Smoothed NPMLEs in nonparametric Poisson mixtures and beyond

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.