Simultaneous Confidence Bands for Linear Regression and Smoothing

Suppose we observe $Y-i = f(x_i) + \varepsilon_i, i = 1, \ldots, n$. We wish to find approximate $1 - \alpha$ simultaneous confidence regions for $\{f(x), x \in \mathscr{X}\}$. Our regions will be centered around linear estimates $\hat{f}(x)$ 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 $\hat{f}$ and the region $\mathscr{X}$. 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.