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

• URL: http://arxiv.org/abs/2607.27532v1
• Date: Wed, 29 Jul 2026 23:51:17 GMT
• Title: HOMER: Huber-of-Means for Efficient and Robust Estimation in Hilbert Spaces
• Abstract summary: We propose emphHOMER, or Huber-of-Means for Robust Estimation.<n>Its canonical and pseudo-Huber forms bound each block score and interpolate between median-like robustness and the empirical mean.<n>We show that HOMER remains stable when a minority of block summaries is displaced.
• Score: 0.26856688022781555
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

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