On confidence bands in nonparametric density estimation and regression

We describe a unified approach to the construction of confidence bands in nonparametric density estimation and regression. Our techniques are based on interpolation formulae in numerical differentiation, and our arguments generate a variety of bands depending on the assumptions one is prepared to make about derivatives of the unknown function. The bands are simultaneous, in the sense that they contain the entire function with probability at least an amount. The order of magnitude of the minimum width of any confidence band is described, and our bands are shown to achieve that order. Examples illustrate applications of the technique.