New criteria for robust integer-valued designs in linear models

We investigate the problem of designing for linear regression models, when the assumed model form is only an approximation to an unknown true model, using two novel approaches. The first is based on a notion of averaging of the mean-squared error of predictions over a neighbourhood of contaminating functions. The other is based on the usual D-optimal criterion but subject to bias-related constraints in order to ensure robustness to model misspecification. Both approaches are integer-valued constructions in the spirit of Fang and Wiens [2000. Integer-valued, minimax robust designs for estimation and extrapolation in heteroscedastic, approximately linear models. J. Amer. Statist. Assoc. 95(451), 807–818]. Our results are similar to those that have been reported using a minimax approach even though the rationale for the designs presented here are based on the notion of averaging, rather than maximizing, the loss over the contamination space. We also demonstrate the superiority of an integer-valued construction over the continuous designs using specific examples. The designs which protect against model misspecification are clusters of observations about the points that would have been the design points for classical variance-minimizing designs.