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原文摘要
Uniform confidence bands are widely used in empirical analysis for uncertainty qualification of nonparametric inference of unknown functions. A variety of simple implementation methods, including 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 the effect of this discretization on coverage accuracy and consider how fine the evaluation grid must be for a multiplier bootstrap procedure over finite grid points to deliver valid uniform confidence bands. Specifically, we first illustrate that coarse grids can invalidate uniform inference. For a nonparametric estimator based on kernel smoothing, we establish conditions under which uniform coverage converges to zero, even when the number of evaluation points diverges. Simulations further show that increasing the sample size can worsen coverage on a fixed coarse grid, whereas approximation errors from sources other than discretization decrease. We then consider general empirical processes and derive an upper bound on the coverage error of uniform confidence bands calibrated using multiplier bootstrap critical values computed on a finite grid. The bound distinguishes discretization from the remaining approximation error on the grid. Also, for a broad class of empirical processes arising from nonparametric estimators based on kernel smoothing, we provide primitive sufficient conditions for negligible discretization error. These conditions yield sufficient grid rules expressed in terms of the sample size and bandwidth.