No need for an oracle: the nonparametric maximum likelihood decision in the compound decision problem is minimax

TL;DR

This paper demonstrates that the nonparametric maximum likelihood decision in the compound decision problem is minimax, providing a straightforward proof that avoids the traditional oracle argument and establishes its optimality.

Contribution

It proves the minimax optimality of the NPMLE decision in the compound decision problem without relying on the invalid oracle argument, offering a simpler and more direct proof.

Findings

NPMLE decision converges at a specific rate in the empirical Bayes setting.

The same NPMLE solution is optimal even when parameters are not assumed random.

The proof avoids complex oracle-based arguments, simplifying the theoretical understanding.

Abstract

We discuss the asymptotics of the nonparametric maximum likelihood estimator (NPMLE) in the normal mixture model. We then prove the convergence rate of the NPMLE decision in the empirical Bayes problem with normal observations. We point to (and heavily use) the connection between the NPMLE decision and Stein unbiased risk estimator (\sure). Next, we prove that the same solution is optimal in the compound decision problem where the unobserved parameters are not assumed to be random. Similar results are usually claimed using an oracle-based argument. However, we contend that the standard oracle argument is not valid. It was only partially proved that it can be fixed, and the existing proofs of these partial results are tedious. Our approach, on the other hand, is straightforward and short.