Bootstrapping the Studentized Sample Mean of Lattice Variables

It is shown by an example that in general, the rate of bootstrap approximation to the studentized sample mean of lattice data is not better than the rate in the case of the normalized sample mean. A modified bootstrap method is proposed and shown to be second-order correct. Unlike the conventional "smoothed" bootstrap methods, the proposed procedure smooths the estimator rather than the resampling distribution. It is observed that irrespective of the lattice or nonlattice character of the data, confidence intervals based on this procedure have more accurate coverage probabilities than the usual bootstrap confidence intervals.