Accuracy of Uniform Inference on Fine Grid Points

Uniform confidence bands are widely used to quantify uncertainty about unknown functions in nonparametric inference. Although various methods, including the multiplier bootstrap, have been developed and theoretically justified, practical implementations typically replace the supremum over a continuum by the maximum over a finite evaluation grid. This paper investigates how this discretization affects coverage accuracy and how fine the evaluation grid must be to ensure valid uniform inference. We first illustrate through simulations of a kernel-based nonparametric estimator that coarse grids can substantially reduce uniform coverage. Surprisingly, increasing the sample size can worsen coverage on a fixed coarse grid, even as the remaining approximation error on the grid decreases. We then derive general upper bounds on the coverage probability and coverage error of uniform confidence bands based on multiplier-bootstrap critical values on the finite grid points, separating discretization error from the remaining approximation error on the grid. For local empirical processes arising from kernel smoothing, we establish primitive conditions under which coverage converges to zero even when the number of grid points diverges, and conditions ensuring asymptotically nominal coverage. The latter results yield rate-sharp grid resolution requirements in terms of the bandwidth. Furthermore, we derive an explicit leading constant in the bound on coverage error and develop a quantitative grid selection rule for a user-specified discretization tolerance level, which can be implemented through a simple one-dimensional search.