Bias constrained minimax robust designs for misspecified regression models

We exhibit regression designs and weights which are robust against incorrectly specified regression responses and error heteroscedasticity. The approach is to minimize the maximum integrated mean squared error of the fitted values, subject to an unbiasedness constraint. The maxima are taken over broad classes of departures from the 'ideal' model. The methods yield particularly simple treatments of otherwise intractable design problems. This point is illustrated by applying these methods in a number of examples including polynomial and wavelet regression and extrapolation. The results apply to generalized M-estimation as well as to least squares estimation. Two Open problems - one concerning designing for polynomial regression and the other concerning lack of fit testing - are given.