A recentred bootstrap procedure for constructing uniformly correct confidence sets under smooth function models

It has been found, under a smooth function model setting, that the n out of n bootstrap is inconsistent at stationary points of the smooth function, but that the m out of n bootstrap is consistent, provided that a correct convergence rate is specified of the plug-in smooth function estimator. By considering a more general moving-parameter framework, we show that neither of the above bootstrap methods is consistent uniformly over neighbourhoods of stationary points, so that anomalies often arise of coverages of bootstrap sets over certain subsets of parameter values. We propose a recentred bootstrap procedure for constructing confidence sets with uniformly correct coverages over compact sets containing stationary points. A weighted bootstrap procedure is also proposed as an alternative under more general circumstances. Unlike the m out of n bootstrap, both procedures do not require knowledge of the convergence rate of the smooth function estimator. Empirical performance of our procedures is illustrated with numerical examples.