Simultaneous Confidence Bands for Linear Regression and Smoothing

Suppose we observe Y-i = f(xi) + εi, i = 1, ..., n. We wish to find approximate 1 - α simultaneous confidence regions for {f(x), x ∈ X}. Our regions will be centered around linear estimates f̂(x) of nonparametric or nonparametric f(x). There is a large amount of previous work on this subject. Substantial restrictions have been usually placed on some or all of the dimensionality of x, the class of functions f that can be considered, the class of linear estimates f̂ and the region X. The method we present is an approximation to the tube formula dn can be used for multidimensional x and a wide class of linear estimates. By considering the effect of bias we are able to relax assumptions on the class of functions f which are considered. Simultaneous and numerical computations are used to illustrate the performance.