On Edgeworth Expansion and Bootstrap Confidence Bands in Nonparametric Curve Estimation

SUMMARY The notion of pivoting and techniques of Edgeworth expansion have become basic tools for using and understanding bootstrap methods in finite parameter problems. This paper shows that these ideas have important contributions to make to infinite parameter problems, e.g. in the construction of simultaneous confidence bands for nonparametric density estimators. We propose a bootstrap method for confidence band construction based on a pivotal function estimator and illustrate its use. We give an Edgeworth expansion argument which demonstrates the efficacy of the technique. The expansion is a power series in {(nh)-1(log h -1)3}1/2. where n is the sample size and h denotes the bandwidth. This should be compared with a power series in n -1/2 in more conventional, finite dimensional problems.