Optimal minimax random designs for weighted least squares estimators

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