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This work studies an experimental design problem where {the values of apredictor variable, denoted by $x$}, are to be determined with the goal ofestimating a function $m(x)$, which is observed with noise. A linear model isfitted 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 theordinary least squares estimator typically leads to an inconsistent estimationof $\ell(x)$, and rather weighted least squares should be considered. Anasymptotic minimax criterion is formulated for this estimator, and a designthat minimizes the criterion is constructed. An important feature of thisproblem is that the $x$'s should be random, rather than fixed. Otherwise, theminimax risk is infinite. It is shown that the optimal random minimax design isdifferent from its deterministic counterpart, which was studied previously, anda simulation study indicates that it generally performs better when $m(x)$ is aquadratic or a cubic function. Another finding is that when the variance of thenoise goes to infinity, the random and deterministic minimax designs coincide.The results are illustrated for polynomial regression models and the generalcase is also discussed.

Optimal minimax random designs for weighted least squares estimators