Analytic construction of optimal designs in rational and exponential regression models

非线性回归模型被广泛用于描述响应和解释变量之间的依赖关系。由于选择适当的实验条件可以显著提高统计推断的质量，许多作者讨论了非线性回归模型的实验设计问题。该项目的目的是研究一类具有三个以上参数和各种最优性准则的非线性模型的最优设计问题。特别是，我们将对理性模型，指数模型和以下最优性准则的最优设计感兴趣：可微最优性准则（例如D-或A-最优性准则），不可微最优性准则（例如C-和E-最优性准则）。大多数关于非线性回归模型最优设计的文献都考虑了一个或两个非线性参数的情况。本计画所考虑的模型包含两个以上的非线性参数，因此很难得到对应设计问题的显式解。在正在考虑的项目中，设计问题将通过应用分析方法来解决，这些方法最近被Dette和Melas成功地应用于线性回归模型的情况下，并允许将最佳设计的权重和支持点表示为非线性参数的分析函数。一方面，这种方法在几种特殊情况下产生一个明确的局部最优设计的建设。另一方面，这些结果将是确定贝叶斯或极大极小最优性标准的最优设计的基础，因此，该项目的工作是在理性和指数模型中构建这些更复杂的标准的最优设计的基础。

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.