Pinsker bound under measurement budget constrain: optimal allocation

TL;DR

This paper investigates optimal measurement allocation under budget constraints in normal means problems, improving Pinsker's classical bounds by demonstrating how non-uniform strategies enhance estimation accuracy for ellipsoids and hyperrectangles.

Contribution

It derives the optimal measurement allocation strategies under budget constraints for ellipsoids and hyperrectangles, extending Pinsker's bounds with practical re-allocation methods.

Findings

Optimal allocation improves estimation risk over uniform allocation.

Re-allocation strategies outperform standard uniform measurement in specific set classes.

Enhanced bounds generalize Pinsker's classical results.

Abstract

In the classical many normal means with different variances, we consider the situation when the observer is allowed to allocate the available measurement 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 measurement budget constrain for two types of sets, ellipsoids and hyperrectangles. By elaborating on the two examples of Sobolev ellipsoids and hyperectangles, 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.