Uniformly consistent confidence intervals under smooth function models

Abstract

It has been found, under a smooth function model setting, that the conventional bootstrap is inconsistent at stationary points of the smooth function and that the m out of n bootstrap succeeds in restoring consistency, provided that a correct convergence rate can be specified of the plug-in smooth function estimator. We re-investigate the properties of the bootstrap in a moving-parameter framework, and show that neither the conventional bootstrap nor the m out of n bootstrap are uniformly consistent over the parameter space. The results reflect to some extent finite-sample anomalies that cannot be explained by conventional, fixed-parameter, asymptotics. We propose a weighted bootstrap procedure for constructing uniformly consistent bootstrap confidence intervals, which does not require explicit specification of the convergence rate of the smooth function estimator. Our findings are illustrated with several numerical examples to compare the coverage accuracies of the different bootstrap procedures.