The Annals of Statistics

Optimal designs which are efficient for lack of fit tests

Wolfgang Bischoff and Frank Miller

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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

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

First available in Project Euclid: 3 November 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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