Abstract

In mixture experiments, sometimes, there are some so-called collapse phenomena among its mixture components. In view of these collapse phenomena, we present a detailed discussion of the influence thereof on the response models and the structures of the corresponding optimal designs. In this paper, we first propose a set of concepts related to the collapse phenomena, including collapsed mixture models, collapsed Scheffé model, multiple mixture simplex, and direct sum measure. For the collapsed Scheffé model, we first prove a new inequality which is based on positive definite quadratic form. And applying this inequality, we prove that the optimal design of any collapsed Scheffé model can be obtained by combining the optimal designs of all its local models. Meanwhile, we also give out the coefficients of all the designs in the combinations. The relevant optimization criteria discussed in this paper include D-, A- and $I_{\lambda }$- optimality criteria. The conclusions of this paper clarify the optimal design structures for the collapsed Scheffé models. Applying these conclusions, a new method for constructing the optimal designs of the collapsed Scheffé models is also given.

摘要

在混合实验中，有时其混合成分之间会出现一些所谓的塌陷现象。鉴于这些坍塌现象，我们详细讨论了其对响应模型和相应优化设计结构的影响。在本文中，我们首先提出了一组与塌陷现象相关的概念，包括塌陷混合模型、塌陷 Scheffé 模型、多重混合单纯形和直接和测度。对于坍缩的 Scheffé 模型，我们首先证明了一个基于正定二次型的新不等式。并且应用这个不等式，我们证明了任何折叠的 Scheffé 模型的最优设计可以通过组合其所有局部模型的最优设计来获得。同时，我们还给出了组合中所有设计的系数。${\mathrm{一世}}_{\lambda }$- 最优标准。本文的结论阐明了折叠 Scheffé 模型的最佳设计结构。应用这些结论，还给出了一种构建折叠 Scheffé 模型优化设计的新方法。