Accuracy of Uniform Inference on Fine Grid Points

TL;DR

This paper investigates how fine a grid must be for multiplier bootstrap methods to produce valid uniform confidence bands over continuous functions, providing explicit bounds and practical guidance.

Contribution

It derives explicit bounds on grid fineness needed for valid uniform inference using bootstrap, separating discretization error from bootstrap approximation error.

Findings

Explicit bound on coverage error for finite grid bootstrap

Guidance on choosing grid size in practice

Illustration with kernel density estimation example

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.