Analytic construction of optimal designs in rational and exponential regression models

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Nonlinear regression models are widely used to describe the dependencies between a response and an explanatory variable. Because an appropriate choice of the experimental conditions can improve the quality of statistical inference significantly, many authors have discussed the problem of designing experiments for nonlinear regression models. It is the purpose of the project to study optimal design problems for a broad class of nonlinear models with more than three parameters and various optimality criteria. In particular we will be interested in optimal designs for rational models, exponential models and the following optimality criteria: differentiable optimality criteria (e.g. D- or A-optimality criteria), non-differentiable optimality criteria (e.g. C- and E-optimality criteria). Most of the literature on optimal designs for nonlinear regression models considers the case of one or two nonlinear parameters. The models under consideration in this project contain more than two nonlinear parameters and explicit solutions of the corresponding design problems are difficult to obtain. In the project under consideration the design problems will be solved by an application of analytical methods, which were recently successfully applied by Dette and Melas in the case of linear regression models and allow to express the weights and support points of the optimal designs as analytic functions of the nonlinear parameters. On the one hand this approach yields an explicit construction of locally optimal designs in several special cases. On the other hand these results will be the basis for a determination of optimal designs with respect to Bayesian or minimax optimality criteria, and therefore the work of this project is fundamental for the construction of optimal designs in rational and exponential models with respect to these more sophisticated criteria.