Robust designs for nearly linear regression

In this paper we seek designs and estimators which are optimal in some sense for multivariate linear regression on cubes and simplexes when the true regression function is unknown. More precisely, we assume that the unknown true regression function is the sum of a linear part plus some contamination orthogonal to the set of all linear functions in the L₂ norm with respect to Lebesgue measure. The contamination is assumed bounded in absolute value and it is shown that the usual designs for multivariate linear regression on cubes and simplices and the usual least squares estimators minimize the supremum over all possible contaminations of the expected mean square error. Additional results for extrapolation and interpolation, among other things, are discussed. For suitable loss functions optimal designs are found to have support on the extreme points of our design space.