The operation of the bootstrap in the context of nonparametric regression is considered. Bootstrap samples are taken from estimated residuals to study the distribution of a suitably recentered kernel estimator. The application of this principle to the problem of local adaptive choice of bandwidth and to the construction of confidence bands is investigated and compared with a direct method based on asymptotic means and variances.
A theorem is proved to establish that the bootstrap distribution approximates the distribution of interest in terms of the Mallows metric. Two applications are considered. The first uses bootstrap sampling to approximate the mean squared error of the nonparametric estimate at some point of interest. This can then be minimized over the smoothing parameter to adapt the degree of smoothing applied at any point to the local behavior of the underlying curve. The second application uses the percentiles of the approximate distribution to construct confidence intervals for the curve at specific design points.