The Annals of Statistics

Optimal designs which are efficient for lack of fit tests

Wolfgang Bischoff and Frank Miller

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Abstract

Linear regression models are among the models most used in practice, although the practitioners are often not sure whether their assumed linear regression model is at least approximately true. In such situations, only designs for which the linear model can be checked are accepted in practice. For important linear regression models such as polynomial regression, optimal designs do not have this property. To get practically attractive designs, we suggest the following strategy. One part of the design points is used to allow one to carry out a lack of fit test with good power for practically interesting alternatives. The rest of the design points are determined in such a way that the whole design is optimal for inference on the unknown parameter in case the lack of fit test does not reject the linear regression model.

To solve this problem, we introduce efficient lack of fit designs. Then we explicitly determine the ek-optimal design in the class of efficient lack of fit designs for polynomial regression of degree k−1.

Article information

Source
Ann. Statist., Volume 34, Number 4 (2006), 2015-2025.

Dates
First available in Project Euclid: 3 November 2006

Permanent link to this document
https://projecteuclid.org/euclid.aos/1162567641

Digital Object Identifier
doi:10.1214/009053606000000597

Mathematical Reviews number (MathSciNet)
MR2283725

Zentralblatt MATH identifier
1246.62175

Subjects
Primary: 62J05: Linear regression 62F05: Asymptotic properties of tests
Secondary: 62F05: Asymptotic properties of tests

Keywords
Linear regression models testing lack of fit efficient maximin power polynomial regression of degree k−1 optimal designs to estimate the highest coefficient

Citation

Bischoff, Wolfgang; Miller, Frank. Optimal designs which are efficient for lack of fit tests. Ann. Statist. 34 (2006), no. 4, 2015--2025. doi:10.1214/009053606000000597. https://projecteuclid.org/euclid.aos/1162567641


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References

  • Biedermann, S. and Dette, H. (2001). Optimal designs for testing the functional form of a regression via nonparametric estimation techniques. Statist. Probab. Lett. 52 215--224.
  • Bischoff, W. (1998). A functional central limit theorem for regression models. Ann. Statist. 26 1398--1410.
  • Bischoff, W. (2002). The structure of residual partial sums limit processes of linear regression models. Theory Stoch. Processes 8 23--28.
  • Bischoff, W. and Miller, F. (2000). Asymptotically optimal tests and optimal designs for testing the mean in regression models with applications to change-point problems. Ann. Inst. Statist. Math. 52 658--679.
  • Bischoff, W. and Miller, F. (2006). Efficient lack of fit designs that are optimal to estimate the highest coefficient of a polynomial. J. Statist. Plann. Inference 136 4239--4249.
  • Biswas, A. and Chaudhuri, P. (2002). An efficient design for model discrimination and parameter estimation in linear models. Biometrika 89 709--718.
  • Box, G. E. P. and Draper, N. (1959). A basis for the selection of a response surface design. J. Amer. Statist. Assoc. 54 622--654.
  • Dette, H. (1993). Bayesian $D$-optimal and model robust designs in linear regression models. Statistics 25 27--46.
  • Dette, H. and Studden, W. J. (1997). The Theory of Canonical Moments With Applications in Statistics, Probability, and Analysis. Wiley, New York.
  • El-Krunz, S. M. and Studden, W. J. (1991). Bayesian optimal designs for linear regression models. Ann. Statist. 19 2183--2208.
  • Ermakov, S. M. and Melas, V. B. (1995). Design and Analysis of Simulation Experiments. Kluwer, London.
  • Federov, V. V. (1972). Theory of Optimal Experiments. Academic Press, New York.
  • Kiefer, J. and Wolfowitz, J. (1959). Optimum designs in regression problems. Ann. Math. Statist. 30 271--294.
  • Krafft, O. (1978). Lineare statistische Modelle und optimale Versuchspläne. Vandenhoeck and Ruprecht, Göttingen.
  • Miller, F. (2002). Optimale Versuchspläne bei Einschränkungen in der Versuchspunktwahl. Ph.D. dissertation, Fakultät für Mathematik, Univ. Karlsruhe. Available at www.ubka.uni-karlsruhe.de/cgi-bin/psview?document=2002/mathematik/10.
  • Montepiedra, G. and Yeh, A. B. (1998). A two-stage strategy for the construction of $D$-optimal experimental designs. Comm. Statist. Simulation Comput. 27 377--401.
  • Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York.
  • Pukelsheim, F. and Rosenberger, J. L. (1993). Experimental designs for model discrimination. J. Amer. Statist. Assoc. 88 642--649.
  • Rivlin, T. J. (1990). Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory, 2nd ed. Wiley, New York.
  • Sacks, J. and Ylvisaker, D. (1966). Design for regression problems with correlated errors. Ann. Math. Statist. 37 66--89.
  • Schwarz, H. R. (1988). Numerische Mathematik, 2nd ed. Teubner, Stuttgart.
  • 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.
  • Wiens, D. P. (1991). Designs for approximately linear regression: Two optimality properties of uniform designs. Statist. Probab. Lett. 12 217--221.