NPMLE attains parametric rates for density and posterior mean estimation in bounded-support Gaussian/Poisson mixtures when mixing is finitely discrete, and the LRT is asymptotically tight iff the mixing distribution is finitely discrete.
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Adaptivity of the NPMLE to finitely discrete mixing distributions in Gaussian/Poisson mixtures
NPMLE attains parametric rates for density and posterior mean estimation in bounded-support Gaussian/Poisson mixtures when mixing is finitely discrete, and the LRT is asymptotically tight iff the mixing distribution is finitely discrete.