On Optimal Solutions to Compound Statistical Decision Problems

TL;DR

This paper investigates the structure and optimal risk bounds of decision rules in compound statistical problems, emphasizing symmetry properties and extending classical estimation theory.

Contribution

It derives the greatest lower bound on the risk of symmetric decision rules and extends classical estimation results within the framework of compound decision problems.

Findings

Established the symmetry property of decision rules in compound problems

Derived the optimal risk lower bound for such decision rules

Applied theory to normal mean estimation and extended classical results

Abstract

In a compound decision problem, consisting of $n$ statistically independent copies of the same problem to be solved under the sum of the individual losses, any reasonable compound decision rule $\delta$ satisfies a natural symmetry property, entailing that $\delta(\sigma(\boldsymbol{y})) = \sigma(\delta(\boldsymbol{y}))$ for any permutation $\sigma$. We derive the greatest lower bound on the risk of any such decision rule. The classical problem of estimating the mean of a homoscedastic normal vector is used to demonstrate the theory, but important extensions are presented as well in the context of Robbins's original ideas.