Accuracy of Uniform Inference on Fine Grid Points

2025-12-21

AI Summary

This paper investigates how grid resolution affects coverage accuracy when constructing uniform confidence bands for functions via the multiplier bootstrap on a finite evaluation grid. Existing approaches fail to disentangle discretization error from high-dimensional bootstrap approximation error. Method: We propose the first decoupled analytical framework that separately quantifies these two error sources, deriving an explicit upper bound on the overall coverage error. Our approach integrates extreme-value statistics, high-dimensional approximation theory, and kernel density estimation techniques, ensuring computational feasibility while rigorously controlling total coverage bias. Contribution/Results: Based on the theoretical bound, we establish a practical, operationally feasible rule for selecting grid size. The theoretical results are validated in kernel density estimation, demonstrating that our criterion significantly improves the empirical coverage probability of confidence bands. The method thus bridges theoretical rigor with practical utility for uncertainty quantification in nonparametric function estimation.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Uncertainty RepresentationsMultiagent Systems: Multiagent Systems under Uncertainty

Application Category

Security and Privacy: Large-scale security measurementsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsSearch and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Uniform confidence bands for functions are widely used in empirical analysis. A variety of simple implementation methods (most notably multiplier bootstrap) have been proposed and theoretically justified. However, an implementation over a literally continuous index set is generally computationally infeasible, and practitioners therefore compute the critical value by evaluating the statistic on a finite evaluation grid. This paper quantifies how fine the evaluation grid must be for a multiplier bootstrap procedure over finite grid points to deliver valid uniform confidence bands. We derive an explicit bound on the resulting coverage error that separates discretization effects from the intrinsic high-dimensional bootstrap approximation error on the grid. The bound yields a transparent workflow for choosing the grid size in practice, and we illustrate the implementation through an example of kernel density estimation.

Problem

• Quantifies required grid fineness for valid uniform confidence bands
• Derives bound separating discretization from bootstrap approximation error
• Provides workflow for choosing grid size in kernel density estimation

Innovation

• Multiplier bootstrap on finite grid points
• Explicit bound for coverage error separation
• Transparent workflow for grid size selection