Connection between Gaussian-smoothed Fisher information and two-point testing rates

Characterize whether and how the Gaussian-smoothed Fisher information quantity used in finite-sample location estimation is connected to the Le Cam two-point testing lower bound for arbitrary densities.

Background

The paper notes that an earlier result obtains a location-estimation rate governed by Gaussian-smoothed Fisher information for arbitrary densities. Its own benchmark is instead based on the inverse Hellinger modulus of continuity, equivalently the critical separation in Le Cam’s two-point testing problem.

The unresolved problem is to determine whether these two descriptions of statistical difficulty are equivalent, comparable, or otherwise related, and under what conditions. Establishing such a connection could clarify when smoothed Fisher-information methods recover the information-theoretic two-point rate.

References

Another earlier paper achieves a rate related to Gaussian-smoothed Fisher information for arbitrary density, whereas it remains unclear whether and how this quantity is connected to the two-point testing lower bound.

— Instance-Optimal Adaptive Location Estimation via Multiscale Mid-Summaries  (2609.20749 - Wang et al., 17 Sep 2026) in Section 1, subsection “Related Work,” paragraph “Location estimation adapted to unknown density shapes”