Asymptotic convergence of the covariate-dependent NPMLE

Determine whether the covariate-dependent iMoE-NPMLE converges asymptotically to the true conditional prior measure.

Background

The paper introduces the infinite Mixture-of-Expertizing (iMoE) class to regularize nonparametric estimation of a covariate-dependent prior measure in unit-level generalized linear models. The iMoE-NPMLE estimates a conditional prior G*(β|z) through a softmax-gated mixture representation and has a discrete solution with at most 2n support points.

Although the paper establishes structural properties of the iMoE class, including Lipschitz continuity and a universal approximation result for aggregated gates, it does not establish consistency of the estimated conditional prior as the sample size increases. The authors explicitly identify convergence to the true conditional prior measure as unresolved. They also mention convergence rates for the fixed-component mixture-of-experts case, but that sentence does not itself contain an explicit uncertainty marker and is therefore not extracted as a separate problem.