D-optimally Lack-of-Fit-Test-efficient Designs and Related Simple Designs


  • Wolfgang Bischoff Catholic University Eichstätt-Ingolstadt, Germany




In practice it is often more popular to use a uniform than an optimal design for estimating the unknown parameters of a linear regression model. The reason is that the model can be checked by a uniform design but it cannot be checked by an optimal design in many cases. On the other hand, however, for important regression models a uniform design is not very efficient to estimate the unknown parameters. Therefore Bischoff and Miller
proposed in a series of papers a compromise. It is suggested there to look for designs that are optimal with respect to a specific criterion in the class of designs that are efficient for lack-of-fit-tests. In this paper we consider the D-criterion and polynomial regression models. For polynomial regression models with degree larger than two D-optimally lack-of-fit-test-efficient designs are difficult to determine. Therefore, in this paper we determine easily to calculate and for estimating the parameters highly efficient designs that are
additionally lack-of-fit-test–efficient.


Biedermann, S., and Dette, H. (2001). Optimal designs for testing the functional form of a regression via nonparametric estimation techniques. Statistics and Probability Letters, 52, 215-224.

Bischoff, W., and Miller, F. (2006a). Optimal designs which are efficient for lack of fit tests. Annals of Statistics, 34, 2015-2025.

Bischoff, W., and Miller, F. (2006b). Efficient lack of fit designs that are optimal to estimate the highest coefficient of a polynomial. Journal of Statistical Planning and Inference, 136, 4239-4249.

Bischoff, W., and Miller, F. (2006c). Lack-of-fit-efficiently optimal designs to estimate the highest coefficient of a polynomial with large degree. Statistics and Probability Letters, 76, 1701-1704.

Bischoff, W., and Miller, F. (2007). D-optimally lack-of-fit-test-efficient designs with an application to a fertilizer-response-relationship. ((Preprint))

Dette, H., and Studden, F. (1997). The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis. New York: Wiley.

Fedorov, V. V. (1972). Theory of Optimal Experiments. New York: Academic Press.

Miller, F. (2002). Versuchspläne bei Einschränkungen in der Versuchspunktwahl (In German). Unpublished doctoral dissertation, Fakultät für Mathematik, Universität Karlsruhe.

Pilz, J. (1991). Bayesian Estimation and Experimental Design in Linear Regression Models (2nd ed.). New-York: Wiley.

Pukelsheim, F. (1993). Optimal Design of Experiments. New-York: Wiley.

Silvey, S. D. (1980). Optimal Design. London: Chapman and Hall.

Wiens, D. P. (1991). Designs for approximately linear regression: Two optimality properties of uniform design. Statistics and Probability Letters, 12, 217-221.




How to Cite

Bischoff, W. (2016). D-optimally Lack-of-Fit-Test-efficient Designs and Related Simple Designs. Austrian Journal of Statistics, 37(3&4), 245–253. https://doi.org/10.17713/ajs.v37i3&4.306