In multi-response regression models, the error covariance matrix is never known in practice. Thus, there is a need for optimal designs which are robust against possible misspecification of the error covariance matrix. In this paper, we approximate the error covariance matrix with a neighbourhood of covariance matrices, in order to define minimax D-optimal designs which are robust against small departures from an assumed error covariance matrix. It is well known that the optimization problems associated with robust designs are non-convex, which makes it challenging to construct robust designs analytically or numerically, even for one-response regression models. We show that the objective function for the minimax D-optimal design is a difference of two convex functions. This leads us to develop a flexible algorithm for computing minimax D-optimal designs, which can be applied to any multi-response model with a discrete design space. We also derive several theoretical results for minimax D-optimal designs, including scale invariance and reflection symmetry.

中文翻译：

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多因素多元回归模型的极小极大 D 最优设计

在多响应回归模型中，误差协方差矩阵在实践中是未知的。因此，需要对可能的误差协方差矩阵的错误指定具有鲁棒性的最佳设计。在本文中，我们用协方差矩阵的邻域来近似误差协方差矩阵，以定义极小极大 D 最优设计，该设计对假设误差协方差矩阵的小偏差具有鲁棒性。众所周知，与稳健设计相关的优化问题是非凸的，这使得以分析或数值方式构建稳健设计具有挑战性，即使对于单响应回归模型也是如此。我们证明了极小极大 D 最优设计的目标函数是两个凸函数的差值。这导致我们开发了一种灵活的算法来计算极小极大 D 最优设计，该算法可以应用于任何具有离散设计空间的多响应模型。我们还推导出了极小极大 D 最优设计的几个理论结果，包括尺度不变性和反射对称性。