Adaptivity of the NPMLE to finitely discrete mixing distributions in Gaussian/Poisson mixtures

TL;DR

This paper investigates the adaptivity and optimal rates of the NPMLE for Gaussian and Poisson mixtures, revealing new phenomena in finite discrete cases and establishing its efficiency.

Contribution

It demonstrates that the NPMLE achieves parametric rates and optimal demixing rates for finitely discrete mixing distributions, and uncovers an adaptivity phenomenon in likelihood ratio testing.

Findings

NPMLE attains exact parametric rates for density estimation and posterior mean.

NPMLE achieves the optimal demixing rate for overparameterized finite mixtures.

Likelihood ratio test statistic is asymptotically tight only for finitely discrete true mixing distributions.

Abstract

We study the nonparametric maximum likelihood estimator (NPMLE) for Gaussian and Poisson mixture models, assuming the support of the true mixing distribution lies in a fixed bounded set. In this setting, we establish exact parametric rates for both, marginal density estimation and the posterior mean when the true mixing distribution is finitely discrete. Moreover, we show that the NPMLE attains the optimal demixing rate previously known for overparameterized finite mixture models. Finally, we identify a new adaptivity phenomenon for inference: the likelihood ratio test statistic is asymptotically tight if and only if the true mixing distribution is finitely discrete.