Optimal Designs for Implicit Models

In this paper the tools provided by the theory of the optimal design of experiments are applied to a model where the function is given in implicit form. This work is motivated by a dosimetry problem, where the dose, the controllable variable, is expressed as a function of the observed value from the experiment. The best doses will be computed in order to obtain precise estimators of the parameters of the model. For that, the inverse function theorem will be used to obtain the Fisher information matrix. Properly the D-optimal design must be obtained directly on the dose using the inverse function theorem. Alternatively a fictitious D-optimal design on the observed values can be obtained in the usual way. Then this design can be transformed through the model into a design on the doses. Both designs will be computed and compared for a real example. Moreover, different optimal sequences and their D-effiencies will be computed as well. Finally, c-optimal designs for the parameters of the model will be provided.