Instance-Optimal Adaptive Location Estimation via Multiscale Mid-Summaries
Abstract
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: sup{r>0:H2(f0, f2r)≲ log(1/δ)/n}. 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.
Related
- Halve-Then-Greedy: Adaptive Sampling for Posterior Success with Near-Optimal Instance Dependence
- Brown's loss function and estimation of the scale parameter and the entropy in scale mixture models with unknown location
- Density Evolution: A Multiscale View of Density Estimation
- Adaptive LASSO Penalized Minimum Density Power Divergence Estimation through Least Squares Approximation: Application to Bone Mineral Density Data from the SWAN Study
- Efficient and Adaptive Estimation of Portfolio Weights with Spectral Risk Measures
BibTeX
@misc{wang2026instanceoptimal,
title = {{Instance-Optimal Adaptive Location Estimation via Multiscale Mid-Summaries}},
author = {Qiaosen Wang and Chao Gao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20749},
eprint = {2609.20749},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20749},
}
RIS
TY - UNPB
TI - Instance-Optimal Adaptive Location Estimation via Multiscale Mid-Summaries
AU - Qiaosen Wang
AU - Chao Gao
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20749
UR - https://arxiv.org/abs/2609.20749
AB - 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: sup{r>0:H^2(f_0, f_2r)≲ log(1/δ)/n}. 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.
ER -