HOMER: Huber-of-Means for Efficient and Robust Estimation in Hilbert Spaces

2026-07-29

AI Summary

This work addresses the limitations of the classical empirical mean under heavy-tailed distributions, which suffers from poor confidence control, and the median-of-means approach, which struggles to adaptively balance robustness and efficiency. We propose HOMER, a novel estimator that aggregates blockwise means via radial Huber centers in Hilbert space, combining canonical and pseudo-Huber losses to simultaneously achieve the robustness of medians and the asymptotic efficiency of the empirical mean. We establish a law of large numbers and non-asymptotic bias bounds in Hilbert space, complemented by sandwich covariance estimation and asymptotic linearity theory. Requiring only a finite third moment, HOMER attains parametric convergence rates for any fixed projection, remains stable under heavy-tailed data, and nearly matches the efficiency of the empirical mean under Gaussian assumptions, though it exhibits sensitivity to contamination affecting a majority of blocks.

Abstract

Heavy tails weaken high-confidence control for the empirical mean. Geometric median-of-means (MOM) also lacks a threshold that moves toward mean efficiency. We propose \emph{HOMER}, or Huber-of-Means for Efficient and Robust Estimation. HOMER aggregates block means through a radial Huber center. Its canonical and pseudo-Huber forms bound each block score and interpolate between median-like robustness and the empirical mean. We establish a Hilbert-space majority theorem and a MOM-order deviation bound under a finite second moment. Canonical HOMER recovers the sample mean inside its quadratic region. Pseudo-HOMER approaches the sample mean as the threshold grows. It also admits asymptotic linearity and consistent sandwich covariance estimation around the population block-Huber target. Under a finite third moment, fixed finite-dimensional projections support mean inference at the usual parametric rate. This result requires growing block sizes and counts, with block sizes increasing faster. Heavy-tailed simulations show that HOMER remains stable when a minority of block summaries is displaced. On clean Gaussian data, both versions closely approach the empirical mean's efficiency. Finite-block sandwich intervals undercovered, especially for skewed functional data. Further studies show failure when contamination affects most blocks or compromises ordinary within-block means.

Problem

• heavy tails
• robust estimation
• Hilbert spaces
• empirical mean
• median-of-means

Innovation

• HOMER
• Huber-of-Means
• robust estimation
• Hilbert spaces
• median-of-means