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, 𝒫, 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.