Simultaneous Confidence Bands for Linear Regression and Smoothing

Suppose we observe Y-i = f (xᵢ) + ᵢ, i = 1, , n. We wish to find approximate 1 - simultaneous confidence regions for \f (x), x X\. Our regions will be centered around linear estimates f (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 f 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.