Conservative Confidence Bands in Curvilinear Regression

Abstract

This paper gives a method for constructing conservative Scheffe-type simultaneous confidence bands for curvilinear regression functions over finite intervals. The method is based on the use of a geometric inequality giving an upper bound for the uniform measure of the set of points within a given distance from y, an arbitrary piecewise differentiable path with finite length in $S^{k-1}$, the unit sphere in $R^k$. The upper bound is obtained by "straightening" the path so that it lies in a great circle in $S^{k-1}$.