In this work we build optimal experimental designs for precise estimation of the functional coefficient of a function-on-function linear regression model where both the response and the factors are continuous functions of time. After obtaining the variance-covariance matrix of the estimator of the functional coefficient which minimizes the integrated sum of square of errors, we extend the classical definition of optimal design to this estimator, and we provide the expression of the A-optimal and of the D-optimal designs. Examples of optimal designs for dynamic experimental factors are then computed through a suitable algorithm, and we discuss different scenarios in terms of the set of basis functions used for their representation. Finally, we present an example with simulated data to illustrate the feasibility of our methodology.

翻译：本文针对响应变量与因子均为时间连续函数的函数对函数线性回归模型，构建了用于精确估计函数系数的最优实验设计。在获得最小化误差平方和积分的函数系数估计量的方差-协方差矩阵后，我们将经典最优设计定义扩展至该估计量，并给出了A-最优与D-最优设计的数学表达式。随后通过特定算法计算了动态实验因子的最优设计实例，并基于不同基函数集合的表示方式讨论了多种应用场景。最后，我们通过模拟数据示例展示了该方法的可行性。