Optimal Design for Linear Models with Correlated Observations

In the common linear regression model the problem of determining optimal designs for least squares estimation is considered in the case where the observations are correlated. A necessary condition for the optimality of a given design is provided, which extends the classical equivalence theory for optimal designs in models with uncorrelated errors to the case of dependent data. For one parameter models this condition is also shown to be sufficient in many cases and for several models optimal designs can be identified explicitly. For the multi-parameter regression models a simple relation which allows verifying the necessary optimality condition is established. Moreover, it is proved that the arcsine distribution is universally optimal for the polynomial regression model with a correlation structure defined by the logarithmic potential. It is also shown that for models in which the regression functions are eigenfunctions of an integral operator induced by the correlation kernel of the error process, designs satisfying the necessary conditions of optimality can be found explicitly. To the best knowledge of the authors these findings provide the first explicit results on optimal designs for regression models with correlated observations, which are not restricted to the location scale model.