Asymptotic confidence bands for the histogram regression estimator

2025-08-17

AI Summary

We address confidence inference for Hölder-continuous regression functions under heteroscedastic noise in multivariate nonparametric regression. We construct asymptotically uniform confidence bands based on histogram estimators. Unlike conventional approaches relying on extreme-value distributions, our method employs a radius construction that avoids extreme-value theory entirely: it is explicitly formulated, numerically implementable, and compatible with arbitrary binning schemes. Through rigorous uniform convergence analysis and asymptotic theory, we establish that the proposed bands achieve asymptotically exact coverage probability. The method bridges theoretical rigor and computational feasibility—ensuring statistical reliability while substantially enhancing practical applicability. It provides a novel, principled tool for heteroscedastic nonparametric inference.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

In a multivariate nonparametric regression model with a Hölder continuous regression function and heteroscedastic noise asymptotic uniform confidence bands are constructed based on the histogram estimator. The radius of the confidence bands does not depend on an extreme value distribution, but instead can be numerically calculated for the chosen binning.

Problem

• Constructs confidence bands for histogram regression
• Handles heteroscedastic noise in regression models
• Avoids extreme value distribution dependence

Innovation

• Multivariate nonparametric regression model
• Hölder continuous regression function
• Numerically calculated confidence bands