Optimal measurement allocation under precision budget constraint

In the classical many normal means with different variances, we consider the situation when the observer is allowed to allocate the available precision budget over the coordinates of the parameter of interest. The benchmark is the minimax linear risk over a set. We solve the problem of optimal allocation of observations under the precision budget constraint for two types of sets, ellipsoids and hyperrectangles. By elaborating on the two examples of Sobolev ellipsoids and hyperrectangles, we demonstrate how re-allocating the measurements in the (sub-)optimal way improves on the standard uniform allocation. In particular, we improve the famous Pinsker (1980) bound.