# Instance-Optimal Adaptive Location Estimation via Multiscale Mid-Summaries

arXiv: https://arxiv.org/abs/2609.20749 · HTML: https://mlsanity.com/paper/2609.20749/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 48 references that are in the map.
- First posted on arXiv 17 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [Recent progress in log-concave density estimation](https://mlsanity.com/paper/1709.03154/) (2017)
- [Communication-constrained hypothesis testing: Optimality, robustness, and reverse data processing inequalities](https://mlsanity.com/paper/2206.02765/) (2022)
- [Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression](https://mlsanity.com/paper/1903.05315/) (2019)
- [Optimal convex M-estimation via score matching](https://mlsanity.com/paper/2403.16688/) (2024)
- [Finite-Sample Symmetric Mean Estimation with Fisher Information Rate](https://mlsanity.com/paper/2306.16573/) (2023)
- [High-dimensional nonparametric density estimation via symmetry and shape constraints](https://mlsanity.com/paper/1903.06092/) (2019)
- [Nonparametric inference under shape constraints: past, present and future](https://mlsanity.com/paper/2509.26040/) (2025)
- [Attainability of Two-Point Testing Rates for Finite-Sample Location Estimation](https://mlsanity.com/paper/2502.05730/) (2025)
- [Bi-log-concave distribution functions](https://mlsanity.com/paper/1508.07825/) (2015)

## Abstract (arXiv)

Location estimation exhibits markedly different finite-sample behavior across noise distributions: regular families typically yield root-(n) rates, whereas compactly supported laws may admit faster, boundary-driven rates. We question whether a single estimator, without knowledge of the density's shape, can adapt to the instance-wise optimal estimation rate, as an oracle that knows the underlying location family can. For a known location family with symmetric log-concave noise density (f), the optimal location estimation error with sample size (n) under failure probability (δ) is known to be Le Cam's two-point rate: [ supleftr>0:mathsfH2left(f0, f2rright)lesssim fraclog(1/δ)nright. ] When the location family is unknown, we propose a shape-agnostic estimator that attains this oracle benchmark simultaneously over all symmetric unimodal densities with non-decreasing hazard rates, a class strictly broader than symmetric log-concave distributions. We establish that the Hellinger-driven two-point rate can be characterized solely by a multiscale function of dyadic quantile gaps. This new structural connection between Hellinger divergence and quantile geometry motivates a simple estimation procedure that aggregates sample mid-summaries with carefully designed data-dependent weights. The resulting estimator is finite-sample instance-optimal and runs in only (O(log(n))) time on sorted samples.