  We consider the problem of designing experiments for the comparison of two
regression curves describing the relation between a predictor and a response in
two groups, where the data between and within the group may be dependent. In
order to derive efficient designs we use results from stochastic analysis to
identify the best linear unbiased estimator (BLUE) in a corresponding
continuous time model. It is demonstrated that in general simultaneous
estimation using the data from both groups yields more precise results than
estimation of the parameters separately in the two groups. Using the BLUE from
simultaneous estimation, we then construct an efficient linear estimator for
finite sample size by minimizing the mean squared error between the optimal
solution in the continuous time model and its discrete approximation with
respect to the weights (of the linear estimator). Finally, the optimal design
points are determined by minimizing the maximal width of a simultaneous
confidence band for the difference of the two regression functions. The
advantages of the new approach are illustrated by means of a simulation study,
where it is shown that the use of the optimal designs yields substantially
narrower confidence bands than the application of uniform designs.
