  Standard approaches to constructing nonparametric confidence bands for
functions are frustrated by the impact of bias, which generally is not
estimated consistently when using the bootstrap and conventionally smoothed
function estimators. To overcome this problem it is common practice to either
undersmooth, so as to reduce the impact of bias, or oversmooth, and thereby
introduce an explicit or implicit bias estimator. However, these approaches,
and others based on nonstandard smoothing methods, complicate the process of
inference, for example, by requiring the choice of new, unconventional
smoothing parameters and, in the case of undersmoothing, producing relatively
wide bands. In this paper we suggest a new approach, which exploits to our
advantage one of the difficulties that, in the past, has prevented an
attractive solution to the problem - the fact that the standard bootstrap bias
estimator suffers from relatively high-frequency stochastic error. The high
frequency, together with a technique based on quantiles, can be exploited to
dampen down the stochastic error term, leading to relatively narrow,
simple-to-construct confidence bands.
