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：本文研究一个实验设计问题，其中需要确定预测变量（记作$x$）的值，目标是估计存在噪声观测的函数$m(x)$。对$m(x)$拟合线性模型，但未假设模型正确指定。因此，关注的量是$m(x)$的最佳线性逼近，记作$\ell(x)$。研究表明，在此框架下，普通最小二乘估计通常导致$\ell(x)$的估计不一致，而应考虑加权最小二乘。本文为该估计量构建了渐近极小极大准则，并构造了最小化该准则的设计。该问题的一个重要特征是$x$应为随机变量而非固定值，否则极小极大风险为无穷大。结果表明，最优随机极小极大设计不同于先前研究的确定性对应设计，模拟研究表明当$m(x)$为二次或三次函数时，该设计通常表现更优。另一个发现是，当噪声方差趋于无穷时，随机设计与确定性极小极大设计趋于一致。本文以多项式回归模型为例说明结果，并讨论了一般情形。