Preprint, Ruhr-Universität Bochum. Available at
  • Dette, H., Melas, V. B. and Pepelyshev, A. (2004). Optimal designs for estimating individual coefficients in polynomial regression---a functional approach. J. Statist. Plann. Inference 118 201--219.
  • Dette, H. and Studden, W. J. (1993). Geometry of $E$-optimality. Ann. Statist. 21 416--433.
  • Dette, H. and Wong, W. K. (1999). $E$-optimal designs for the Michaelis--Menten model. Statist. Probab. Lett. 44 405--408.
  • DeVore, R. A. and Lorentz, G. G. (1993). Constructive Approximation. Springer, New York.
  • Dudzinski, M. L. and Mykytowycz, R. (1961). The eye lens as an indicator of age in the wild rabbit in Australia. CSIRO Wildlife Research 6 156--159.
  • Elfving, G. (1952). Optimum allocation in linear regression theory. Ann. Math. Statist. 23 255--262.
  • Ford, I. and Silvey, S. D. (1980). A sequentially constructed design for estimating a nonlinear parametric function. Biometrika 67 381--388.
  • 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.
  • He, Z., Studden, W. J. and Sun, D. (1996). Optimal designs for rational models. Ann. Statist. 24 2128--2147.
  • Heiligers, B. (1994). $E$-optimal designs in weighted polynomial regression. Ann. Statist. 22 917--929.
  • Imhof, L. A. and Studden, W. J. (2001). $E$-optimal designs for rational models. Ann. Statist. 29 763--783.
  • Karlin, S. and Studden, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience, New York.
  • Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 849--879.
  • Kitsos, C. P., Titterington, D. M. and Torsney, B. (1988). An optimal design problem in rhythmometry. Biometrics 44 657--671.
  • Melas, V. B. (1978). Optimal designs for exponential regression. Math. Operationsforsch. Statist. Ser. Statist. 9 45--59.
  • Melas, V. B. (2000). Analytical theory of $E$-optimal designs for polynomial regression. In Advances in Stochastic Simulation Methods (N. Balakrishan, V. B. Melas and S. Ermakov, eds.) 85--115. Birkhäuser, Boston.
  • Melas, V. B. (2001). Analytical properties of locally $D$-optimal designs for rational models. In MODA 6---Advances in Model-Oriented Design and Analysis (A. C. Atkinson, P. Hackel and W. G. Müller, eds.) 201--210. Physica, Heidelberg.
  • Petrushev, P. P. and Popov, V. A. (1987). Rational Approximation of Real Functions. Cambridge Univ. Press.
  • Pronzato, L. and Walter, E. (1985). Robust experimental design via stochastic approximation. Math. Biosci. 75 103--120.
  • Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York.
  • Pukelsheim, F. and Studden, W. J. (1993). $E$-optimal designs for polynomial regression. Ann. Statist. 21 402--415.
  • Pukelsheim, F. and Torsney, B. (1991). Optimal designs for experimental designs on linearly independent support points. Ann. Statist. 19 1614--1625.
  • Ratkowsky, D. A. (1983). Nonlinear Regression Modeling: A Unified Practical Approach. Dekker, New York.
  • Ratkowsky, D. A. (1990). Handbook of Nonlinear Regression Models. Dekker, New York.
  • Seber, G. A. J. and Wild, C. J. (1989). Nonlinear Regression. Wiley, New York.
  • Silvey, S. D. (1980). Optimal Design. Chapman and Hall, London.
  • Studden, W. J. (1968). Optimal designs on Tchebycheff points. Ann. Math. Statist. 39 1435--1447.
  • Studden, W. J. and Tsay, J. Y. (1976). Remez's procedure for finding optimal designs. Ann. Statist. 4 1271--1279.
  • Szegö, G. (1975). Orthogonal Polynomials, 4th ed. Amer. Math. Soc., Providence, RI.
  • Wu, C.-F. J. (1985). Efficient sequential designs with binary data. J. Amer. Statist. Assoc. 80 974--984.