Asymptotic efficiency of simple decisions for the compound decision problem

TL;DR

This paper analyzes the asymptotic efficiency of simple symmetric estimators in the compound decision problem, showing they are nearly as effective as permutation-invariant estimators under mild conditions.

Contribution

It demonstrates that simple symmetric procedures achieve asymptotic optimality comparable to more complex permutation-invariant estimators in the compound decision setting.

Findings

Asymptotic equivalence of risks for simple and permutation-invariant estimators

Minimal total squared error risks differ by at most a constant asymptotically

Conditions under which simple estimators are nearly optimal

Abstract

We consider the compound decision problem of estimating a vector of $n$ parameters, known up to a permutation, corresponding to $n$ independent observations, and discuss the difference between two symmetric classes of estimators. The first and larger class is restricted to the set of all permutation invariant estimators. The second class is restricted further to simple symmetric procedures. That is, estimators such that each parameter is estimated by a function of the corresponding observation alone. We show that under mild conditions, the minimal total squared error risks over these two classes are asymptotically equivalent up to essentially O(1) difference.