  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.
