Conservative Confidence Bands in Curvilinear Regression

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 Sᵏ⁻¹, the unit sphere in Rᵏ. The upper bound is obtained by "straightening" the path so that it lies in a great circle in Sᵏ⁻¹.