Smoothed NPMLEs in nonparametric Poisson mixtures and beyond

TL;DR

This paper investigates the estimation of nonparametric mixing distributions using Gaussian-smoothed optimal transport distance, demonstrating near-parametric convergence rates for Poisson and other mixture models.

Contribution

It proves that the Poisson NPMLE attains near root-n convergence under GOT distance and extends this result to other estimators and mixture models beyond Poisson.

Findings

Poisson NPMLE achieves near root-n convergence rate under GOT distance.

Extension of convergence results to other minimum-distance estimators.

Discussion of mixture models beyond the Poisson case.

Abstract

We discuss nonparametric mixing distribution estimation under the Gaussian-smoothed optimal transport (GOT) distance. It is shown that a recently formulated conjecture -- that the Poisson nonparametric maximum likelihood estimator can achieve root-$n$ rate of convergence under the GOT distance -- holds up to some logarithmic terms. We also establish the same conclusion for other minimum-distance estimators, and discuss mixture models beyond the Poisson.