Typical optimal design methods for inverse or parameter estimation problems are designed to choose optimal sampling distributions through minimization of a specific cost function related to the resulting error in parameter estimates. It is hoped that the inverse problem will produce parameter estimates with increased accuracy using data collected according to the optimal sampling distribution. Here we formulate the classical optimal design problem in the context of general optimization problems over distributions of sampling times. We present a new Prohorov metric based theoretical framework that permits one to treat succinctly and rigorously any optimal design criteria based on the Fisher Information Matrix (FIM). A fundamental approximation theory is also included in this framework. A new optimal design, SE-optimal design (standard error optimal design), is then introduced in the context of this framework. We compare this new design criteria with the more traditional D-optimal and E-optimal designs. The optimal sampling distributions from each design are used to compute and compare standard errors; the standard errors for parameters are computed using asymptotic theory or bootstrapping and the optimal mesh. We use three examples to illustrate ideas: the Verhulst-Pearl logistic population model [13], the standard harmonic oscillator model [13] and a popular glucose regulation model [16, 19, 29].

中文翻译：

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反问题优化设计方法比较


反演或参数估计问题的典型优化设计方法旨在通过最小化与参数估计中产生的误差相关的特定成本函数来选择最佳采样分布。希望反问题能够使用根据最佳抽样分布收集的数据产生更准确的参数估计。在这里，我们在采样时间分布的一般优化问题的背景下制定了经典的优化设计问题。我们提出了一种新的基于 Prohorov 度量的理论框架，该框架允许人们简洁而严格地处理基于 Fisher 信息矩阵（FIM）的任何最优设计标准。该框架还包含基本的近似理论。然后在此框架的背景下引入了一种新的优化设计，即 SE 优化设计（标准误差优化设计）。我们将这个新的设计标准与更传统的 D 最优和 E 最优设计进行比较。每个设计的最佳抽样分布用于计算和比较标准误差；使用渐近理论或自举法和最佳网格计算参数的标准误差。我们用三个例子来说明想法：Verhulst-Pearl 逻辑总体模型 [13]、标准谐振子模型 [13] 和流行的葡萄糖调节模型 [16,19,29]。