Optimal Designs Efficient

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 e k-optimal design in the class of efficient lack of fit designs for polynomial regression of degree k − 1. 1. Introduction. Linear regression models are among the models most used in practice. Such a parametric assumption for the regression function is very attractive among practitioners, although they are often not sure whether their assumed linear regression model is at least approximately true. Therefore, if a design can be chosen (according to which the data are sampled), the practitioners spread out the design points over the whole experimental region. For important linear regression models such as polynomial regression, such designs and classical optimal designs are quite different. Even more serious when using such an optimal design, deviations from the assumed polynomial regression model are not detectable. In this paper we address these concerns.