This work studies an experimental design problem where {the values of a predictor variable, denoted by $x$}, are to be determined with the goal of estimating a function $m(x)$, which is observed with noise. A linear model is fitted to $m(x)$ but it is not assumed that the model is correctly specified. It follows that the quantity of interest is the best linear approximation of $m(x)$, which is denoted by $\ell(x)$. It is shown that in this framework the ordinary least squares estimator typically leads to an inconsistent estimation of $\ell(x)$, and rather weighted least squares should be considered. An asymptotic minimax criterion is formulated for this estimator, and a design that minimizes the criterion is constructed. An important feature of this problem is that the $x$'s should be random, rather than fixed. Otherwise, the minimax risk is infinite. It is shown that the optimal random minimax design is different from its deterministic counterpart, which was studied previously, and a simulation study indicates that it generally performs better when $m(x)$ is a quadratic or a cubic function. Another finding is that when the variance of the noise goes to infinity, the random and deterministic minimax designs coincide. The results are illustrated for polynomial regression models and the general case is also discussed.

翻译：这项工作研究一个实验设计问题, { 预测变量的值, 以 $x$ 表示 }, 要确定一个实验设计问题, 以估算一个函数$( x) $( x) 的值来确定, 以噪声观测。 线性模型符合 $( x) 美元, 但没有假设该模型的指定正确。 由此可见, 利息的数量是美元( x) 美元的最佳线性近似值, 以 $\ ell( x) 美元 表示 。 显示在此框架中, 普通最小正方位估计值通常导致对美元( x) 的估算不一致, 并应考虑相当加权的最小方块值。 用于此估计值的微缩缩缩缩图标准被设计为淡淡淡微缩图, 而在普通正缩略图中, 缩略图式的缩略图的缩略图显示, 也就是当一个任意的缩略图和缩图时, 矩形的缩图的计算结果会更好。