Closing the logarithmic gap in NPMLE Hellinger rates

Ascertain whether the Hellinger risk bound for the constrained nonparametric maximum likelihood estimator (NPMLE) over families such as Bdd(M) or 𝒫_α(β) can be sharpened from O(m*(H, 𝒫, n^{-1/2}) log n / n) to the minimax-optimal O(m*(H, 𝒫, n^{-1/2}) / n), thereby removing the extra log n factor; alternatively, prove a lower bound showing the gap is unavoidable.

Background

The authors derive upper bounds for the Hellinger error of the constrained NPMLE that scale as (m*(H, đť’«, n{-1/2}) log n)/n, where m* denotes the finite mixture complexity at accuracy n{-1/2}. They note that existing minimax lower bounds match m*/n without the logarithmic factor, which indicates a gap between upper and lower rates.

The open question is whether this log n factor can be eliminated or whether intrinsic obstacles prevent matching the minimax lower bound, thus determining the optimal statistical rate for the NPMLE in these mixture families.

References

We note that existing minimax lower bounds in [PW21] agree with mH,đť’«,n{-1/2}/n. However, bridging this gap remains an open problem.

— On the best approximation by finite Gaussian mixtures  (2404.08913 - Ma et al., 2024) in Section 6.1 (Convergence rates of nonparametric maximum likelihood estimator)

The logarithm in our upper bound comes from a finite-grid comparison; its necessity remains open.

— Sieve Estimation of Optimal Transport Maps from Paired Data in Gaussian Spaces  (2609.09089 - Jin et al., 8 Sep 2026) in Section 7, Discussion, paragraph beginning “Two extensions would broaden these results”