{"ID":23507425,"CreatedAt":"2026-09-18T02:21:44.056544415Z","UpdatedAt":"2026-09-20T18:11:56.143995915Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20749","arxiv_id":"2609.20749","title":"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\\left\\{r\u003e0:\\mathsf{H}^2\\left(f_0, f_{2r}\\right)\\lesssim \\frac{\\log(1/δ)}{n}\\right\\}. \\] 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.","short_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 t...","url_abs":"https://arxiv.org/abs/2609.20749","url_pdf":"https://arxiv.org/pdf/2609.20749v1","authors":"[\"Qiaosen Wang\",\"Chao Gao\"]","published":"2026-09-17T17:37:15Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"stat.ME\",\"stat.ML\"]","methods":"[]","has_code":false}